On Advanced Analytic Number Theory

By C.L. Siegel

Tata Institute of Fundamental Research Mumbai, India i

FOREWORD

“Advanced Analytic Number Theory” was ﬁrst published by the Tata Insti- tute of Fundamental Research in their Lecture Notes series in 1961. It is now being made available in book form with an appendix–an English translation of Siegel’s paper “Berechnung von Zetafunktionen an ganzzahligen Stellen” which appeared in the Nachrichten der Akademie der Wissenschaften in G¨ottingen, Math-Phy. Klasse. (1969), pp. 87-102. We are thankful to Professor Siegel and to the Gottingen¨ Academy for according us permission to translate and publish this important paper. We also thank Professor S. Raghavan, who originally wrote the notes of Profes- sor Siegel’s lectures, for making available a translation of Siegel’s paper.

K. G. Ramanathan ii

PREFACE

During the winter semester 1959/60, I delivered at the Tata Institute of Fundamental Research a series of lectures on Analytic Number Theory. It was may aim to introduce my hearers to some of the important and beautiful ideas which were developed by L. Kronecker and E. Hecke. Mr. S. Raghavan was very careful in taking the notes of these lectures and in preparing the manuscript. I thank him for his help.

Carl Ludwig Siegel Contents

1 Kronecker’s Limit Formulas 1 1 Theﬁrstlimitformula...... 1 2 The Dedekind η-function ...... 13 3 The second limit formula of Kronecker ...... 20 4 The elliptic theta-function ϑ1(w, z)...... 29 5 TheEpsteinZeta-function ...... 41

2 Applications of Kronecker’s Limit Formulas to Algebraic Number Theory 52 1 Kronecker’s solution of Pell’s equation ...... 52 2 Class number of the absolute class field of Q( √d)(d < 0) . . . 69 3 The Kronecker Limit Formula for real quadratic fields and its applications ...... 75 4 Ray class fields over Q( √d), d < 0...... 89 5 Ray class fields over Q( √D), D > 0...... 102 6 SomeExamples...... 133

3 Modular Functions and Algebraic Number Theory 149 1 Abelian functions and complex multiplications ...... 149 2 Fundamental domain for the Hilbert modular group ...... 165 3 Hilbert modular functions ...... 184 1 EllipticModularForms ...... 220 2 Modular Forms of Hecke ...... 225 3 Examples ...... 231

iii Chapter 1

Kronecker’s Limit Formulas

1 The ﬁrst limit formula 1 Let s = σ + it be a complex variable, σ and t being real. The Riemann zeta- function ζ(s) is deﬁned for σ > 1 by

∞ s ζ(s) = k− , k=1

s s log k here k− stands for e− , with the real value of log k. The series converges absolutely for σ > 1 and uniformly in every s-half-plane deﬁned by σ 1 + ǫ(ǫ > 0). It follows from a theorem of Weierstrass that the sum-function≥ ζ(s) is a regular function of s for σ > 1. Riemann proved that ζ(s) possesses an analytic continuation into the whole s-plane which is regular except for a simple pole at s = 1 and satisﬁes the well-known functional equation

s/2 s ((1 s)/2) 1 s π− Γ ζ(s) = π− − Γ − ζ(1 s), 2 2 − Γ(s) being Euler’s gamma-function. We shall now prove Proposition 1. The function ζ(s) can be continued analytically into the half- plane σ > 0 and the continuation is regular for σ > 0, except for a simple pole at s = 1 with residue 1. Further, at s = 1, ζ(s) has the expansion 1 ζ(s) = C + a (s 1) + a (s 1)2 + − s 1 1 − 2 − − 1 Kronecker’s Limit Formulas 2

C being Euler’s constant. We first need to prove the simplest form of a summation formula due to Euler, namely. Lemma. If f (x) is a complex-valued function having a continuous derivative f (x) in the interval 1 x n, then ′ ≤ ≤ 1 n 1 − 1 f (1) + f (n) f ′(x + k) x dx +  − 2 2 0 k=1     n n = f (k) f (x)dx. (1) − k=1 1 2 Proof. In face, if a complex-valued function g(x) defined in the interval 0 ≤ x 1, has a continuous derivative g′(x), then we have from integration by parts,≤ 1 1 1 1 g′(x) x = dx = (g(0) + g(1)) g(x)dx. (2) 0 2 2 − 0 (Formula (2) has a simple geometric interpretation in that the right hand side of (2) represents the area of the portion of the (x, y)-plane bounded by the curve y = g(x) and the straight lines y g(0) = (g(1) g(0))x, x = g(0) and x = (g1)). In (2), we now set g(x) = f (x + k−), successively− for k = 1, 2,..., n 1. We then have for k = 1, 2,..., n 1, − − 1 1 1 k+1 f ′(x + k) x dx = ( f (k) + f (k + 1)) f (x)dx. 0 − 2 2 − k Adding up, we obtain formula (1). This is Euler’s summation formula in its simplest form, with the remainder term involving only the first derivative of f (x). It is interesting to notice that if one uses the fact that the Fourier series e2πinx (n = 0, omitted) converges uniformly to x 1/2 in any closed in- − n 2πin − terval (ǫ, 1 ǫ)(0 < ǫ < 1) one obtains from (2) the following useful result, namely, − If f (x) is periodic in x with period 1 and has a continuous derivative, then

1 ∞ 2πinx 2πinx f (x) = e f (x)e− dx. n= 0 −∞ Kronecker’s Limit Formulas 3

Proof of Proposition 1. Let us now specialize f (x) to be the function f (x) = s x log x s x− = e− (log x taking real values for x > 0). The functionx− is evidently 3 continuously diﬀerentiable in the interval 1 x n; and applying (1) to the s ≤ ≤ function x− , we get

n 1 1 − s 1 1 1 s s (x + k)− − x dx + (1 + n− ) − − 2 2 k=1 0 n n s s = k− x− dx − k=1 1 1 s n k s 1 n − (if s , 1) = k=1 − −s 1 n 1 − − (3)  k− log n(if s = 1).  k=1 −  Let us now observe that the right-hand side of (3) is an entire function of s.

n σ > s We suppose now that 1. When n tends to infinity 1 x− dx tends to n s 1/(s 1). Further, as n tends to infinity, k− converges to ζ(s). Thus for − k=1 σ > 1, the right hand side of (3) converges to ζ(s) 1/(s 1) as n tends to infinity. − − On the other hand, let the left-hand side of (3) be denoted by ϕn(s). Then ϕ (s) is an entire funciton of s and further, for σ ǫ > 0, n ≥ n ǫ 1 1 ǫ 1 + n− ϕ (s) s k− − + | n | ≤ 2| | 2 k=1

1 ∞ 1 ǫ s k− − + 1. ≤ 2| | k=1

Thus, as n tends to inﬁnity, ϕn(s) converges to a regular function of s in the half-plane σ > 0; this provides the analytic continuation of ζ(s) 1/(s 1) for σ > 0. − − Now the constant a0 in the power-series expansion at s = 1 of 1 ζ(s) = a + a (s 1) + a (s 1)2 + − s 1 0 1 − 2 − − is nothing but lim ϕn(1). In other words, n →∞ 1 1 a0 lim 1 + + + log n n 2 n →∞ − Kronecker’s Limit Formulas 4

= C (Euler’s constant).

Proposition 1 is thus proved. 4 The constant C lies between 0 and 1. It is not known whether it is rational or irrational; very probably, it is irrational. One could determine the constants a1, a2,... also explicitly but this is more complicated. We shall consider now an analogous problem leading to the Kronecker limit formula (Kroneckersche Grenzformle). Instead of the simple linear function x, we consider a positive-definite binary quadratic form Q(u, v) = au2 + 2buv + cv2 in the real variables u and v and with real coefficients a, b, c (we then have a > 0 and ac b2 = d > 0). Associated with Q(u, v), let us define −

∞ s ζQ(s) = (Q(m, n))− , (4) m,n= −∞ where ′ denotes summation over all pairs of integers (m, n), except (0, 0). Q(u, v) being positive-deﬁnite, there exists a real number λ > 0, such that Q(u, v) λ(u2 + v2) and it is an immediate consequence that the series (4) converges≥ absolutely for σ > 1 and uniformly in every half-plane deﬁned by σ 1 + ǫ(ǫ > 0). Thus ζQ(s) is a regular function of s, for σ > 1. ≥As in the case of ζ(s) above, we shall obtain an analytic continuation of ζQ(s) regular in the half-plane σ > 1/2, except for a simple pole at s = 1. The constant a0 in the expansion at s = 1 of ζQ(s), viz.

a 1 ζ (s) = − + a + a (s 1) + Q s 1 0 1 − − is given precisely by the Kronecker limit formula. The constant a 1 which is − the residue of ζQ(s) at s = 1 was found by Dirichlet in his investigations on the class-number of quadratic fields. Letus first effect some simplifications. 5 Since Q(u, v) = Q( u, v), we see that − −

∞ s ∞ ∞ s ζQ(s) = 2 (Q(m, 0))− + 2 (Q(m, n))− . (5) m=1 n=1 m= −∞ Further

b 2 b2 Q(u, v) = a u + v + c v2 a − a Kronecker’s Limit Formulas 5

b 2 d b + √ d b √ d = a u + v + v2 = a u + − v u + − − v a a  a   a      = a(u + zv)(u + zv)     where arg( √ d) = π/2 and z = (b + √ d)/a = x + iy with y > 0. − − Also, we could, without loss of generality, suppose that d = 1; if Q1(u, v) = √ 2 2 2 2 s/2 (1/ d)(au + 2buv + cv ) = a1u + 2b1uv + c1v , then ζQ1 (s) = d ζQ(s) and a c b2 = 1. The function ds/2 (choosing a ﬁxed branch) is a simple entire 1 1 − 1 function of s and therefore, to study ζQ(s), it is enough to consider ζQ1 (s). 1 1 1 2 Then we have a = y− and Q(u, v) = y− (u + zv)(u + zv) = y− u + zv . Now (5) becomes | |

s ∞ 2s s ∞ ∞ 2s ζ (s) = 2y m− + 2y m + nz − Q | | m=1 n=1 m= −∞ s s ∞ ∞ 2s = 2y ζ(2s) + 2y m + nz − (6) | | n=1 m= −∞ We know from Proposition 1 that ζ(2s) has an analytic continuation in the half-plane σ > 0, regular except for a simple pole at s = 1/2. To obtain an analytic continuation for ζQ(s), we have, therefore, only to investigate the nature of the second term on the right hand side of (6), as a function of s. For this purpose, we need the Poisson summation formula.

Proposition 2. Let f (x) be continuous in ( , ) and let ∞ f (x + m) be −∞ ∞ m= uniformly convergent in 0 x 1. Then −∞ ≤ ≤ ∞ ∞ ∞ 2πikx 2πikξ f (x + m) = e− f (ξ)e dξ. m= k= −∞ −∞ −∞ 6

Proof. The function ϕ(x) = ∞ f (x + m) is continuous in ( , ) and pe- m= −∞ ∞ −∞1 = ϕ ξ 2πikξ ξ riodic in x, of period 1. If ak 0 ( )e d then, by Fej´ er’s´ Theorem, the ∞ 2πikx ∞ (C, 1) sum of ake− is equal to ϕ(x). In particular, if ak converges, k= k= | | then for x in ( −∞, ) −∞ −∞ ∞ 1 ∞ ∞ 2πikx 2πikξ f (x + m) = ϕ(x) = e− ϕ(ξ)e dξ. m= k= 0 −∞ −∞ Kronecker’s Limit Formulas 6

In view of the uniform convergence of ∞ f (x + m)in0 x 1, m= ≤ ≤ −∞ 1 1 ∞ ∞ f (ξ + m)e2πikξdξ = f (ξ + m)e2πikξdξ 0 m= m= 0 −∞ −∞ = ∞ f (ξ)e2πikξdξ. −∞ Thus ∞ ∞ 2πikx ∞ 2πikξ f (x + m) = e− f (ξ)e dξ, (7) m= k= −∞ −∞ −∞ which is the Poisson summation formula. Now we set f (x) = x + iy 2s = z 2s for x in ( , ). Then we see | |− | |− −∞ ∞ ∞ 2s that the series m + z − converges absolutely, uniformly in every interval m= | | N x N, for−∞σ > 1/2. For this purpose, it clearly suﬃces to consider the interval− ≤ 0≤ x 1, since the series remains unchanged when x is replaces by x + 1. For≤ 0 x≤ 1, ≤ ≤

∞ 2s ∞ 2σ m + z − m + z − | | ≤ | | m= m= −∞ −∞ 2σ 2σ ∞ 2σ 2σ z − + z 1 − + (m− + m− ) ≤ | | | − | m=1

2σ ∞ 2σ 1 < 2y− + 2 m− σ > . 2 m=1 7 Thus, by (7), for σ > 1/2,

∞ 2s ∞ 2πikx ∞ 2s 2πikξ m + z − = e− ξ + iy − e dξ | | | | m= k= −∞ −∞ −∞ ∞ 2πikx ∞ 2 2 s 2πikξ = e− (ξ + y )− e dξ k= −∞ −∞ 1 2s ∞ 2πikx ∞ 2 s 2πikξ = y − e− (1 + ξ )− e dξ, (8) k= −∞ −∞ whenever the series (8) converges. Actually, we shall prove that it converges absolutely for σ > 1/2. For this purpose, let us consider for k > 0, ∞ (1 + −∞ Kronecker’s Limit Formulas 7

2 s 2πikζ ζ )− e dζ as a line integral in the complex ζ-plane; let ζ = ξ + iη. The 2 s function (1 + ζ )− has a logarithmic branch-point at ζ = i and ζ = i. In the complex ζ-plane cut along the η-axis from η = 1 to η = and− from η = 1 to η = , let ABCDEFG denote the contour consisting of∞ the straight line −GA(η = 0−∞, ξ R), the arc AB(ζ = Reiθπ θ π/2), the straight line BC(ξ = 0, 3/2 | η| ≤ R) on the left bank of the cut,≥ the≥ circle ≤ ≤ CDE(ζ = i + (i/2)eiϕ, 0 ϕ 2π), ≤ ≤ the straight line EF on the right bank of the cut (ξ = 0, 3/2 η R) ≤ ≤

Fig. 1

andthe arc FG(ζ = Reiθ, π/2 θ 0). If s lies in a compact set K of the 8 s-plane with σ > 0, then on the arcs≥ AB≥ and FG,

2 s 2πikyζ 2 σ 2πky (1 + ζ )− e 1 + ζ − e− R sin θ | | ≤ | | 2σ 2πky R sin θ = 0(R− e− ), since π arg(1 + ζ2) π on the whole contour. The constants in the O- estimate− depend≤ only on K≤. Thus

π/2 2 s 2πikyζ 2σ 2πky R sin θ (1 + ζ )− e dζ = 0 R− e− Rdθ AB 0 π/2 1 2σ 4kyRθ = 0 R − e− dθ 0 2σ = 0(R− ),

+ ζ2 s 2πikζ ζ and hence AB(1 )− e d tends to zero as R tends to infinity. Similarly + ζ2 s 2πikζ ζ Γ+ FG(1 )− e d also tends to zero as R tends to infinity. If denotes the Kronecker’s Limit Formulas 8 contour BCDEF in the limiting position as R tends to infinity, then for σ > 0, we have, by Cauchy’s Theorem,

∞ 2 s 2πikyζ 2 s 2πikyζ (1 + ζ )− e dζ = (1 + ζ )− e dζ. (9) Γ+ −∞

Let Γ− denote the limiting position of B′C′D′E′F′ (which is the reﬂection of BCDEF with respect to ξ-axis), as R tends to inﬁnity. We can show similarly that when k < 0,

∞ 2 s 2πikyζ 2 s 2πikyζ (1 + ζ )− e dζ = (1 + ζ )− e dζ. (10) Γ −∞ − We shall now show that the right hand side of (??) deﬁnes an entire function of s. In fact, if s = σ+it lies in a compact set K in the s-plane with σ > d(d > + −2 σ 0), then, on Γ we have for a constant c1 depending only on K, 1 + ζ − 2d | | ≤ c1η . Thus for k > 0,

2 s 2πikyζ 2 s 2πkyη (1 + ζ )− e dζ (1 + ζ )− e− dζ Γ+ ≤ | | | | Γ+ 2 σ π t 2πkyη 2d π t 2πkyη 1 + ζ − e | |− dζ c1 η e | |− dζ ≤ Γ+ | | | | ≤ Γ+ | | πky 2d 2πky(η 1 ) πky 2d 2πy(η 1 ) c2e− η e− − 2 dζ c2e− η e− − 2 dζ ≤ Γ+ | | ≤ Γ+ | | πky c e− , (11) ≤ 3 + 2 s 2πkyζ the constants c2 and c3 depending only on K. Hence Γ+ (1 ζ )− e dζ con- 9 verges absolutely and uniformly in K and therefore deﬁnes an analytic function of s in any domain contained in K. Since d is an arbitrary positive number, our assertion above is proved. Similar to (11), we have also for k < 0,

2 s 2πikyζ π k y (1 + ζ )− e dζ c3′ e− | | , (12) Γ ≤ − 2 s 2πikyζ and therefore (1 + ζ )− e dζ deﬁnes for k < 0, an entire function of s. Γ− In view of (??), (10), (11) and (12), we see that in order to prove the absolute convergence of the series (8), for σ > 1/2, we need to prove only that for 2 s σ > 1/2, ∞ (1 + ζ )− dζ exists. But, in fact, for σ > 1/2, −∞ ∞ 2 s ∞ 2 s (1 + ζ )− dζ = 2 (1 + ζ )− dζ 0 −∞ Kronecker’s Limit Formulas 9

∞ s 1 2 = (1 + λ)− λ− 2 dλ (setting ζ = λ) 0 1 1 s 3/2 = − 2 (1 ) − d setting λ = − 1 0 − 1 1 = B , s , 2 − 2 where B(a, b) is Euler’s beta-function. Hence

1 1 2 Γ ∞ 2 s π (s 2 ) (1 + ζ )− dζ = − Γ(s) −∞ Thus, for σ > 1,

1 1 2 Γ ∞ ∞ 2s π (s 2 ) 1 2s 1 2s 2πimx ∞ 2 s 2πimyζ m + z − = − y − + y − ′ e− (1 + ζ )− e dζ, | | Γ(s) m= m= −∞ −∞ −∞ where the accent on Σ indicates the omission of m = 0. Substituting this in (6), 10 we have for σ > 1,

s ζQ(s) = 2y ζ(2s)+ 1 1 2 Γ s ∞ π (s 2 ) 1 2s 1 2s 1 2s 1 2s + 2y − n − y − + n − y −  Γ(s) × n=1  ∞  2πimnx ∞ 2 s 2πimnyζ ′ e− (1 + ζ )− e dζ ×  m=  −∞ −∞   By (??) and (10), therefore, for σ > 1, 

1 1 2 Γ s 1 s π (s 2 ) ζ (s) = 2y ζ(2s) + 2y − − ζ(2s 1)+ Q Γ(s) −

1 s ∞ 1 2s ∞ 2πimnx ∞ 2 s 2πimnyζ + 2y − n − e− (1 + ζ )− e dζ; n=1 m= −∞ −∞ i.e.

1 1 2 Γ s 1 s π (s 2 ) 1 s ∞ 1 2s ζ (s) = 2y ζ(2s) + 2y − − ζ(2s 1) + 2y − n − Q Γ(s) − × n=1

∞ 2πimnx 2 s 2πimnyζ e− (1 + ζ )− e dζ+ ×  Γ+ m=1  Kronecker’s Limit Formulas 10

∞ 2πimnx 2 s 2πimnyζ e (1 + ζ )− e− dζ . (13) Γ  m=1 −   The double series in (13) converges absolutely, uniformly in every compact subset of the s-plane, in view of the fact that by (11) and (12), it is majorised by ∞ ∞ ∞ 2y1 σ n1 2σ c e πmny + c e πmny , − −  3 − 3′ −  n=1  m=1 m=1    whichclearly converges. Since we have seen earlier that the terms of the double 11 series deﬁne entire functions of s, it follows by the theorem of Weierstrass that the double series in (13) deﬁnes an entire function of s. Formula (13) gives us an analytic continuation of ζQ(s) for σ > 1/2. For, ζ(2s) is regular for σ > 1/2 and from the fact that ζ(s) can be analytically continued into the half-plane σ > 0 such that ζ(s) 1/(s 1) is regular for σ > 0, we see on replacing s by 2s 1 that ζ(2s 1) can− be− continued analytically in the half-plane σ > 1/2 such− that ζ(2s − 1) 1/2(s 1) regular for σ > 1/2 : Γ(s 1 )/Γ(s) is regular for σ > 1/2 and− non-zero− at− s = 1. Hence from − 2 (13), we see that ζQ(s) can be continued analytically in the half-plane σ > 1/2 and its only singularity in this half-plane is a simple pole at s = 1 with residue 1 1 equal to 2 π 2 π 2 1/2 = π. ζ (s) π/(s 1) has therefore a convergent power-series expansion a + Q − − 0 a1(s 1) + at s = 1. We shall now determine the constant a0. Clearly, from (13),−

1 1 2 Γ 1 s π (s 2 ) π a0 = 2yζ(2) + lim 2y − − ζ(2s 1) + s 1  Γ(s) − − s 1 →  −    ∞ 1 ∞ 2πimnx 2 1 2πimnyζ  + 2 e− (1 + ζ )− e dζ+ n  + n=1 m=1 Γ   ∞ 2πimnx 2 1 2πimnyζ + e (1 + ζ )− e− dζ Γ  m=1 −   Now from 1 ζ(s) = + C + a (s 1) + s 1 1 − − we have 1 ζ(2s 1) = + C + 2a (s 1) + . − 2(s 1) 1 − − Kronecker’s Limit Formulas 11

Also replacing s by s 1/2 in the formula −

1 1 2s Γ s + Γ(s) = √π 2 − Γ(2s). 2 wehave 12 1 2 2s Γ s Γ(s) = √π2 − Γ(2s 1). − 2 − At s = 1, Γ(s) has the expansion

Γ(s) = 1 + a(s 1) + − and hence Γ2(s) = 1 + 2a(s 1) + − Further at s = 1, Γ(2s 1) = 1 + 2a(s 1) + − − Hence we have at s = 1, by Cauchy multiplication of power-series,

2 2 Γ(2s 1)Γ− (s) = 1 + b(s 1) + − − and thus Γ 1 1 s 1 (s 2 ) 2 2s Γ(2s 1) 2y − π 2 − ζ(2s 1) = 2π(2 √y) − − ζ(2s 1) Γ(s) − Γ2(s) − = 2π(1 + b(s 1)2 + ) − × (1 2 log(2 √y)(s 1) + ) × − − × 1 + C + . × 2(s 1) − In other words, Γ 1 1 s 1 (S 2 ) π lim 2y − π 2 − ζ(2s 1) = 2π(C log(2 √y))/ s 1  Γ(s) − − s 1 − →  −   2 1  Now (see Fig. 1), for mn > 0, since (1 + ζ )− has no branch-point at ζ = i or ζ = i, − 2 1 2πimnyζ 2 1 2πimnyζ (1 + ζ )− e dζ = (1 + ζ )− e dζ Γ+ CDE e2πimnyζ = 2πi Residue of at ζ = i 1 + ζ2 Kronecker’s Limit Formulas 12

2πmny = πe− .

Similarly, for mn > 0, 13

e 2πimnyζ (1 + ζ2) 1e 2πimnyζ dζ = 2πi Residue of − at ζ = i − − + 2 Γ− − 1 ζ − 2πmny = πe−

Hence

2 π ∞ 1 ∞ 2πimnx 2πmny a = y + 2π(C log(2 √y)) + 2π e− − + 0 3 − n n=1 m=1

∞ 1 ∞ 2πimnx 2πmny + 2π e − n n=1 m=1 These series converge absolutely and it is practical to sum over n ﬁrst. Then

2 π ∞ ∞ 1 2πimnz a = y + 2π(C log 2 √y) + 2π e− + 0 3 − n m=1 n=1 ∞ ∞ 1 + 2π e2πimnz n m=1 n=1 2 π ∞ 2πimz = y + 2π(C log(2 √y)) 2π log(1 e− ) 3 − − − − m=1 ∞ 2π log(1 e2πimz) − − m=1

(πi/12)(z z) ∞ 2πimz = 2π(C log(2 √y)) 2π(log e − + log (1 e )+ − − − m=1

∞ 2πimz + log (1 e− )). − m=1 For complex z = x + iy with y > 0, Dedekind deﬁned the η-function,

∞ η(z) = eπiz/12 (1 e2πimz). − m=1

Inthe notation of Dedekind, then, 14 Kronecker’s Limit Formulas 13

a = 2π(C log 2 √y) 2π log η(z)η( z)) 0 − − − = 2π(C log 2 log( √y η(z) 2)). − − | | Thus we have proved Theorem 1. Let z = x + iy, y > 0 and let Q(u, v) = y 1(u + vz) (u + vz). − Then the zeta function ζQ(s) associated with Q(u, v) and deﬁned by ζQ(s) = s ′(Q(m, n))− ,s = σ + it, σ > 1, can be continued analytically into a function m,n of s regular for σ > 1/2 except for a simple pole at s = 1 with residue π and at s = 1, ζQ(s) has the expansion π ζ (s) = 2π(C log 2 log( √y η(z) 2)) + a (s 1) + Q − s 1 − − | | 1 − − This leads us to the interesting ﬁrst limit formula of Kronecker, viz.

π 2 lim ζQ(s) = 2π(C log 2 log √y η(z) ). s 1 − s 1 − − | | → − We make a few remarks. It is remarkable that the residue of ζQ(s) at s = 1 does not involve a, b, c and this was utilized by Dirichlet in determining the class-number of positive binary quadratic forms. Of course, we had supposed that d = ac b2 = 1; in the general case, the residue of ζ (s) at s = 1 would − Q be π/ √d. This limit formula has several applications; Kronecker himself gave one, namely that of ﬁnding solutions of Pell’s (diophantine) equation x2 dy2 = 4, by means of elliptic functions. It has sereval other applications− and it can be generalized in many ways. In the next section, we shall show that as an application of this formula, the transformation-theory of η(z) under the elliptic modular group can be developed.

2 The Dedekind η-function

Let H denote the complex upper halt-plane, namely the set of z = x + iy with y > 0. For z H, Dedekind defined the η-function ∈ ∞ η(z) = eπiz/12 (1 e2πimz). − m=1 15 This infinite product converges absolutely, uniformly in every compact subset of H. Thus, as a function of z, η(z) is regular in H. Since none of the factors Kronecker’s Limit Formulas 14 of the convergent infinite product is zero in H, it follows that η(z) , 0, for z in H. If g2 and g3 are the usual constants occurring in Weierstrass’ theory of 12 24 3 2 elliptic functions with the period-pair (1, z), then (2π) η (z) = g2 27g3. The function η(z) is a “modular form of dimension 1/2”. − Let α, β, γ, δ be rational integers such that αδ− βγ = 1. The transformation z z = (αz + β)(γz + δ) 1 takes H onto itself; for,− if z = x + iy , then → ∗ − ∗ ∗ ∗ 1 1 αz + β αz + β y∗ = (z∗ z∗) = 2i − 2i γz + δ − γz + δ

1 2 2 = (z z) γz + δ − = y γz + δ − > 0, (14) 2i − | | | | 1 and clearly z = (δz∗ β)( γz∗+α)− . These transformations are called “modular transformations” and− they− form a group called the “elliptic modular group”. It is known that the elliptic modular group is generated by the simple modular transformations, z z + 1 and z 1/z. In other words, any general modular transformation can→ be obtained by→ iterating − the transformations z z 1 and z 1/z. → ± →We − shall give, in this section, two proofs of the transformation-formula for the behaviour of η(z) under the modular transformation z 1/z. The first proof is a consequence of the Kronecker limit formula proved→in − 1. With z = x + iy H, we associate the positive-definite binary§ quadratic 1 ∈ form Q(u, v) = y− (u + vz)(u + vz). By the Kronecker limit formula for ζQ(s) = ∞ s ′ (Q(m, n))− m,n= −∞ π 2 lim ζQ(s) = 2π(C log 2 log( √y η(z) )). (15) s 1 − s 1 − − | | → − 1 Let z∗ = (αz + β)(γz + δ)− = x∗ + iy∗ be the image of z under a modular transformation. Then with z∗, let us associate the positive-definite binary quadratic form 1 Q∗(u, v) = y∗− (u + vz∗)(u + vz∗). Again, by the Kronecker limit formula for 16

∞ = ′ s ζQ∗ (s) (Q∗(m, n))− , m,n= −∞ π 2 lim ζQ (s) = 2π(C log 2 log y∗ η(z∗) ). (16) s 1 ∗ − s 1 − − | | → − Now, by (??), 1 2 1 2 1 2 y∗− m + nz∗ = y− γz + δ m + n(αz + β)(γz + δ)− | | | | | | Kronecker’s Limit Formulas 15

1 2 = y− m(γz + δ) + n(αz + β) | | 1 2 = y− (mδ + nβ) + (mγ + nα)z . | | Since α, β, γ, δ are integral and αδ βγ = 1, when m, n run over all pairs of rational integers except (0, 0), so do− (mδ + nβ, mγ + nα). Thus, in view of absolute convergence of the series, for σ > 1,

(ζQ(s) is what is called a “non-analytic modular function of z”).

Since ζQ(s) and ζQ∗ (s) can be analytically continued in the half-plane σ > = 1/2, ζQ(s) ζQ∗ (s) even for σ > 1/2. Hence, from (15) and (16),

αz + β η = η(z) γz + δ . (17) γz + δ | || | Let us consider the function η((αz + β)(γz + δ) 1) f (z) = − γz + δη(z) where the branch of √γz + δ is chosen as follows; namely, if γ = 0, then α = δ = 1 and assuming without loss of generality that α = δ = 1, we choose √γz + δ ±= 1. If γ , 0, we might suppose that γ > 0 and then √γz + δ is that branch whose argument always lies between 0 and π for z H. Since η(z) and √γz + δ never vanish in H, (17) means that the funciton f (∈z) which is regular in H is of obsolute value 1. By the maximum modulus principle, f (z) = ǫ, a constant of absolute value 1. We have thus Kronecker’s Limit Formulas 16

Proposition 3. The Dedekind η-function

∞ η(z) = eπiz/12 (1 e2πimz), z H − ∈ m=1

1 satisﬁes, under a modular transformation z (αz + β) (γz + δ)− the transformation formula →

1 η((αz + β)(γz + δ)− ) = ǫ γz + δη(z) with ǫ = ǫ(α,β,γ,δ) and ǫ = 1. | | We shall determine ǫ for the special modular transformations, z z + 1 and z 1/z. → From→ − the very deﬁnition of η(z),

η(z + 1) = eπi/12η(z) (18) and so here ǫ = eπi/12. Also if we set z = i in the formula η( 1/z) = ǫ √zη(z), we have η(i) = πi/4 − ǫ √iη(i). Since η(i) , 0, ǫ = 1/ √i = e− . Thusη( 1/z) = e πi/4 √zη(z). We rewrite this as 18 − − 1 z η = η(z), (19) − z i

√z/i being that branch taking the value 1 at z = i. To determine ǫ in the general case, we only observe that any modular transformation is obtained by iteration of the transformations z z 1 and z 1/z. Since every time we apply these transformations we get,→ in± view of → − πi/12 πi/4 th (18) and (19), a factor which is e± or e− , we see that ǫ is a 24 root of unity; ǫ can be determined this way, by a process of “reduction”. The question arises as to whether there exists an explicit expression for ǫ, in terms of α, β, γ and δ. This problem was considered and solved by Dedekind who for this purpose, had to investigate the behaviour of η(z) as z approaches the ‘rational points’ on the real line. This problem has also been considered recently by Rademacher in connection with the so-called ‘Dedekind sums’. We now give a very simple alternative proof of formula (19). Kronecker’s Limit Formulas 17

Fig. 2

Letus consider the integral 19 1 πζ dζ cot πζ cot 8 Ct z ζ where Ct is the contour of the parallelogram in the ζ-plane with vertices at ζ = t, tz, t and tz (t positive and not integral). The integrand− − is a meromorphic function of ζ. It has simple poles at ζ = k 1 πk 1 ± and ζ = kz(k = 1, 2, 3,...) with residues cot and cot πkz respec- ± πk z πk tively, as one can readily notice from the expansion 1 πu cot πu = + πu − 3 valid for 0 < u < 1. Also, the integrand has a pole of the third order at ζ = 0, | 1| with residue 3 (z + 1/z). Let t = h−+ 1/2, h being a positive integer. Then there are no poles of the integrand on Ct and inside Ct there are poles at ζ = 0, k and kz(k = 1, 2,..., h). By Cauchy’s theorem on residues, ± ±

h h 1 ζ dζ 2πi 1 πk 1 1 1 cot πζ cot π = ′ cot + ′ cot πkz z + , 8 C z ζ 8  πk z πk − 3 z  h+ 1 k= h k= h  2  − −    the accent on Σ indicating the omission of k = 0 from the summation. In other Kronecker’s Limit Formulas 18

words,

1 ζ dζ 1 i h 1 πk cot πζ cot π z + = cot πkz + cot . (20) 8 C z 12 z 2  k z  h+ 1 k=1  2     We wish to consider the limit of the relation (20) as h tends to inﬁnity through the sequence of natural numbers. For this purpose, let us ﬁrst observe that if ζ = ξ + iη, then

eπiζ + e πiζ cot πζ = i − eπiζ e πiζ − − tendsto i if η tends to , and tends to +i, if η tends to . Similarly cot π(ζ/z) 20 tends to− i, when the imaginary∞ part of ζ/z tends to −∞and to i, when the latter tends to − . If O is the origin and ζ tends to infinity∞ along a ray OP with −∞ P on one of the open segments AB, BC, CD, DA of C 1 (see figure), then h+ 2 cot πζ π(ζ/z) tends to the values +1, 1, +1, 1 respectively. Moreover, this process of tending to the respective values− is− uniform, when the ray OP lies between two fixed rays from O, which lie in one of the sectors AOB, BOC, COD or DOA and neither of which coincides with the lines η = 0 or ζ = λz (λ real). This means that the sequence of functions cot π(h + 1/2)ξ cot π(h + 1/2)(ζ/z) (h = 1, 2,...) tends to the limits +1 or 1{ uniformly on any proper } − closed subsegment of a side of C1. Moreover, this sequence of functins is uniformly bounded on C1. This can be seen as follows. Let K j denote the discs ζ j < 1/4, j = 0, 1, 2,.... In view of the periodicity of cot πζ, let us| − confine| ourselves to± the ±strip, 0 ξ 2. We then readily see that in the complement of the set- ≤ ≤ union of the discs K0, K1 and K2 with respect to this vertical strip, the function cot πζ is bounded; for cot πζ tends to πi as η tends to and is bounded away from its poles at ζ = 0, 1 and 2. Indeed therefore,±∞ cot πζ is bounded in the complement of the set union of the discs K , j = 0, 1,... with respect to the j ± entire plane. Since the contours C 1 , for h = 1, 2,... lie in the complement, h+ 2 we have cot πζ α for ζ Ch+ 1 and α independent of h and ζ. By a similar | | ≤ ∈ 2 argument for cot(πζ/z) concerning its poles at kz, we see that for ζ C 1 , h+ 2 h = 1, 2,... cot(πζ/z) β, β independent of±h and ζ. In other words,∈ the sequence of functions| cot| ≤π(h+ 1 )ζ cot π(h+ 1 )(ζ/z) is uniformly bounded on { 2 2 } c1. Now πζ dζ 1 1 ζ dζ cot πζ cot = cot π h + ζ cot π h + , (21) z ζ 2 2 z ζ C + 1 C1 h 2 Kronecker’s Limit Formulas 19 and in view of the foregoing considerations, we can interchange the passage to the limit (as h tends to infinity) and the integration, on the right side of (21). Thus, letting h tend to infinity, we obtain from (20),

πi 1 1 z 1 z 1 dζ z + + − + − 12 z 8 1 − z 1 − z ζ − − i ∞ 1 πk = cot πkz + cot . (22) 2  k z  k=1    Now   21 1 z 1 z 1 dζ 1 z z dζ − + − = + . 8 1 − z 1 − z ζ 4 1 1 ζ − − − For z in the ζ-plane cut along the negative ξ-axis, let log z denote the branch that is real on the positive ξ-axis. Then we see

z dζ z dζ = log z and = log z πi. 1 ζ 1 ζ − − Thus from (22),

πi 1 1 πi i ∞ 1 πk z + + log z = cot πkz + cot . (23) 12 z 2 − 2 2  k z  k=1      We insert in (23), the expansions 

e2πikz ∞ cot πkz = i 1 + 2 = i 1 + 2 e2πimkz − 1 e2πikz −   m=1 −     and 2πik/z πk e− ∞ 2imπk/z cot = i 1 + 2 = i 1 + 2 e− ; z 1 e 2πik/z   − −  m=1    then we see that the resulting double series is absolutely convergent. We are therefore justiﬁed in summing over k ﬁrst; we see as a result that the series (23) goes over into

1 η ∞ (1 e2πimz) − z πi 1 log − = log + z + . − (1 e 2πim/z) η(z) 12 z m=1 − − Kronecker’s Limit Formulas 20

Finally 1 η z 1 πi z log − = log z = log , η(z) 2 − 2 i where √z/i is that branch which takes the value 1 at z = i i.e. η( 1/z) = 22 √(z/i)η(z). − We wish to remark that it was only for the sake of simplicity that we took a parallelogram C1 and considered ‘dilatations’ C 1 of C1. It is clear that h+ 2 instead of the parallelogram, we could have taken any other closed contour passing through 1, z, 1, z and through no other poles of the integrand and worked with corresponding− − ‘dilatations’ of the same.

3 The second limit formula of Kronecker

We Shall consider now some problems more general than the ones concerning the analytic continuation of the function ζQ(s) in the half-plane σ > 1/2 and the Kronecker limit formula for ζQ(s). In the ﬁrst place, we ask whether it is possible to continue ζQ(s) analytically in a larger half-plane; this question shall be postponed for the present. We shall show later ( 5) that ζQ(s) has an analytic continuation in the entire plane which is regular except§ for the simple pole at s = 1 and satisﬁes a functional equation similar to that of the Riemann ζ-function. ∞ s Instead of ζQ(s) = ′ (Q(m, n))− , one might also consider m,n= −∞

∞ s (Q(m + , n + ν))− m,n= −∞ where and ν are non-zero real numbers which are not both integral and where m and n run independently from to + . Or, instead of the positive-definite binary quadratic form Q(u, v), we−∞ might take∞ a positive-definite quadratic form in more than two variables. Let, in fact, S = (s ), 1 i, j p be the matrix of i j ≤ ≤ a positive-definite quadratic form si j xi x j in p variables. Let us consider 1 i, j p ≤≤

∞ = ζS (S )  S i j xi x j x1,...,xp= i, j  −∞     x1,..., xp running over all p-tuples of integers, except (0,..., 0). The func- tionζS (S ), known as the Epstein zeta-function, is regular for σ?p/2. It was 23 Kronecker’s Limit Formulas 21

shown by Epstein that ζS (S ) can be continued analytically in the entire s-plane, into a function regular except for a simple pole at S = p/2 and satisfying the functional equation

s 1 (p/2 s) p p π− Γ(S )ζS (S ) = S − 2 π− − Γ s ζS 1 s | | 2 − − 2 − Epstein also obtained for ζS (s), an analogue of the first limit formula of Kro- necker but this naturally involves more complicated functions than η(z). We shall now consider a generalization of ζQ(s) which shall lead us to the second limit formula of Kronecker. namely, let u and v be independent real parameters. Let, as before, Q(m, n) = am2 + 2bmn + cn2 be a positive-definite binary quadratic form and let ac b2 = 1, without loss of generality. We then define, for σ > 1, −

′ 2πi(mu+nv) s ζQ(s, u, v) = e (Q(m, n))− (24) m,n m, n running over all ordered pairs of integers except (0, 0). The series converges absolutely for σ > 1 and converges uniformly in every half-plane σ 1 + ǫ(ǫ > 0). Thus ζ (s, u, v) is a regular function of s for σ > 1. If ≥ Q u and v are both integers, then ζQ(s, u, v) = ζQ(s) which has been already considered. We shall, suppose, in the following, that u and v are not both integers. ( ) ∗ Furthermore, on account of the periodicity of ζQ(s, u, v) in u and v, we can clearly suppose that 0 u, v < 1. The condition ( ) then means that at least one of the two conditions≤ 0 < u < 1, 0 < v < 1 is satisﬁed.∗ We might, without loss of generality, suppose that 0 < u < 1. For, otherwise, if u = 0, then necessarily 0 < v < 1. In this case, let us take the positive-deﬁnite binary 2 2 quadratic form Q1(m, n) = cm + 2bmn + an ; we see then that, for σ > 1,

′ 2πi(mv+nu) s ζQ(s, u, v) = e (Q1(m, n))− . m,n And here, since 0 < v < 1, v shall play the role of u in (24). Weshall prove now that ζQ(s, u, v) can be continued analytically into a func- 24 tion regular in the half-plane σ > 1/2 and then determine its value at s = 1; this will lead us to the second limit formula of Kronecker. For σ > 1, by our earlier notation,

s ∞ 2πimu 2s s ∞ 2πinv ∞ 2πimu 2s ζ (s, u, v) = y e m − + y e e m + nz − . (25) Q | | | | m= n= m= −∞ −∞ −∞ Kronecker’s Limit Formulas 22

The first series in (25) converges absolutely for σ > 1/2 and uniformly in every compact subset of this half-plane. Hence it defines a regular function of s for σ > 1/2. The double series in (25) again defines a regular function of s in the half-plane σ > 1. To obtain its analytic continuation in a larger half- plane, we shall carry out the summation of the double series in the following particular way. Namely, we consider the finite partial sums

2πi(mu+nv) 2s ′ e m + nz − . | | n1n n2 m1 m m2 ≤ ≤ ≤ ≤

We shall show that when m1 and n1 tend to independently and m2 and n2 tend to independently, then these partial sums−∞ which are entire functions of s converge∞ uniformly in every compact subset of the half-plane σ > 1/2. The proof is based on the method of partial summation in Abel’s Theorem. e2πiku Let us deﬁne for any integer k, c = . (Recall that 0 < u < 1). k e2πiu 1 2πimu Then e = c + c . Moreover, − m 1 − m 2πiu 1 c 1 e − = α, | m| ≤ | − | where α is independent of m. Now

2s 2σ 2σ c + m + nz − α n − y− m2 1| 2 | ≤ | | and 2s 2σ 2σ c m + nz − α n − y− . m1 | 1 | ≤ | | Further 2s 2s m 1 + nz − m + nz − || − | − | | | Kronecker’s Limit Formulas 23

m 2 2 2 s 1 2s (( + nx) + n y )− − ( + nx)d − m 1 − m 2 2 2 (σ+ 1 ) 2 s (( + nx) + n y )− 2 du. ≤ | | m 1 − Thus (26) is majorised by

n2 m2 m n2 2 2 2 (σ+ 1 ) 2σ 2σ 2 s α ′ (( + nx) + n y )− 2 d + 2αy− ′ n − . (27) | | m 1 | | n=n1 m=m1+1 − n=n1

In (27), we sum m from to + instead of summing from m1 + 1 to m2 and then we see that (26) has−∞ the majorant∞

n2 n2 ∞ 2 2 2 (σ+ 1 ) 2σ 2σ 2α s ′ (( + nx) + n y )− 2 d + 2αy− n − . | | | | n=n1 −∞ n=n1 Now 26

∞ 2 2 2 (σ+ 1 ) ∞ 2 2 2 (σ+ 1 ) (( + nx) + n y )− 2 d = ( + n y )− 2 d −∞ −∞ 2σ 2σ ∞ 2 (σ+ 1 ) = n − y− (1 + )− 2 d. (28) | | −∞ Since the integral in (28) converges for σ > 1/2, we see that (26) is majorised n2 2σ by β n − , β depending only on the compact set in which s lies in the n=n1 | | half-plane σ > 1/2 and on α and y. If, in (26), we now carry out the passage of m1 and n1 to and of m2 and n to + independently, then we see that the double series in−∞ (25) summed 2 ∞ ∞ 2σ in this particular manner, is majorised by the series 2β n− . Hence it con- n=1 verges uniformly when s lies in a compact set in the half-plane σ > 1/2. Since each partial sum (26) is an entire function of s, we see, by Weierstrass’ Theo- rem that the double series in (25) converges uniformly to a function which is regular for σ > 1/2 and which provides the necessary analytic continuation in this half-plane. Thus under the assumption that u and v are not both integers, ζQ(s, u, v) has an analytic continuation which is regular for σ > 1/2. We shall now determine its value at s = 1. We wish to make it explicit that hereafter we shall make only the assumption ( ) and not the speciﬁc assumption 0 < u < 1. From the convergence∗ proved above, it follows that

z z ∞ e2πimu ζ (1, u, v) = − ′ + Q 2i m 2 m= −∞ | | Kronecker’s Limit Formulas 24

z z ∞ ∞ e2πimu + − ′ e2πinv (29) 2i (m + nz)(m + nz) n= m= −∞ −∞ where the accents on Σ have the usual meaning. e2πimu e2πimu ∞ ∞ 27 In order to sum 2 , we observe that since the series m= m m= m converges uniformly to−∞ 2π(1| /|2 u) in any closed interval. 0 < p u−∞ q < 1, we have − ≤ ≤

u u ∞ e2πimu ∞ e2πimu du = lim ′ du  m  ǫ 0  m  0 m=  → ǫ m=   −∞   −∞     u    ∞  e2πimu  = lim ′ du, ǫ 0 m → m= ǫ −∞ i.e. 1 ∞ e2πimu ∞ e2πimu 2πi(u u2) = lim ′ ′ , − 2πi ǫ 0  m2 − m2  → m= m=   −∞ −∞  i.e.   ∞ e2πimu π2 2π2(u2 u) = ′ , − − m2 − 3 m= −∞ 2πimu ∞ e since ′ 2 is continuous in u. In other words, m= m −∞ ∞ e2πimu 1 ′ = 2π2 u2 u + . m2 − 6 m= −∞ We shall sum the double series in (29) more generally with z replaced by r, where r H. Leter, we shall set r = z. We remark that with slight changes, − ∈ 2πi(mu+nv) 2s our arguments concerning the series ′ e m + nz − above will also 2πi(mu+nv) | | s go through for the double series ′ e ((m + nz) (m + nr)) . Now for − summing the series

z r ∞ ∞ e2πimu − ′ e2πinv 2i (m + nz)(m + nr) n= m= −∞ −∞ 1 ∞ e2πinv ∞ 1 1 = ′ e2πimu . (30) 2i n m + nr − m + nz n= m= −∞ −∞ we could apply the Poisson summation formula with respect to m to the inner sum on the left side of (30), but we shall adopt a diﬀerent method here. Kronecker’s Limit Formulas 25

Letus deﬁne, for complex z = x + iy, 28

1 ∞ ∞ e2πimu e 2πimu f (z) = + + + − z z + m z m m=1 m=1 − and consider f (z) in the strip 1/2 x 1/2. It is not hard to see that whenever z lies in a compact set− not containing≤ ≤ z = 0 in this strip, the series converges uniformly and hence f (z) is regular in this strip except at z = 0, where it has a simple pole with residue 1. Moreover, one can show that f (z) 1/z is bounded in this strip. Further −

2πiu f (z + 1) = e− f (z). Consider now the function e 2πiuz g(z) = 2πi − ; 1 e 2πiz − − g(z) is again regular in this strip except at z = 0 where it has a simple pole with residue 1. Moreover g(z) = 1/z is bounded in this strip; as a matter of fact, if 0 < u < 1, g(z) tends to 0 if y tends to + or and if u = 0, then g(z) tends ∞ −∞ 2πiu to 0 or 2πi according as y tends to + or . Further g(z + 1) = e− g(z). As a consequence, the function ∞f (z) −∞g(z) is regular and bounded in this 2πiu− strip and since f (z + 1) g(z + 1) = e− ( f (z) g(z)), it is regular in the whole z-plane and bounded, too.− By Liouville’s theorem,− f (z) g(z) = c, a constant. It can be seen that, if u , 0, c = 0 and if u = 0, c = π−i, by using the series expansion for the function π cot πz. In any case, f (nr) f (nz) = g(nr) g(nz). (31) − − In view of the convergence-process of the series (29) described above, we can rewrite the double series (30) as

1 ∞ e2πinv 1 1 ∞ 1 1 1 1 ′ + e2πimu + 2i n nr − nz m + nr m + nr − m + nz − m + hz n=  m=1  −∞  − −  2πinv 2πinv 2πiunr 2πiunz  1 ∞ e  ∞ e e e  = ′ ( f (nr) f (nz)) = π ′ − − , 2i n − n 1 e 2πinr − 1 e 2πinz n= n= − − − − −∞ −∞ (32) by (31). Now, we insert in (32), the expansions 29

e2πimnz, (n > 0) 2πinz 1 = m∞=1 (1 e− )− − 2πmnz −  ∞ e− , (n < 0)  m=0  Kronecker’s Limit Formulas 26

e 2πimnr, (n > 0) 2πinr 1 = m∞=0 − (1 e− )− 2πimnr −  ∞ e , (n < 0) − m=1 valid for z, r H and then the series (32) goes over into − ∈

∞ 1 2πin(v ur) 2πin(v uz) ∞ 2πin(v uz mz) π e − + e− − + (e− − − + n  n=1  m=1  + 2πin(v ur mr) + 2πin(v uz+m′z) + 2πin(v ur+mr) e − − e − e− − ) (33)  If 0 < u < 1, then this series converges absolutely, since z, r H. Carrying out the summation over n ﬁrst, we see that this double series− is∈ equal to

∞ 2πi(v uz mz) 2πi(v ur mr) π log (1 e− − − )(1 e − − ) −  − − × m=0  ∞ 2πi(v uz+mz) 2πi(v ur+mr) (1 e − )(1 e− − ) . (34) × − −  m=1  If u = 0, then necessarily 0 < v < 1 and in this case, 

∞ 1 2πinv 2πinv 2πiv 2πiv π (e + e− ) = π log(1 e )(1 e− ). n − − − n=1 Therest of the series (33) is absolutely convergent and summing over n first, 30 we see that the value of the series (33) is given by the expression (34). Let us now set v uz = w, v ur = q. Then replacing z by r in (29), we have finally, for 0 u−, v < 1 and u−and v not simultaneously zero, ≤ + z r ∞ e2πi(mu nv) − 2i (m + nz)(m + nr) m,n= −∞ 2 2 1 ∞ 2πi(w mz) = π i(z r) u u + π log (1 e− − ) − − − 6 −  − × m=0  2πi(q mr) ∞ 2πi(w+mz) 2πi(q+mr) (1 e − ) (1 e )(1 e− ) (35) × − × − −  m=1   For z H and arbitrary complex w, the elliptic theta-functionϑ1(w, z) is defined by∈ the infinite series,

∞ πi(n+ 1 )2z+2πi(n+ 1 )(w 1 ) ϑ1(w, z) = e 2 2 − 2 . n= −∞ Kronecker’s Limit Formulas 27

It is clear that this series converges absolutely, uniformly when z lies in a compact set in H and w in a compact set of the w-plane. Hence ϑ1(w, z) is an analytic function of z and w for z H and complex w. It is proved in the ∈ theory of the elliptic theta-functions that ϑ1(w, z) can also be expressed as an absolutely convergent inﬁnite product, namely,

πi(z/4) πiw πiw ϑ (w, z) = ie (e e− ) 1 − − × ∞ 2πi(w+mz) 2πi(w mz) 2πimz (1 e )(1 e− − )(1 e ). (36) × − − − m=1

We shall identify these two forms of ϑ1(w, z) later on. For the present, we shall consider ϑ1(w, z) as deﬁned by the inﬁnite product. Now a simple rearrangement of the expression (35) gives

z r ∞ e2πi(mu+nv) − ′ 2i (m + nz)(m + nr) m,n= −∞ 1 1 = π2i(z r) u2 u + + π2i (w q) + (z r) − − − 6 − 6 − − ϑ (w, q)ϑ (q, r) π log 1 1 − − η(z)η( r) − (w q)2 ϑ (w, q)ϑ (q, r) = π2i − π log 1 1 − − z r − η(z)η( r) −2πi(mu+nv) − z r e ϑ1(w, z)ϑ1(q, r) πi((w q)2/(z r)) − ′ = log − e − − . (37) 2πi (m + nz)(m + nr) η(z)η( r) − m,n −

Setting r = z again, we have q = w and then 31

(w w)2 ϑ (w, z)ϑ (w, z) ζ (1, u, v) = π2i − π log 1 1 − Q − z z − η(z) 2 − | | One can see easily from (36) that ϑ (w, z) = ϑ (w, z). Hence 1 1 − ϑ (w, z) ζ (1, u, v) = π2iu2(z r) 2π log 1 , (38) Q − − − η(z) for 0 u, v < 1 and u and v not both zero. We≤ contend that formula (38) is valid for all u and v not both integral. Since ζQ(1, u + 1, v) = ζQ(1, u, v + 1) = ζQ(1, u, v) and w goes over into w + 1 and Kronecker’s Limit Formulas 28 w z respectively under the transformations v v + 1 and u u + 1, it would su−ﬃce for this purpose to prove that → → ϑ (w + 1, z) ζ (1, u, v) = π2iu2(z r) 2π log 1 Q − − − η(z) and 2 2 ϑ1(w z, z) ζQ(1, u, v) = π i(u + 1) (z r) 2π log − . − − − η(z) The ﬁrst assertion is an immediate consequence of the fact that ϑ1(w + 1, z) = ϑ (w, z). To prove the second, we observe that from (36), we have − 1 ϑ (w z, z) (eπi(w z) e πi(w z))(1 e2πiw) 1 = − − − − πiw − πiw 2−πi(w z) ϑ1(w, z) (e e− )(1 e− − ) +2πiw −πiz − = e − and hence 32 ϑ (w z, z) ϑ (w, z) 2π log 1 − log 1 = +π2i (2(w w) (z z) η(z) − η(z) { − − − } = π2i(z z)((u + 1)2 u2). − − − Thus the second assertion is proved and we have

Theorem 2. If u and v are real and not both integral, then the Epstein zeta- function ζQ(s, u, v) deﬁned by (24) for σ > 1, can be continued analytically into a function regular for σ > 1/2 and its value at s = 1 is given by (38).

We are ﬁnally led to the following formula: viz. for all u and v not both integral,

z z e2πi(mu+nv) (w w)2 ϑ (w, z) − ′ = π2i − 2π log 1 . 2i m + nz 2 − z z − η(z) m,n | | − This is the second limit formula of Kronecker. We may rewrite it as

2πi(mu+nv) 2 z z ′ e ϑ (v uz, z) 2 − = log 1 − eπizu . (39) 2πi m + nz 2 η(z) − | | If u and v are both integral, then v uz is a zero of ϑ1(w, z) and then both sides are inﬁnite. − Kronecker’s Limit Formulas 29

4 The elliptic theta-function ϑ1(w, z)

For z = x + iy H and arbitrary complex w, the elliptic theta-function ϑ (w, z) ∈ 1 was deﬁned in 3, by the inﬁnite product (36). ϑ1(w, z) is a regular function of z and w, for§z in H and arbitrary complex w. We shall now develop the transformation-theory of ϑ1(w, z) under modular transformations, as a consequence of the second limit formula of Kronecker. Letu and v be arbitrary real numbers which are not simultaneously integral. 33 Let z z = (αz + β)(γz + δ) 1 = x + iy be a modular transformation. Then → ∗ − ∗ ∗ u∗ = δu + γv and v∗ = βu + αv are again real numbers, which are not both integral. For σ > 1, we have

2πi(mu∗+nv∗) 2πi((mδ+nβ)u+(mγ+nα)v) s ∞ e s e y∗ = y ′ m + nz 2s (mδ + nβ) + (mγ + nα)z 2s m,n= ∗ m,n −∞ | | | | e2πi(mu+nv) = ys ′ . m + nz 2s m,n | | From 3, we know that both sides, as functions of s, have analytic continua- tions regula§ in the half-plane σ > 1/2 and hence the equality of the two sides is valid even for σ > 1/2. In particular, for s = 1,

e2πi(mu∗+nv∗) e2πi(mu+nv) y∗ ′ = y ′ . (40) m + nz 2 m + nz 2 m,n | ∗| m,n | | By Kronecker’s second limit formula,

2 2 2 (w∗ w∗) ϑ1(w∗, z∗) (w w) ϑ1(w, z) π i − 2π log = π∗i − 2π log − z∗ z∗ − η(z∗) − z z − η(z) − − where w = v uz and w = v u z =w/(γz + δ). Therefore, − ∗ ∗ − ∗ ∗ 2 2 ϑ (w , z ) ϑ (w, z) πi (ww) πi (w w∗) log 1 ∗ ∗ log 1 = ∗ − η(z ) − η(z) 2 z z − 2 z z∗ ∗ − ∗ − 2 2 πiγ w w 2 = = log e(πiγw )/(γz+δ) 2 γz + δ − γz + δ Thus w αz+β ϑ1 γz+δ , γz+δ ϑ (w, z) 2 = 1 e(πiγw )/(γz+δ) ω, (41) αz+β η(z) η γz+δ where ω = 1 and ω might depend on every one of w, z, α, β, γ and δ. Butfrom 34 | | Kronecker’s Limit Formulas 30

(41), we observe that ω is a regular function of z in H and w, except possibly when w is of the form m + nz with integral m and n. But since ω = 1, we see, by applying the maximum modulus principle to ω as a function| of| w and of z, that ω is independent of w and z. Hence ω = ω(α,β,γ,δ). Now from the fact that

ϑ (w, z) = 2πη3(z)w + 1 we note that ϑ1′ (0, z), the value of the derivative of ϑ1(w, z) with respect to w at w = 0, is equal to 2πη3(z). Diﬀerentiating the relation (41) with respect to w at w = 0, we get αz+β 1 ϑ1′ 0, γz+δ ϑ′ (0, z) = 1 ω, γz + δ αz+β η(z) η γz+δ i.e. 1 2 αz + β 2 (γz + δ)− η = ωη (z). γz + δ We have thus rediscovered the formula αz + β η = ǫ γz + δη(z) γz + δ with ǫ being a 24th root of unity depending only on α, β, γ and δ and ω = ǫ2. Rewriting (41), we obtain Proposition 4. The elliptic theta-function has the transformation formula

w αz + β 3 (πiγw2)/(γz+δ) ϑ1 , = ǫ γz + δe ϑ1(w, z) γz + δ γz + δ 1 3 th under a modular substitution z (αz + β)(γz + δ)− , ǫ being an 8 root of unity depending only on α, β, γ and→ δ. From the transformation-theory of η(z) we know that corresponding to the modular transformations z z+1 and z 1/z, ǫ3 has respectively the values πi/4 3πi/4 → → − e and e− . Hence

πi/4 ϑ1(w, z + 1) = e ϑ1(w, z)  (42) w 1 πiw2/z 1 z  ϑ1 , = e ϑ1(w, z). z − z i i    Once again using the fact that the modular transformations z z + 1 and 35 → Kronecker’s Limit Formulas 31 z 1/z generate the elliptic modular group, we can determine ǫ3, with the help→ of − (42), by a process of ‘reduction’. Hermite found an explicit expression for ǫ3, involving the well-known Gaussian sums. We now wish to make another remark which is of function-theoretic signif- icance and which is again, a consequence of Kronecker’s second limit formula. Let us deﬁne for z H and real u and v (not both integral), the function ∈ ϑ (v uz, z) 2 f (z, u, v) = log 1 − eπizu η(z) choosing a ﬁxed branch of the logarithm in H. Then from (40) and (39),

ϑ1(v∗ u∗z∗, z∗) 2 ϑ1(v uz, z) 2 log − eπiz∗u∗ = log − eπizu . η(z∗) η(z) Hence αz + β f ,δu + γv, βu + αv = f (z, u, v) + iλ, γz + δ where λ is real and might depend on z, u, v, α, β, γ and δ. But since λ is a regular function of z, whose imaginary part is zero in H, λ is independent of z. Thus αz + β f ,δu + γv, βu + αv = f (z, u, v) + iλ(u, v,α,β,γ,δ). (43) γz + δ Let now q > 1, be a ﬁxed integer and let u and v be proper rational fractions with reduced denominator q i.e. u = a/q, v = a/q, (a, b, q) = 1 and 0 u, v < 1. Since αδ βγ = 1, we have again. ≤ − (δa + γb, βa + αb, q) = 1.

ϑ1(v uz, z) 2 Nowfrom (39), we see that log − eπizu is invariant under the 36 η(z) transformations u u + 1 and v v + 1. Hence f (z, u, v) picks up a purely imaginary additive→ constant under→ these transformations. If now, (δa + γb)/q ≡ (a∗/q)( mod 1) and (βa + αb)/q (b∗/q)( mod 1) where a∗/q, b∗/q are proper rational fractions with reduced≡ denominator q, then

αz + β δa + γb βa + γb αz + β a∗ b∗ f , , = f , , + iλ′ γz + δ q q γz + δ q q where λ′ is real and depends on a, b, α, β, γ and δ. Writing f (z, a/q, b/q) as fa,b(z), we get from (43), αz + β fa ,b = fa,b(z) + iλ∗(a, b,α,β,γ,δ) (44) ∗ ∗ γz + δ Kronecker’s Limit Formulas 32

λ∗(a, b,α,β,γ,δ) being a real constant depending on a, b, α, β, γ and δ. The number of reduced fractions a/q, b/q with reduced denominator q is given by v(q) = q2 (1 1/p2)p running through all prime divisors of q. Corre- p q − | sponding to each pair a/q, b/q, we have a function fa,b(z) and the relation (44) asserts that when we apply a modular transformation on z, these functions are permuted among themselves, except for a purely imaginary additive constant. 1 α β 1 0 The modular transformations z to(αz + β)(γz + δ)− for which γ δ 0 1 ( mod q), form a subgroup M (q) of the elliptic modular group called the≡principal congruence subgroup of level (stufe) q. The group M (q) has index qv(q) in the elliptic modular group and acts discontinuously on H. We can construct in the usual way, a fundamental domain for M (q) in H. This can be made into a compact Riemann surface by identify- ing the points on the boundary which are ‘equivalent’ under M (q) and ‘adding the cusps’. Complex-valued functions f (z) which are meromorphic in H and invariant under the modular transformations in M (q) and have at most a pole in the ‘local uniformizer’ at the ‘cusps’ of the fundamental domain are called modular functions of level q. They form an algebraic function field of one variable (over the field of complex numbers) which coincides with the field of meromorphic functions on the Riemann surface. Inthe case when the transformation z (αz + β)(γz + δ) 1 is in M (q), then 37 → − referring to (44), we have a∗ = a, b∗ = b and

αz + β fa,b = fa,b(z) + iλ∗(a, b,α,β,γ,δ). (45) γz + δ

The function fa,b(z) is regular everywhere in H. To examine its singularities (in the local uniformizers) at the ‘cusps’ on the Riemann surface, it suffices in virtue of (44) to consider the singularity of fa,b(z) at the ‘cusp at infinity’. It can be seen that at the ‘cusp at infinity’, fa,b(z) has the singularity given by πiz(a2q 2 aq 1+1/6) log e − − − . In view of (45), then fa,b(z) is an abelian integral on the Riemann surface associated with M (q) and the quantities λ∗(a, b,α,β,γ,δ) are ‘periods’. Since the expression u2 u + 1/6 does not vanish for rational u, − fa,b(z) is an abelian integral strictly of the third kind. It might be an interesting problem to determine the ‘periods’, λ∗(a, b,α,β,γ,δ). We had defined the function ϑ1(w, z) by the infinite product (36); now, we shall identify it with the infinite series expansion by which it is usually defined. For this purpose, we define for z H and complex w, ∈

∞ πi(n+ 1 )2z+2πi(n+ 1 )(w 1 ) f (w, z) = e 2 2 − 2 . (46) n= −∞ Kronecker’s Limit Formulas 33

Clearly f (w, z) is a regular function of w and z. Moreover, it is easy to see that

f (w + 1, z) = f (w, z). (47) − Also replacing n by n + 1 on the right-hand side of (46), we have

∞ πiz(n+1+ 1 )2+2πi(n+1+ 1 )(w 1 ) f (w, z) = e 2 2 − 2 n= −∞ = e2πiw+πiz f (w + z, z), − i.e. 2πiw πiz f (w + z, z) = e− − f (w, z). (48) − We know already that ϑ (w + 1, z) = ϑ (w, z) 1 − 1 and 2πiw πiz ϑ (w + z, z) = e− − ϑ (w, z). 1 − 1 f (w, z) Hence,for ﬁxed z H, the function h(w, z) = is a doubly periodic 38 ∈ ϑ1(w, z) function of w with independent periods 1 and z and is regular in w except possibly when w = m + nz with integral m and n, since ϑ1(w, z) has simple zeros at these points. But f (w, z) also has zeros at these points, since, replacing n by n 1 on the right-hand side of (46), we have f ( w, z) = f (w, z) and hence−f (0−, z) = 0 and from (47) and (48), f (m + nz, z) =− 0 for integral− m and n. Thus h(w, z) is a doubly periodic entire function of w and by Liouville’s theorem, it is independent of w, i.e. h(w, z) = h(z). The function h(z) is a regular function of z in H and to determine its behaviour under the modular transformations, it suﬃces to consider its behaviour 1 under the transformations z z + 1 and z z− . We can show easily that h(z + 1) = h(z). Moreover, by→ (42), → −

w 1 πiw2/z 1 z ϑ1 , = e ϑ1(w, z). z − z i i Let us assume, for the present, that we have proved the formula

w 1 2 1 z f , = eπiw .z f (w, z). (49) z − z i i It will then follow that h( 1/z) = h(z). Hence h(z) is a modular function and it is indeed regular everywhere− in H. Kronecker’s Limit Formulas 34

Now, a fundamental domain of the elliptic modular group in H is given by the set of z = x + iy H for which z 1 and 1/2 x 1/2. Let z tend to infinity in this fundamental∈ domain.| | Then ≥ it is− easy to≤ see≤ that the functions πiz/4 πiz/4 πiw πiw ϑ1(w, z)e− and f (w, z)e− tend to the same limit i(e e− ). Hence h(z) 1 tends to zero as z tends to infinity in the fundamental− − domain. But then,− being regular in H and invariant under modular transformations, it attains its maximum at some point in H. By the maximum modulus principle, h(z) 1 is a constant and since h(z) 1 tends to zero as z tends to infinity, h(z) = 1.− In − other words, ϑ1(w, z) = f (w, z). To complete the proof, all we need to do is to prove (49). We may first rewrite f (w, z) as

πi(w 1 )2/z ∞ πiz(n+ 1 +w/z 1/2z)2 f (w, z) = e− − 2 e 2 − n= −∞ 39 πξx2 ∞ Let ξ > 0 and let g(x) = e− , for < x < . Then the series g(x+ −∞ ∞ n= n) converges uniformly in the interval 0 x 1 and by the Poisson summation−∞ formula (7), ≤ ≤

∞ πξ(x+n)2 ∞ 2πinx ∞ πξx2 2πinx e− = e e− − dx, (50) n= n= −∞ −∞ −∞ (provided the series on the right-hand side converges). Now

πn2/ξ ∞ πξx2 2πinx e− ∞ π(x+in/ √ξ)2 e− − dx = e− dx, ( ξ > 0). √ξ −∞ −∞ By the Cauchy integral theorem, one can show that

∞ π(x+in/ √ξ)2 ∞ πx2 e− dx = e− dx = γ(say). −∞ −∞ It is now obvious that the series on the right-hand side of (50) converges absolutely. Hence

∞ πξ(x+n)2 γ ∞ πn2/ξ+2πinx e− = e− . (51) n= √ξ n= −∞ −∞ Setting ξ = 1 and x = 0 in (51), we have immediately γ = 1. Both sides of (51) represent, for complex x, analytic functions of x and since they are equal for real x, formula (51) is true even for complex x. Moreover, both sides of (51) represent analytic functions of ξ, for complex ξ with Re ξ > 0. Since they Kronecker’s Limit Formulas 35 coincide for real ξ > 0, we see again that (51) is valid for all complex ξ with Re ξ > 0. We have thus the theta-transformation formula

∞ πξ(n+v)2 1 ∞ πn2/ξ+2πinv e− = e− (52) n= √ξ n= −∞ −∞ valid for all complex v and complex ξ with Re ξ > 0, √ξ denoting that branch which is positive for real ξ > 0. Settingξ = iz and v = (1/2) + (w/z) (1/2z) in (53), we have 40 − − πi(w 1 )2/z e− − 2 ∞ πin2/z+2πin((1/2)+(w/z) (1/2z)) f (w, z) = e− − . z n= i −∞ Now

πin2/z+2πin((1/2)+(w/z) (1/2z)) πi(w 1 )2/z e− − − − 2 πi(n+ 1 )2/z+2πi(n+ 1 )(w/z 1/2) πiw2/z+πi/2 = e− 2 2 − − .

Thus ie πiw2/z w 1 f (w, z) = − f , , z z − z i which is precisely (49). Thus

Proposition 5. For the elliptic theta-function ϑ1(w, z) deﬁned by the inﬁnite product (36), we have the series-expansion

∞ πi(n+ 1 )2z+2πi(n+ 1 )(w 1 ) ϑ1(w, z) = e 2 2 − 2 . n= −∞ Using the method employed above, we shall also prove Proposition 6. The Dedekind η-function η(z) has the inﬁnite series expansion

πiz/12 ∞ λ πiz(3λ2 λ) η(z) = e ( 1) e − − λ= −∞ Proof. Let us deﬁne for z H, ∈

πiz/12 ∞ πiz(3λ2 λ) f (z) = e ( 1)∗e − . − λ= −∞ Kronecker’s Limit Formulas 36

The function f (z) is regular in H and moreover, it is clear that f (z + 1) = f (z)eπi/12. Again,

πiz/12 ∞ 2πiz(λ 1 (1 1/z))2 πiz(1 1/z)2/12 f (z) = e e − 6 − − − λ= −∞ πi/12z πi/6 ∞ 3πiz(λ 1 (1 1/z))2 = e− e e − 6 − λ= −∞ πi/12z πi/6 i ∞ πiλ2/3z πiλ(1 1/z)/3 = e− e e− − − , 3z λ= −∞ using formula (52). Here √i/3z denotes that branch which is positive for z = i. 41 ∞ πi/3zλ2 πiλ/3(1 1/z) Now we split up e− − − as g0(z) + g1(z) + g2(z), where for λ= k = 0, 1, 2, −∞ ∞ πi(3+k)2/3z πi(3+k)(1 1/z)/3 gk(z) = e− − − . n= −∞ Clearly,

∞ πi(32 )/z g (z) = ( 1) e− − , 0 − = −∞ πi/3 g1(z) = e− g0(z),

∞ πi(92+12+4)/3z πi(3+2)(1 1/z)/3 g2(z) = e− − − = −∞ 2πi/3z 2πi/3 ∞ 3πi(+1)/z = e− − ( 1) e− . − = −∞ Now ∞ 3πi(+1)/z ∞ 3πi(+1)/z ( 1) e− = ( 1) e− , − − − = = −∞ −∞ on replacing by 1 , and hence g (z) = 0. Thus − − 2

πi/12z i πi/6 πi/6 ∞ πi(32 )/z f (z) = e− (e + e− ) ( 1) e− − 3z − = −∞ i 1 = f . z − z Kronecker’s Limit Formulas 37

Asa consequence, the function h(z) = f (z)/n(z) is invariant under the transfor- 42 mations z z + 1 and z 1/z and hence under all modular transformations. Moreover,→ since η(z) does→ not − vanish anywhere in H, h(z) is regular in H; further, as z tends to inﬁnity in the fundamental domain, h(z) 1 tends to zero. − Now by the same argument as for ϑ1(w, z) above, we can conclude that h(z) = 1 and the proposition is proved.

The integers (1/2)(3λ2 λ) for λ = 1, 2,... are the so-called ‘pentagonal numbers’. − We now give another interesting application of Kronecker’s second limit formula. Let us consider, once again, formula (37). The left-hand side may be looked upon as a trigonometric series in u and v; it is true, of course, that u and v are not entirely independent. We shall now see how, using the inﬁnite series expansion of ϑ1(w, z), Kronecker showed that the function

(πi(w q)2)/(z r) ϑ (w, z)ϑ (q, r)e − − 1 1 − has a valid Fourier expansion in u and v and derived a beautiful formula from (37). First,

∞ πiz(n+ 1 )2+2πi(n+ 1 )(w 1 ), ϑ1(w, z) = e 2 2 − 2 n= −∞ ∞ πir(m+ 1 )2+2πi(m+ 1 )(q 1 ) ϑ (q, r) = e− 2 2 − 2 , 1 − m= −∞ replacing m by m 1, we have − −

∞ πir(m+ 1 )2 2πi(m+ 1 )(q 1 ) ϑ (q, r) = e− 2 − 2 − 2 . 1 − m= −∞ Hence

ϑ (w, z)ϑ (q, r) 1 1 − ∞ ∞ πi z(n+ 1 )2+2(n+ 1 )(w 1 ) r(m+ 1 )2 2(m+ 1 )(q 1 ) = e { 2 2 − 2 − 2 − 2 − 2 }. m= n= −∞ −∞ This double series converges absolutely and replacing n by n + m in the inner sum, we have Kronecker’s Limit Formulas 38

ϑ (w, z)ϑ (q, r) 1 1 − ∞ ∞ πi(z r)(m+ 1 )2+2πi(m+ 1 )(w+nz q)+πin2z+2πin(w 1 ) = e − 2 2 − − 2 m= n= −∞ −∞ ∞ n πin2z+2πinw πi(w+nz q)2/(z r) ∞ πi(z r)(m+ 1 +(w+nz q)/(z r))2 = ( 1) e − − − e − 2 − − − n= m= −∞ −∞ Applying the theta-transformation formula to the inner sum, we have 43

πi(w q)2/(z r) ϑ (w, z)ϑ (q, r)e − − 1 1 − i ∞ n πin2z+2πinw πi(n2z2+2nz(w q))/(x r) = ( 1) e − − − z r − × n= − −∞ ∞ m πim2/(z r) 2πim(w q+nz)/(z r) ( 1) e− − − − − × − m= −∞ i mn+m+n πi(m+nz)(m+nr)/(z r)+2πi((m+nz)q (m+nr)w)/(z r) = ( 1) e− − − − z r − − m,n √i/z r being that branch which assumes the value 1 for z r = i. Let− − i Q(ξ,η) = (ξ + ηz)(ξ + ηr) = aξ2 + bξη + cηn, z r − where i i(z + r) izr a = , b = and c = . z r z r z r − − − Then

πi(w q)2/(z r) ϑ (w, z)ϑ (q, r)e − − 1 1 − i mn+m+n πQ(m,n)+2πi((m+nz)q (m+nr)w)/(z r) = ( 1) e− − − z r − − m,n Formula (37) now becomes

i mn+m+n πQ(m,n)+2πi(mu+nv) + 1 e2πi(mu nv) z r m,n( 1) e− ′ = log − − −2π Q(m, n) η(z)η( r) m,n − It is remarkable that the quadratic form Q(m, n) emerges undisturbed on the 44 Kronecker’s Limit Formulas 39 right-hand side; the right-hand side is free from z and r, except for the factors √i/(z r) and η(z)η( r). To eliminate z and r therefrom, we diﬀerentiate − πi(w q)2/(−z r) ϑ1(w, z)ϑ1(q, r)e − − once with respect to w and q at w = 0, q = 0, and then we get −

i mn+m+n πQ(m,n) ϑ′ (0, z)ϑ′ (0, r) = ( 1) e− × 1 1 − z r − − m,n 2πi(m + nz) 2πi(m + nr) − , z r z r − − i.e. 3 3 3 i (m+1)(n+1) πQ(m,n) η (z)η ( r) = ( 1) Q(m, n)e− , −  z r  −  −  m,n   i.e.   1 3 i (m+1)(n+1) πQ(m,n) η(z)η( r) = ( 1) Q(m, n)e− − z r  −  − m,n  the branch of the cube root on the right-hand side being determined by the condition that it is real and positive when r = z, since η(z)η( z) > 0. We have then the following formula due to Kronecker, namely−

2πi(mu+nv) mn+m+n πQ(m,n)+2πi(mu+nv) 1 e ( 1) e− ′ m,n = log − 1 . (53) −2π Q(m, n) 3 m,n ( 1)(m+1)(n+1)Q(m, n)e πQ(m,n) m,n − − It is remarkable that, eventually, on the right-hand side of (53), z and r do not appear explicitly and only the quadratic form Q(m, n) appears on the right. The quadratic form Q(ξ, n) = dξ2 +bξη+cη2 introduced above is a complex quadratic form with discriminant

(r + z)2 + 4rz b2 4ac = − = 1. − (z r)2 − − Moreover,its real part is positive-deﬁnite for real ξ and η. For proving this, 45 i we have to show that for (ξ, n) , (0, 0), Re (ξ + zη) (ξ + rη) > 0, z r 1 − , i.e. Im z r (ξ + zn)(ξ + rη) > 0. If η = 0, this is trivial. If η 0, we may take η =−1− without loss of generality and then we have only to show that

z r 1 1 Im − > 0, i.e., Im > 0. (ξ + z)(ξ + r) ξ + r − ξ z − Kronecker’s Limit Formulas 40

But this is obvious, since z, r H and ξ being real, ξ + z, (ξ + r) and hence, 1/(ξ + z), 1/(ξ + r) are all− in H∈. − − Conversely, if Q(ξ,η) = aξ2 + bξη + cη2 is a complex quadratic form with real part positive-definite for real ξ and η and discriminant 1 and if Re a > 0, i − it can be written as (ξ +ηz)(ξ +ηr) for some z, r H. In fact, if Re a > 0, z r − ∈ Q(ξ, η) = a(ξ + ηz)(ξ−+ ηr), where z, r are the roots of the quadratic equation aλ2 bλ + c = 0. Since the real part of Q(ξ,η) is positive definite, z and r have− both got to be complex. Now the discriminant of Q(ξ,η) is 1 and hence a2(z r)2 = 1, i.e. a = i/(z r). We may take a = i/(z r) and then− z r H. − − i ± − − − ∈ Now Q(ξ, η) = (ξ + ηz)(ξ + ηr). We have only to show that z, r H. z r − ∈ − i Consider the expression Q(ξ, 1) = (ξ + z)(ξ + r). If both z and r are in H z r then as ξ varies from to + , the− argument of Q(ξ, 1) decreases by 2π and if both z and r are−∞ in H then∞ as ξ varies from to + , the argument of Q(ξ, 1)− increases− by 2π. But neither case is possible,−∞ since∞ Re Q(ξ, 1) > 0 and so Q(ξ, 1) lies always in the right half-plane for real ξ. Hence either z, r or z, r H. But, again, since z r H we see that z, r H. − − When∈ we subject the quadratic− ∈ form Q(ξ,η) to− the∈ unimodular transformation (ξ, η) (αξ + βη, γξ + δη) where α, β, γ, δ are integers such that αδ βγ = →1, then Q(ξ,η) goes over into the quadratic form Q (ξ, η) = − ± ∗ Q(αξ + βη,γξ + δη). The quadratic forms Q∗(ξ,η) and Q(ξ, n) are said to be equivalent. Q∗(ξ, η) again has discriminant 1 and its real part is positive- definite for real ξ and η. − Let u∗ = αu + γv and v∗ = βu + δv. Then we assert that the right-hand side of (53) is invariant if we replace Q(m, n), u and v respectively byQ∗(m, n), 46 u∗ and v∗. This is easy to prove, for the series on the right-hand side converge absolutely and when m, n run over all integers independently, then so do m∗ = (m +1)(n +1) αm + βn, n∗ = γm + δn. And all we need to verify is that ( 1) ∗ ∗ = ( 1)(m+1)(n+1). This is very simple to prove; for ( 1)(m+1)(n+1)−= +1 unless m − − and n are both even; and m and n are both even if and only if m∗ and n∗ are both even. The invariance of the left hand side of (53) under the transformation Q(m, n), u, v to Q∗(m, n), u∗, v∗ could be also directly proved by observing that for e2πi(mu+nv) Re s > 1, ′ is invariant under this transformation and hence, by (Q)(m, n)s analytic continuation, the invariance holds good even for s = 1. Kronecker’s Limit Formulas 41

5 The Epstein Zeta-function

We shall consider here a generalization of the function ζQ(s, u, v) introduced in 3. Let Q be the matrix of a positive-definite quadratic form Q(x1,..., xn) in § x1 the n variables x ,..., x (n 2). Further let x denote the n-rowed column . . 1 n ≥ . xn A homogeneous polynomial P(x) = P(x1,..., xn) of degree g in x1,..., xn is called a spherical function (Kugel-funktion) of order g with respect to Q(x1,..., xn), if it satisfies the differential equation

∂2P(x) q∗ = 0, i j ∂x ∂x 1 i, j n i j ≤ ≤ 1 where the matrix (qi∗ j) = Q− . Let, for a matrix A, A′ denote its transpose; we shall abbreviate A′QA as Q[A], for an n-rowed matrix A. By an isotropic vector of Q, we mean a complex column w satisfying Q[w] = 0. It is easily veriﬁed that if w is an isotropic g vector of Q, then the polynomial (x′Qw) is a spherical function of order g with respect to Q(x1,..., xn). Moreover, it is known thatif P(x) is a spherical 47 M P g function of order g with respect to Q(x1,..., xn), then (x) = (x′Qwm) , m=1 w1,..., wM being isotropic vectors of Q. Let u, v be two arbitrary n-rowed real columns and g, a non-negative integer. Further, let P(x) be a spherical function of order g with respect to Q(x1,..., xn). We deﬁne, for σ > n/2, the zeta-function P(m + v) ζ(s, u, v, Q, P) = e2πim′u (Q[m + v])s+g/2 m+v,0 where m runs over all n-rowed integral columns such that m + v , 0 (0 being the n-rowed zero-column). By the remark on P(x) above, ζ(s, u, v, Q, P)isa linear combination of the series of the form g ((m + v)′Qw) e2πim′u , (Q[m + v])s+g/2 m+v,0 w being an isotropic vector of Q. These series have been investigated in de- tail by Epstein; in special cases as when g = 0 and Q is diagonal, Lerch has considered them independently and has derived for them a functional equation which he refers to as the “generalized Malmsten-Lipschitz relation.” Kronecker’s Limit Formulas 42

The series introduced above converge absolutely for σ > n/2, uniformly in every half-plane σ n/2+ǫ(ǫ > 0), as can be seen from the fact that they have a majorant of the form≥

n g mi + vi i=1 | | c n σ+g/2 m+v,0 2 mi + vi i=1 | | for a constant c depending only on Q, w and g. Hence they represent analytic functions of s for σ > n/2. Thus ζ(s, u, v, Q, P) is an analytic function of s for σ > n/2. We shall study the analytic continuation and the functional equation of ζ(s, u, v, Q, P). The proof will be based on Riemann’s method of obtaining the 48 functional equation of the ζ-function using the theta-transformation formula. First we obtain the following generalization of (52). Proposition 7. If Q is the matrix of a positive-deﬁnite quadratic form in m variables with real coeﬃcients and v an n-rowed complex column, then

πq[m+v] 1 πQ 1 m +2πm v e− = Q − 2 e− − | | ′ , (54) | | m m

1 where, on both sides m runs over all n-rowed integral columns and Q 2 is the positive square root of Q , the determinant of Q. | | | | Proof. We shall assume formula (54) proved for Q and v of at most n 1 rows and uphold it for n. For n = 1, the formula has been proved already. − Now, it is well known that 1 P q P 0 E P− q A = = 1 (55) q′ r 0′ r P− [q]0′ 1 − where P is (n 1)-rowed and symmetric and E, the (n 1)-rowed identity − − = 1 = k = u matrix. Setting λ r P− [q] and writing m 1 and v w with k and u of n 1 rows, we have, in− view of (55), − 2 1 πQ[m+v] πλ(l+w) πP[k+u+P− q(l+w)] e− = E− e− , m 1 k m, k running over all n-rowed and (n 1)-rowed integral columns respectively and l over all integers. By induction hypothesis,−

1 1 1 1 πP[k+u+P− q(l+w)] πP− [k]+2πik′(u+P− q(l+w)) e− = P − 2 e− | | k k Kronecker’s Limit Formulas 43

Thus

1 1 2 1 (l+w) πQ[m+v] πP πλ(l+w) +2πik′ P− q e− = P − 2 e− − [k] + 2πik′u e− | | m k l

Now, by (52), 49

∞ 2 1 πλ(l+w) +2πik′ P q(l+w) e− − l= −∞ 2 1 w ∞ 2 1 πλw +2πik′ P q πλl 2πil(k′ P q+iλw) = e− − e− − − l= −∞ 1 2 1 w ∞ 1 1 2 πλw +2πik′ P− q πλ− (l k′ P− q iλω) = λ− 2 e− e− − − l= −∞ 1 ∞ 1 1 2 πλ− (l k′ P− q) +2πilw = λ− 2 e− − l= −∞ 1 1 1 1 2 But Q = P λ and Q [m] = P [k] + λ ( k′P q + 1) and hence | | | | − − − − −

1 1 πQ[m+v] πQ m +2πi(k′u+lw) e− = Q − 2 e− − | | | | m k,l and (54) is proved. Formula (54) can also be upheld by using the Poisson summation formula in n variables. Further, it can be shown to be valid even for complex symmetric Q whose real part is positive (i.e. the matrix of a positive-deﬁnite quadratic 1 form). For, Q− again has positive real part and the absolute convergence of the series on both sides is ensured. Moreover, considered as functions of the n(n + 1)/2 independent elements of Q, they represent analytic functions and since they are equal for all real positive Q, we see, by analytic continuation, that they are equal for all complex symmetric Q with positive real part. Let u be another arbitrary complex n-rowed column. Replacing v by v 1 − iQ− u in (54), we obtain

πQ[m+v iQ 1u] 1 πQ 1[m]+2πim (v iQ 1u) e− − − = Q − 2 e− − ′ − − , | | m m i.e. πQ[m+v]+2πim u 1 πQ 1[m u]+2πi(m u) v e− ′ = Q − 2 e− − − − ′ . (56) | | m m Kronecker’s Limit Formulas 44

Let w be an isotropic vector of Q and λ, an arbitrary complex number. Replac- ingv by v + λw in (56), we have 50

πQ[m+v] 2πλ(m+v) Qw+2πim u e− − ′ ′ m 1 πQ 1[m u]+2πi(m u) v+2πiλ(m u) w = Q − 2 e− − − − ′ − ′ . | | m

Both sides represent analytic functions of λ and diﬀerentiating them both, g times with respect to λ at λ = 0, we get

πQ[m+v]+2πim u g e− ′ ((m + v)′Qw) m

2πiu′v e− 1 πQ 1[m u]+2πim v g = Q − 2 e− − − ′ ((m u)′w) . (57) ig | | − m

Associated with a spherical function P(x) with respect to Q[x], let us de- 1 ﬁne P∗(x) = P(Q− x); P∗(x) is a spherical function of order g with respect m 1 P P P g to Q− [x]. Moreover ∗∗(x) = (x). Further, if (x) = (x′Qwi) , then i=1 m P g ∗(x) = (x′wi) . Thus, if we set i=1 πQ[m+v]+2πim u f (Q, u, v, P) = e− ′ P(m + v), m where m runs over all n-rowed integral columns, then we see at once from (57) that f (Q, u, v, P) satisﬁes the functional equation

g 2πiu v 1 1 i e ′ f (Q, u, v, P) = Q − 2 f (Q− , v, u, P∗). | | − If now we replace Q by xQ for x > 0, then P(x) goes into xgP(x) and P∗(x) remains unchanged and we have

g 2πiu v n/2 g 1 1 1 i e ′ f (xQ, u, v, P) = x− − Q − 2 f (x− Q− , v, u, P∗). (58) | | − Thus we have Proposition 8. Let Q be an m-rowed complex symmetric matrix with positive real part, P(x) a spherical function of order g with respect to Q and P (x) ∗ − Kronecker’s Limit Formulas 45

1 P(Q− x). Let u and v be two arbitrary m-rowed complex columns and x > 0. Then

2πiu v g πxQ[m+v]+2πim u e ′ i e− ′ P(m + v) m n/2 g 1 πx 1 Q 1[m u]+πim v = x− − Q − 2 e− − − − ′ P∗(m u). | | − m 51 To study the analytic continuation of ζ(s, u, v, Q, P), we obtain an integral representation of the same, by using the well-known formula due to Euler, namely, for t > 0 and Re s > 0,

s s ∞ s πtx dx π− Γ(s)t− = x e− . 0 x For σ > n/2, we then have

(s+g/2) g π− Γ s + ζ(s, u, v, Q, P) 2 2πim u ∞ s+g/2 πxQ[m+v] dx = e ′ P(m + v) x e− x m+v,0 0

∞ s+g/2 πxQ[m+v]+2πim u dx = x  e− ′ P(m + v) , (59) 0   x m+v,0    in view of the absolute convergence of the series, uniform for σ n/2 + ǫ(ǫ > 0). ≥ Let now, for an n-rowed real column x,

0, if x is not integral, ρ(x, g) = 1, if x is integral and g = 0,  0, if x is integral and g > 0.   Then 

πxQ[m+v]+2πim u 2πiu v e− ′ P(m + v) = f (xQ, u, v, P) e− ′ ρ(v, g). − m+v,0 From (59), we see, as a consequence, that

(s+g/2) g π− Γ s + ζ(s, u, v, Q, P) 2 Kronecker’s Limit Formulas 46

∞ s+g/2 πxQ[m+v]+2πim u dx = x  e− ′ P(m + v) + 1   x m+v,0  1   1 s+g/2 dx 2πiuv s+g/2 dx + x f (xQ, u, v, P) ρ(v, g)e− ′ x . 0 x − 0 x In view of (58), 52 1 dx xs+g/2 f (xQ, u, v, P) 0 x 1 1 Q 2 dx = − xs g/2 n/2 f (x 1Q 1, v, u, P ) . g| 2|πiu v − − − − ∗ i e ′ 0 − x Again, since

1 1 f (x− Q− , v, u, P∗) − πx 1 Q 1[m u]+2πim v 2πiu v = e− − − − ′ P∗(m u) + ρ(u, g)e ′ , − mu,0 − we have 1 s g/2 n/2 1 1 dx x − − f (x− Q− , v, u, P∗) 0 − x 1 s g/2 n/2 πx 1 Q 1[m u]+2πim v dx = x − − e− − − − ′ P∗(m u) +  −  x 0 mu,0   −   1  2πiu v s (g/2) (n/2) dx  + ρ(u, g)e ′ x − − 0 x

∞ (n/2) s+(g/2) πxQ 1[m u]+2πim v dx = x − e− − − ′ P∗(m u) +  −  x 1 mu,0   −  2πiu v   ρ(u, g)e ′   + g n . s − 2 − 2 Thus, for σ > n/2,

(s+g/2) g π− Γ s + ζ(s, u, v, Q, P) 2 1 2πiu v Q 2 ρ(u, g) ρ(v, g)e− ′ = | |− + g n g ig s − s + − 2 − 2 2 Kronecker’s Limit Formulas 47

∞ s+g/2 πxQ[m+v]+2πim u dx + x e− ′ P(m + v) +    x 1 m+v,0         1   Q − 2 ∞ 1  dx + (n/2) s+(g/2) πxQ− [m u]+2πim′v P . | 2|πiu v x −  e− − ∗(m u)  ige ′ × − x 1 mu,0    −     (60)    53 One veriﬁes easily that the functions within the square brackets on the πs+g/2 right-hand side of (60) are entire functions of s. Since is also an en- Γ(s + g/2) tire function of s, formula (60) gives the analytic continuation of ζ(s, u, v, Q, P) into the whole s-plane. The only possible singularities arise from those of

1 s+g/2 2πiu′v π Q − 2 ρ(u, g) ρ(v, g)e−  | |  g  g g n g  Γ s + i s − s +   − 2 − 2 2  2     Now   πs+g/2 πs+g/2 = g g Γ s + g s + Γ s + + 1 2 2 2 is an entire function of s and so s = g/2 cannot be a singularity of ζ(s, u, v, Q, P). Moreover, if g > 0 or if u is not− integral, then ρ(u, g) = 0 and hence in these cases, s = (g + n)/2 can not be a pole of ζ(s, u, v, Q, P). If g = 0, and u is integral, ζ(s, u, v, Q, P) has a simple pole at s = n/2 with residue n/2 1 π Q − 2 /Γ(n/2). Moreover, from (60) it is easy to verify that ζ(s, u, v, Q, P) satisﬁes| | the functional equation

sΓ g π− s + ζ(s, u, v, Q, P) 2 2πiu′v e− 1 (n/2 s) n g n 1 = Q − 2 π− − Γ s + ζ s, v, u, Q− , P∗ (61) ig | | 2 − 2 2 − − Wehave then ﬁnally. 54 Theorem 3. The function ζ(s, u, v, Q, P) has an analytic continuation into the whole s-plane, which is an entire function of s if either g > 0 or if g = 0 and u is not integral. If g = 0 and u is integral, then ζ(s, u, v, Q, P) is meromorphic in the entire s-plane with the only singularity at s = n/2 where it has a simple n/2 1 pole with the residue π /( Q 2 Γ(n/2)). In all cases, ζ(s, u, v, Q, P) satisﬁes the functional equation (61)|. | Kronecker’s Limit Formulas 48

1 2 Let now n = 2 and Q(x1, x2) = y− x1 + x2z , in our earlier notation. x1 m1 u2 | v1 | z z Let x = x , m = m , u = u and v = v . Both −1 and −1 are isotropic 2 2 − 1 2 g vectors of Q[x] but we shall take the spherical function P (x) = ( i(x1 + x2z)) , corresponding to the former. Then by the above, the function −

s+g/2 g y 2πi(m u m u ) (m1 + v1 + z(m2 + v2)) ζ(s, u, v, Q, P) = e 1 2− 2 1 ig m + v + z(m + v ) 2s+g m+v,0 | 1 1 2 2 | satisﬁes the functional equation

s g π− Γ s + ζ(s, u, v, Q, P) 2 2πiu′v e− (1 s) g 1 = π− − Γ 1 s + ζ(1 s, v, u, Q− , P∗). ig − 2 − − 1 1 2 Now Q− [x] has the simple form y− x1z x2 and moreover, P∗(x) = ( x1z+ g | − | − x2) . Thus for σ > 1,

1 ζ(s, v, u, Q− , P∗) − g ( (m1 u2)z + m2 + u1) = ys+g/2 e2πi(m1v1+m2v2) − − z(m u ) + m + u 2s+g mu,0 1 2 2 1 − | − − | g ((m1 + u1) + z(m2 + u2)) = s+g/2 2πi(m1v2 m2v1) . y e − 2s+g (m1 + u1) + z(m2 + u2) m1 + u1 ,0 (m2) (u2) | |

u1 v1 Let us now deﬁne for u∗ = , v∗ = , the function u2 v2 g (m1 + v1 + z(m2 + v2)) ζ(s, u , v , z, g) = ys e2πi(m1u2 m2u1) , ∗ ∗ − + + + 2s+g + , m1 v1 z(m2 v2) mv∗ 0 | |

g g/2 forσ > 1. It is clear that ζ(s, u∗, v∗, z, g) = i y− ζ(s, u, v, Q, P) and from 55 g/2 1 (??) we see that ζ(s, v∗, u∗, z, g) = y− ζ(s, v, u, Q− , P∗). If now we deﬁned s − ϕ(s, u∗, v∗, z, g) = π− Γ(s + g/2)ζ(s, u∗, v∗, z, g), we deduce from above that ϕ(s, u∗, v∗, z, g) satisﬁes the nice functional equation

2πi(u v u v ) ϕ(s, u∗, v∗, z, g) = e 1 2− 2 1 ϕ(1 s, v∗, u∗, z, g). −

In the case g > 0, ζ(s, u∗, v∗, z, g) is an entire function of s and one can ask for analogues of Kronecker’s second limit formula even here. For even g, one gets by applying the Poisson summation formula, a limit formula which is connected with elliptic functions. For odd g, the limit formulas which one gets Kronecker’s Limit Formulas 49 are more complicated and involve Bessel functions. If we take the particular case g = 1, there occurs in the work of Hecke, a limit formula (as s tends to 1/2) which has an interesting connection with the theory of complex multiplication. For n > 2, Epstein has obtained for u = 0, v = 0 and g = 0, an analogue of the ﬁrst limit formula of Kronecker. This formula involves more complicated functions that the Dedekind η-function. Perhaps, in general for n > 2, one cannot expect to get a limit formula which would involve analytic functions of several complex variables. Bibliography

[1] S. Bochner: Vorlesungen ¨uber Fouriersche Integrale, Chelsea, New York, (1948). [2] R. Dedekind: Erl¨auterungen zu den Fragmenten XXVIII, Collected Works of Bernhard Riemann, Dover, New York, (1953), 466-478. [3] P. Epstein: Zur Theorie allgemeiner Zeta funktionen, I, II, Math, Ann. 56 (1903), 615-644, ibid, 63 (1907), 205-216. [4] E. Hecke: Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenkorper, Werke. Gottingen (1959), 505-524. [5] F. Klein: Uber die elliptischen Normalkurven der n-ten Ordnung, Gesamm. Math. Abhandlungen. Bd. III. Berlin. (1923). 198-254.

[6] Kronecker:Zur Theorie der Clliptischen Funktionen, Werke. Vol. 4. 347- 56 495, vol. 5, 1-132, Leipzig, (1929). [7] M. Lerch: Zakladove´ Theorie Malmstenovskych rad, Rozpravy ˇceske akademie Ciˇsare Frantiska Josefa, Bd. 1, No. 27, (1892), 525-592. [8] C. Meyer: Die Berechnung der Klassenzahl abelscher Körper über quadratischen Zahlkörpern, Akademie Verlag, Berlin, 1957. [9] H. Rademacher: Zur Theorie der Modulfunktionen, J. für die reine u. angew. Math., vol. 167 (1931), 312-336. Collected papers. [10] B. Schoenesberg: Das Verhalten von mehrfachen Thetareihen bei Modul- substitutionen, Math. Ann. 116 (1939), 511-523. 1 τ [11] C. L. Siegel: A simple proof of η = η(τ) , Mathematika 1 −η i (1954), 4. Ges Abhand.

50 Bibliography 51

[12] E. T. Whittaker and G. N. Watson: Course of Modern Analysis, Cam- bridge, (1946). Chapter 2

Applications of Kronecker’s Limit Formulas to Algebraic Number Theory

1 Kronecker’s solution of Pell’s equation 57 Let K be an algebraic number field of degree n over Q, the field of rational numbers. Let, for any ideal a in K, N(a) denote its norm. Further let s = σ + it be a complex variable. Then for σ > 1, we define after Dedekind, the zeta function s ζK(s) = (N(a))− , a where the summation is over all non-zero integral ideals of K. More generally, with a character χ of the ideal class group of K, we associate the L-series

s s 1 LK(s, χ) = χ(a)(N(a))− = (1 χ(p)(N(p))− )− , a p − for σ > 1. The product on the right runs over all prime ideals p of K. We have h such series associated with all the h characters of the ideal class group of K. It has been proved by Hecke that these functions LK(s, χ) can be continued analytically into the whole plane and they satisfy a functional equation for s 1 s. → −

52 Applications to Algebraic Number Theory 53

Now, χ being an ideal class character, we may write

s LK(s, χ) = χ(A) (N(a))− , A a A ∈ s A running over all the ideal classes of K. If we define ζ(s, A) = (N(a))− , a A then it can be shown that lim(s 1)ζ(s, A) = κ, a positive constant∈ depending 58 s 1 − only upon the field K and→not upon the ideal class A. (For example, in the case of an imaginary quadratic field of discriminant d, κ = 2/w √ d, w being the number of roots of unity in K and κ = 2 log ǫ/ √d in the case− of a real quadratic field of discriminant d, where ǫ is the fundamental unit and ǫ > 1). Since ζK(s) = ζ(s, A), we have A

lim(s 1)ζK(s) = lim(s 1)ζ(s, A) = κ h s 1 − s 1 − → A → where h denotes the class number of K. From now on, we shall be concerned only with quadratic ﬁelds of discriminant d. We have then, on direct computation,

ζK(s) = ζ(s)Ld(s), (63) where ζ(s) is the Riemann zeta-function and Ld(s) is deﬁned as follows:

∞ d s Ld(s) = n− , (σ > 1) n n=1

d being the Legendre-Jacobi-Kronecker symbol. n Then

lim(s 1)ζK(s) = lim(s 1)ζ(s)Ld(s) = lim Ld(s) = Ld(1), s 1 − s 1 − s 1 → → → since Ld(s) converges in the half-plane σ > 0 and lim(s 1)ζ(s) = 1. s 1 − We obtain therefore, →

∞ d 1 Ld(1) = = κ h. (64) n n n=1 Applications to Algebraic Number Theory 54

For the determination of the left-hand side, we consider more generally, for any character χ(, 1) of the ideal class group of K, χ(a) lim LK(s, χ) = LK(1, χ) = . s 1 N(a) → a

Now, 59 κ ζ(s, A) = + ρ(A) + , s 1 − and so

LK(s, χ) = χ(A)ζ(s, A) A κ = χ(A) + ρ(A) + s 1 A − = χ(A)ρ(A) + terms involving higher powers of (s 1), − A since, χ being , 1, χ(A) = 0. On taking the limit as s 1, we have A →

LK(1, χ) = χ(A)ρ(A). (65) A

Therefore, the problem of determination of LK(1, χ) has been reduced to that of ρ(A). We now study LK(s, χ) for a special class of characters, the so-called genus characters (due to Gauss) to be deﬁned below. We call a discriminant, a prime discriminant, if it is divisible by only one prime. In that case, d = p if d is odd, or if d is even, d = 4 or 8. ± − ± Proposition 9. Every discriminant d can be written uniquely as a product of prime discriminants.

Proof. It is known that if d is odd, then d = p1 ... pk where p1,..., pk are mutually distinct odd primes. Let P = p according± as p 1( mod 4); 1 ± 1 1 ≡ ± then P1 is a prime discriminant and d/P1 is again an odd discriminant. Using induction on the number k of prime factors of d (odd!) it can be shown that d/P1 = P2,..., Pk where Pi are odd prime discriminants. If d is even, then we know that

(a) d = 8p ,..., p , if d/4 2( mod 4), ± 2 k ≡ Applications to Algebraic Number Theory 55

(b) d = 4p ,..., p , if d/4 3( mod 4), ± 2 k ≡ p2,..., pk being odd primes. Now we may write d = P1d1 where P1 is chosen to be 4, +8 or 8 such that d is an odd discriminant. From the above it is 60 − − 1 clear that d = P1P2 ... Pk where P1,..., Pk are all prime discriminants. This decomposition is seen to be unique. We may now introduce the genus characters. Let d1 be the product of any of the factors P1,..., Pk of d. Then d1 is again a discriminant and d d. Let d = d d ; d is also a discriminant. Indentifying 1| 1 2 2 the decompositions d = d1d2 and d = d2d1, we note that the number of such k 1 decompositions (including the trivial one, d = 1 d) is 2 − where k is the number of diﬀerent prime factors of d. For any such decomposition d = d1d2 of d and for any prime ideal p not dividing d, we deﬁne d1 χ(p) = χd (p) = , 1 N(p) d where 1 is the Legendre-Jacobi-Kronecker symbol. We shall show that N(p)

d2 d1 χd (p) = = . 2 N(p) N(p)

In face, since p d, p belongs to one of the following types. (a) pp′ = (p), × 2 N(p) = p or (b) p = (p), N(p) = p . If pp′ = (p), N(p) = p then d d d = 1 = 1 2 p N(p) N(p)

d d which means that either both are +1 or both are 1. i.e. 1 = 2 . If − N(p) N(P) 2 d p = (p), N(p) = p then p = 1; but − d d d 1 2 = = 1, N(p) N(p) p2

d d so that we have again 1 = 2 . N(p) N(p) d d When p d, one of the symbols 1 , 2 is zero, and the other non- | N(p) N(p) zero, we take χ(p) to be the non-zero value. In any case, χ(p) = 1. This ± deﬁnition can then be extended to all ideals a of K as follows: If a = pkql,..., 61 Applications to Algebraic Number Theory 56 we deﬁne χ(a) = (χ(p))k(χ(q))l,..., so that

χ(ab) = χ(a)χ(b) for any two ideals a, b of K. By a genus character, we mean a character χ defined as above, correspond- k 1 ing to any one of the 2 − different decompositions of d as product of two mutually coprime discriminants. We shall see later that the genus characters form an abelian group of order k 1 2 − . Now we shall obtain an interesting consequence of our definition of the character χ, with regard to the associated L-series. Consider the L-series, for s = σ + it, σ > 1

s s 1 LK(s, χ) = χ(a)(N(a))− = (1 χ(p)(N(p))− )− a p − s 1 = (1 χ(p)(N(p))− )− . − p p(p) | The prime ideals p of K are distributed as follows: d (a) p = (p), = 1, N(p) = p2, p − d (b) pp′ = (p), = +1, N(p) = p, p d (c) p2 = (p), = 0, N(p) = p. p d d d d d In case (a), = 1 = 1 2 implies that one of 1 , 2 is +1 and the p − p p p p other 1. So − s 1 2s 1 (1 χ(p)(N(p))− )− = (1 p− )− − − p(p) | s 1 s 1 = (1 p− )− (1 + p− )− − 1 1 d1 s − d2 s − = 1 p− 1 p− − p − p

In case (b), 62 Applications to Algebraic Number Theory 57

1 1 s 1 d1 s − d1 s − (1 χ(p)(N(p))− )− = 1 p− 1 p− − − p − p p(p) | 1 1 d1 s − d2 s − = 1 p− 1 p− , − p − p d d d d d since 1 = = 1 2 implies that 1 = 2 . In case (c) p d implies p p p p p | that p d1 or p d2. We shall assume without loss of generality that p d2. Then d | | | 2 = 0 and p

1 1 s 1 d1 s − d2 s − (1 χ(p)(N(p))− )− = 1 p− 1 p− . − − p × − p p(p) | From all cases, we obtain

s 1 L (s, χ) = (1 χ(p)(N(p))− )− K − p p(p) | 1 1 d1 s − d2 s − = 1 p− 1 p− − p − p p

= Ld1 (s)Ld2 (s). We have therefore, Theorem 4 (Kronecker). For a genus character χ of a K corresponding to the decomposition d = d1d2, we have,

LK(s, χ) = Ld1 (s)Ld2 (s). (66) Let us consider the trivial decomposition d = 1 d or d = 1 and d = d. 1 2 Then Ld1 (s) = ζ(s) and Ld2 (s) = Ld(s). In other words, (66) reduces to (63). If d , 1, ζ (s) = ζ(s)L (s), from (63). But L (s) can then be continued 1 Q( √d1) d1 d1 analytically into the whole plane and it is an entire function. Two ideals a and b are said to be equivalent in the “narrow sense” if a = b(γ) with N(γ) > 0. (For α K, N(α) denotes its norm over Q.) ∈ This definition coincides with the usual definition of equivalence in the case 63 of an imaginary quadratic field since the norm of every element is positive. In the case of a real quadratic field, if there exists a unit ǫ in the field with N(ǫ) = 1, then both the equivalence concepts are the same, as can easily be seen. − Otherwise, if h0 denotes the number of classes under the narrow equivalence and h, the number of classes under the usual equivalence, we have h0 = 2h. Applications to Algebraic Number Theory 58

Proposition 10. For a genus character χ, χ(a) = χ(b), if a and b are equivalent in the narrow sense. Proof. We need only to prove that χ((α)) = 1 for α integral and N(α) > 0.

d1 (a) Let (α, d1) = (1). We have to show that = 1. Since N(α) > 0, N((α)) d this is the same as proving that 1 = 1. N(α)

d1 (1) Suppose d1 is odd. Then = 1. If d is even, d = 4m (say). Then 4 α = x + y √m with x, y rational integers. If not, 2α = x′ + y′ √m with x′ and y′ rational integers. In any case, since (d1/4) = 1, it is d d suﬃcient to prove that 1 = 1. i.e. 1 = 1 for two N(2α) a2 mb2 2 2 −2 2 rational integers a, b with a mb > 0 and (a mb , d1) = 1, in both cases. Now d d implies− that d m so that from− the periodicity 1| 1| d1 of the Jacobi symbol, this is the same as = 1 for (a, d1) = 1, a2 which is obvious.

(2) Suppose d1 is even. Then d1 = (even discriminant) (odd discriminant) and we need consider only the even part, since× we have already disposed of the odd part in 1). We have then three possibilities d = 4, +8 and 8. 1 − − (i) d1 = 4. Then d = ( 4)( m) so that m is again a discriminant and− 1( mod 4).− − − Consider≡ now d 4 1 = − ; N(α) x2 my2 − (x2 my2) 1( mod 4) since both x, y cannot be even or odd, − ≡ for (α, d1) = (1). From the periodicity of the Jacobi symbol, 64 follows then that 4 4 − = − = 1. x2 my2 1 − (ii) d1 = +8. Here d = 8 m/2; m/2 is then an odd discriminant and 1( mod 4) or equivalently,× m 2( mod 8). Now,≡ ≡ d 8 1 = ; N(α) x2 my2 − Applications to Algebraic Number Theory 59

(x2 my2) 1( mod 8) so that − ≡ ± 8 8 8 1 = = 1 or = = = 1. x2 my2 1 7 7 − If d = 8, m 2( mod 8) and x2 my2 1, 3( mod 8) 1 − ≡ − − ≡ d1 8 1 and = − = = 1. So, if (α, d1) = (1), χ((α)) = N(α) 3 3 1. If(α, d1) , (1) and (α, d2) = (1), we can apply the same arguments for d2 instead of d1 and prove χ((α)) = 1.

(b) If (α, d1) , (1) and (α, d2) , (1), we decompose (α) as follows: (α) = p1p2 ... plq where pi d and (q, d) = (1). We choose in the narrow class of 1 | p1− , an integral ideal q1 with (d, q1) = (1). Then for the element α1 with (α ) = p q , N(α ) > 0, χ((α )) = 1, since (α , d )or(α , d ) = (1). 1 1 1 1 1 1 1 1 2 Similarly for p2, construct q2 with (d, q2) = 1 and for α2 with (α2) = p2q2 and N(α2) > 0, χ((α2)) = 1 and so on. Finally, we obtain

2 2 2 (αα1α2 ...) = p1p2 ... pl p where (b, d) = (1). But p d imply that p2 = (p ) with p d. Therefore (αα α ...) = i| i i i| 1 2 (p1 p2 ...)b so that b is a principal ideal = (ρ) (say). Now N(αα1α2 ...) = 2 2 2 p1 p2 ... p1N(ρ) and χ((αα1 ...αl)) = χ((ρ)) = 1 since N(ρ) > 0 and (ρ, d) = (1). But χ((α1)) = 1,...χ((αl)) = 1 so that χ((α)) = 1. The proof is now 65 complete.

We have just proved that χ(a) depends only on the narrow class of a, say A χ is therefore a character of the ideal class group in the “narrow sense”. Then, s LK(s, χ) = χ(A) (N(a))− = χ(A)ζ(s, A) (67) A a A A ∈ where A runs over all the ideal classes in the ‘narrow sense’. Using a method of Hecke, we shall now give an alternative proof of the fact that χ((α)) = 1 for principal ideals (α) with N(α) > 0 and (α, d) , (1). Denote by χ0 that character of the ideal class group in the narrow sense, such that χ0(a) = χ(a) for all ideals a in whose prime factor decomposition, only prime ideals p which do not divide d, occur. We need only to prove that χ(a) = χ0(a) for all ideals a. Applications to Algebraic Number Theory 60

s s Let fχ0 (s) = χ0(a)(N(a))− and fχ(s) = χ(a)(N(a))− . Then a a s fχ (s) (1 χ(p)(N(p)) ) 0 = − − , f (s) (1 χ (p)(N(p)) s) χ p d 0 − | − since the remaining factors cancel out. We shall show that the product on the right is = 1, or fχ0 (s) = fχ(s).

We know that fχ(s) = Ld1 (s) = Ld2 (s) from (66) and fχ0 (s) = χ0(A). A ζ(s, A) from (67). Both have functional equations of the same type, so that if we denote by R(s), the quotient, the functional equation is simply R(s) = R(1 s). − We shall now arrive at a contradiction, by supposing that χ , χ0. For, then, there exists a prime ideal p with p d and χ(p) = 1, χ0(p) = 1. log( 1) + 2kπi | ± ∓ Choose s = ∓ (k, such that s , 0). This means that p s = − log p − 1. Consider the product ∓ (1 χ(p)(N(p)) s) R(s) = − − . (1 χ (p)(N(p)) s) p d 0 − | − The above value of s is a zero of the denominator and it cannot be cancelled by any factor in the numerator, for that would mean

log( 1) + 2kπi log( 1) + 2lπi ∓ = ± , log p log q or in other words, λ = log p/ log q is rational; i.e. qλ = p holds for p, q 66 primes and λ rational. This is not possible since p , q. By the same argument, the zeros and poles of R(s) cannot cancel with those on the other side of the equation R(s) = R(1 s). But this is a contradiction. In other words, χ(a) = − χ0(a) for all ideals a of K. Tow ideals a and b are in the same genus, if χ(a) = χ(b) for all genus characters χ defined as above, associated with the decompositions of d. The ideals a for which χ(a) = 1 for all genus characters χ, constitute the principal genus. Then the narrow classes A for which χ(A) = 1 are the classes in the principal genus. If H denotes the group of narrow classes and G, the subgroup of classes lying in the principal genus, the quotient group G = H/G k 1 gives the group of different genera. We claim that the order of G is 2 − , k being the number of different prime factors of d. For, suppose we have proved k 1 that the genus characters form a group of order 2 − . From the definition of a genus and from the theory of character groups of abelian groups, we know that Applications to Algebraic Number Theory 61 this group is the character group of G. By means of the isomorphism between k 1 G and its character group, it would then follow that the order of G is also 2 − . If now remains for us to prove

k 1 Proposition 11. The genus characters form an abelian group of order 2 − , where k is the number of distinct prime factors of d. Proof. Now, χ(a) = 1 implies that χ2 = 1 for all genus characters χ. So, 1 ± the character χ− exists and = χ. We shall prove that the genus characters and closed under multiplication and that they are all diﬀerent for diﬀerent decompositions of d.

Let d = d1d2 = d1∗d2∗. Set d1 = qu1 and d1∗ = qu1∗ so that q = (d1, d1∗) and (u1, u1∗) = 1. If d = u u then d d = q2d . Also for the characters χ and χ associ- 3 1 1∗ 1 1∗ 3 d1 d1∗ ated with the two decompositions d d and d d of d, we have χ χ = χ and 1 2 1∗ 2∗ d1 d1∗ d3 d3 d, since both d1 d and u1∗ d and they are coprime. Hence χd3 is again a genus character.| Now, if |d , d ,| we have to show that χ , χ or χ χ , χ (the 1 1∗ d1 d1∗ d1 d1∗ 1 identity character); or in other words, we need only to prove that for a proper 67 decomposition d = d3d3∗, the associated character χ is not equal to χ1. We have, from (66), for σ > 1,

L (s, χ) = L (s)L (s) = χ(A)ζ(s, A). K d3 d3∗ A

Comparing the residue at s = 1 in the two forms for LK(s, χ), we have χ(A) = A 0, which implies that χ , χ1. Thus the genus characters are different for different decompositions of d k 1 and they form a group. Since there are 2 − different decompositions of d, this k 1 group of genus characters is of order 2 − . A genus character is a narrow class character of order 2. Conversely, we can show that every narrow class character of order 2 is a genus character. For this, we need the notion of an ambiguous (narrow) ideal class. A narrow ideal class A satisfying A = A′ (the conjugate class of A in K) or equivalently A2 = E (the principal narrow class) is called an ambiguous ideal class. The ambiguous classes clearly form a group. Proposition 12. The group of ambiguous ideal classes in Q( √d) is of order k 1 2 − , k being the number of distinct prime factors of d.

Proof. We shall pick out one ambiguous ideal b (i.e. such that b = b′, the conjugate ideal) from each ambiguous ideal class A and show that there are k 1 2 − such inequivalent ideals. Applications to Algebraic Number Theory 62

Let a be any ideal in the ambiguous ideal class A. Since a a′, we may 1 ∼ take aa′− = (λ) with N(λ) = 1 and λ totally positive (in symbols; λ > 0). (For d < 0, the condition “totally positive” implies no restriction). Let ρ = 1 + λ. Then ρ , 0 since λ , 1. Further λ = (1 + λ)/(1 + λ′) = ρ/ρ′. If we deﬁne b = a/(ρ), the b = b , for− b/b = a/a (ρ )/(ρ) = (1). Further b A, since ρ > 0 ′ ′ ′ ′ ∈ (i.e. ρ > 0, ρ′ > 0). We may take b to be integral without loss of generality. We call b primitive, if the greatest rational integer r dividing b is 1. For any ambiguous integral ideal b we have b = rb1 with r, the greatest integer dividing b and b primitive. If b A, b A. 1 ∈ 1 ∈ We shall now prove that any primitive integral ambiguous ideal b is of the 68 form pλ1 ,..., pλk with λ = 0 or 1 and p ϑ, (p , p ), where ϑ is the diﬀerent of 1 k i i| i j √ 1 s 1 a K = Q( d) over Q. Let b = q1 ,..., qs . Now b = b′ implies that q1 ,..., qs = 1 s 1 t q′1 ,..., q′s . We have then q1 = q′t or 1 = t and q1 = qt′. We assert then 1 t 1 t = 1, for if not, q1 = qt′ implies that q1′ = qt and the factor q1 qt = (q1q1′ ) is a rational ideal , (1). This is a contradiction to the hypothesis that b is primitive.

Hence t = 1 and q1 = q1′ . The same argument applies to all prime ideals qi. 2 Since qi = (pi) with prime numbers pi, the exponents i are either 0 or 1. for otherwise they bring in rational ideals which are excluded by the primitive λ1 λk nature of b. Therefore, we have b = p1 ,..., pk with pi ϑ and λi = 0 or 1. k 1 | Now, we shall show that there are exactly 2 − inequivalent (in the narrow sense) such ideals. For the same, it is enough to prove that there is only one l1 lk non-trivial relation of the form p1 ,..., pk (1) in the narrow sense, for, then, k ∼ k 1 it would imply that among the 2 such ideals, there are 2 − inequivelent ones, which is what we require. We shall ﬁrst prove uniqueness, namely that if there is one non-trivial relation, then it is uniquely determined. Later, we show the existence of a non- trivial relation.

l1 lk , (a) Let p1 ,..., pk = (ρ) with ρ > 0 and li > 0or(ρ) (1). Then the set i (l1,..., lk) is uniquely determined.

For, (ρ) = (ρ′) implies that ρ = η ρ′ with a totally positive unit η. The group of totally positive units in K being cyclic, denote by ǫ, the generator of this group. (Note that, for d < 0, the condition “totally positive” imposes no restrictions). Since (ρ) = (ρǫn), when ρ is replaces by ρǫn, η goes over to ηǫ2n. Choosing n suitably, we may assume without loss of generality η = 1 or ǫ. We shall see that (ρ) , (1) implies that η , 1. For, if η = 1, then ρ = ρ′ = a natural number. Now (N(ρ)) = 2 2l1 2lk l1 lk , (ρ ) = p1 ,..., pk = (p1 ,..., pk ), if N(pi) = pi, pi p j and ρ being a Applications to Algebraic Number Theory 63

natural number, this is possible only if all li = 0, in which case (ρ) = (1). But that is a contradiction to the hypothesis (ρ) , (1). We are therefore left with the case η = ǫ. Then set = 1 η = 1 ǫ , 0. − − We have = η′. Denoting by δ = √d, (ρ/δ)′ = ρ/δ = r = a rational number−, 0. Hence (ρ) = (δ)(r) and the decomposition 1 1 (δ) = (ρ) = pl1 ,..., plk r r 1 k

is uniquely determined or in other words, the set (l1,..., lk) is uniquely 69 ﬁxed.

(b) We shall now prove the existence of a non-trivial relation pl1 ,..., plk 1 k ∼ (1)(li = 0 or 1) in the narrow sense. Consider = 1 ǫ. Then δ is in general not primitive. Now choose a rational number− r suitably so that rδ is primitive and denote it by ρ. Then rδ = ρ = ǫρ′. Now if d < 0, N(δ) > 0 clearly and if d > 0, 2 N(δ) = dǫ′ > 0, again. Hence N(ρ) > 0. We may assume that ρ > 0 and since ǫ is not a square, ρ is not a unit, so that (ρ) , (1). Since (ρ) is primitive, (ρ) , rational ideal. Now (ρ) = (ρ′), ρ > 0 and (ρ) is primitive l1 lk so that p1 ,..., pk = (ρ) (1) in the narrow sense. Further this relation is non-trivial as we have∼ just shown. Proposition 12 is thus completely proved. Now, the group of narrow class characters χ with χ2 = 1 is again of k 1 order 2 − , since it is isomorphic to the group of ambiguous (narrow) ideal classes in K. But the genus characters from a subgroup (of order k 1 2 2 − ) of the group of narrow class characters χ with χ = 1. Hence every narrow class character of order 2 is a genus character. Or, equivalently every real narrow class character is a genus character. For any genus character χ, χ(i2) = 1 for every ideal i. In other words, i2 is in the principal genus. Or, for any two ideals a and b with a bi2 (in the narrow sense), we have χ(a) = χ(b) for every genus character∼ χ. That the converse is also true is shown by Theorem 5 (Gauss). If two ideals a and b are in the same genus, there exists an ideal i such that a bi2, in the narrow sense. In particular, if a is in the principal genus, a i2 ∼for an ideal i. ∼ Proof. Two narrow classes A and B are by deﬁnition, in the same genus if and only if χ(A) = χ(B) for all genus characters χ or by the foregoing, for all narrow class characters χ with χ2 = 1. But from the duality theory of subgroups of Applications to Algebraic Number Theory 64

1 abelian groups and their character groups, we deduce immediately that AB− = C2 for a narrow ideal class C. This is precisely the assertion above.

Remarks. (1) The notion of genus in the theory of binary quadratic forms 70 is the same as above, if one carries over the deﬁnition by means of the correspondence between ideals and quadratic forms.

(2) Hilbert generalized the notion of genus to arbitrary algebraic number ﬁelds and used it for higher reciprocity laws and class ﬁeld theory.

We now come back to formulas (66) and (67) (d < 0). If d = d1d2

Ld1 (s)Ld2 (s) = χ(A)ζ(s, A), A

the summation running over all (narrow) ideal classes A in K = Q( √d). 1 Consider any ideal b A− . For a A, ab = (γ) with N(γ) > 0 we have then ∈ ∈ s (N(b)) s ζ(s, A) = (N(γ))− , w bγ,0 | where w denotes the number of units in K = Q( √d). Let [α, β] be an integral basis of b. Then we may write b = [α, β] = (α)[1, β/α] so that β/α = z = x + iy with y > 0, i.e. we may suppose that b = [1, z]. Then, for γ b, if γ = m + nz with m, n integers , (0, 0), N(γ) = m + nz 2. We obtain∈ | | s N([1, z]) 2s ζ(s, A) = ′ m + nz − w | | m,n

1 z Now, abs. = z′ z = 2y = N(b) √ d , so that 1 z′ | − | | | s 1 2y 2s ζ(s, A) = ′ m + nz − w √ d | | | | m,n s 1 1 2 2 − s ′ 2s = y m + nz − . w √ d √ d m,n| | | | | | From Kronecker’s ﬁrst limit formula, we obtain the following expansion:

s 2s 1 2 y ′ m + nz − = π + 2C 2 log 2 2log( √y η(z) ) + . | | s 1 − − | | m,n − Applications to Algebraic Number Theory 65

Further 71 s 1 2 − (s 1) log 2/ √ d 2 = e − | | = 1 + (s 1) log + . √ d − √ d | | | | On multiplication, we obtain 2π 1 ζ(s, A) = + 2C log d 2log( N([1, z]) η(z) 2)+ w √ d s 1 − | | − | | | | − + terms with higher powers of (s 1) . − We now set F(A) = N([1, z]) η(z) 2. ( ) | | ∗ Clearly, F(A) depends only upon the class A. In case d < 4, w = 2, so that − 2π Ld (s)Ld (s) = − χ(A) log F(A)+ 1 2 √ d | | A + terms with higher powers of (s 1). − Taking s = 1, we have 2π Ld (1)Ld (1) = − χ(A) log F(A). 1 2 √ d | | A Let us assume without loss of generality, d1 > 0, d2 < 0. Then, by Dirich- let’s class number formula for a quadratic ﬁeld, we have

2h1 log ǫ Ld1 (1) = √d1 whre ǫ is the fundamental unit and h1 the class-number of Q( √d1) and

2πh2 Ld (1) = , 2 w √ d | 2| h2 denoting the class-number and w, the number of roots of unity in Q( √d2). 72 On taking the product of these two, we obtain 2h h 1 2 log ǫ = χ(A) log F(A). w − A Summing up, we have the following. Let d = d d < 4 be the discrimi- 1 2 − nant of an imaginary quadratic ﬁeld over Q and let d1 > 0, d2 < 0 be again discriminants of quadratic ﬁelds over Q with class-numbers h1, h2, respectively. Let w be the number of roots of unity in Q( √d2). Then one has Applications to Algebraic Number Theory 66

Theorem 6. For the fundamental unit ǫ of Q( √d1), we have the formula

2h h /w χ(A) ǫ 1 2 = (F(A))− , (68) A where A runs over all the ideal classes in Q( √d) and F(A) has the meaning given in ( ) above. ∗ If w = 2, the exponent of ǫ in (68) is a positive integer. 2 2 Now ǫ satisﬁes Pell’s diophantine equation x d1y = 1. And F(A) involves the values of η(z) . So we have a solution of− Pell’s equation± by means of “elliptic functions”.| This| was found by Kronecker in 1863. An Example. We shall take d = 20, d1 = 5, d2 = 4 so that d = d1d2. We know that h = h = 1, w = 4. The− class number of Q−( √ 20) is 2 and for the 1 2 − two ideal classes, say A1, A2, we can choose as representatives the ideals

1 + √ 5 b = [1, √ 5] = (1) and b = 1, − . 1 − 2  2      Now N(b1) = 1 and N(b2) = 1/2. From the deﬁnition of F(A), we have

2 2 1 1 + √ 5 F(A1) = η( √ 5) and F(A2) = η − . | − | √2  2      If χ(, 1) be the genus character associated with the above decompos ition of d, 5 then necessarily χ(A1) = 1 and χ(A2) = = 1. Formula (68) now takes the 2 − form 4 1 + √ 5 η − 1  2  ǫ =   . (69) 2 η( √ 5) 4  | − | From the product expansion of the η-function, we obtain 73

π √5/12 2 4 6 η( √ 5) = e− (1 q )(1 q )(1 q ) ... , − | − − − | 1 + √ 5 where q = e π √5 and for η − , the following expansion: −  2        1 + √ 5 π √5/24 2 3 4 η − = e− (1 + q)(1 q )(1 + q )(1 q ) ... .  2  | − − |       Applications to Algebraic Number Theory 67

On substituting these expansions in (69), we have 1 ǫ = eπ √5/6(1 + q)4(1 + q3)4(1 + q5)4 .... 2

π √5/3 3 3 9 Now, e > 10 so that q < 10− or q < 10− . Hence the product on the right converges rapidly and if we replace the inﬁnite product by 1, we have

1 π √5/6 2 ǫ = e + O(10− )− 2 i.e. 1 + √5 1 π √5/6 2 ǫ = e (upto an error of the order of 10− ). 2 ∼ 2 It is to be noted that in the expression for ǫ as an infinite product, if one cuts off at any finite stage, the resulting number on the right is always transcendental, but the limiting value ǫ on the left is algebraic. We consider now the trivial decomposition

ζd(s) = ζ(s)Ld(s) = ζ(s, A). A On substituting the expansion of ζ(s, A) on the right side in powers of (s 1), we have − 2πh 1 2 ζ (s) = + 2C log d log F(A) + . (70) d  s 1 − | | − h  w √ d A | |  −    ∞ d s On the other hand, ζ(s) = 1/(s 1) + C + and Ld(s) = n− = 74 − n=1 n L(1) + (s 1)L (1) + . From (70), we have now the equation, − ′ 2πh 1 2 + 2C log d log F(A) +  s 1 − | | − h  w √ d A | |  −  1  = + C + (L(1) + (s 1)L′(1) + ). s 1 − − On comparing coefficients of 1/(s 1) and constant terms on both sides, we obtain − 2πh L(1) = , w √ d | |   (71) 2πh 2  L′(1) = C log d log F(A) , w √ d  − | | − h   A   | |       Applications to Algebraic Number Theory 68 or we have an explicit expression for L (1) 2 ′ = C log d log F(A). (71) L(1) − | | − h ′ A We shall obtain now for the left side, an infinite series expansion. Let us consider the infinite product

1 d s − L(s) = 1 p− . − p p this product converges absolutely for σ > 1, and on taking logarithmic deriva- tives, we obtain

1 L′(s) d s − d s = 1 p− p− log p L(s) − − p p p d log p p = . (72) − d p ps − p

On passing to the limit as s 1, if the series on the right converges at 75 → s = 1, then by an analogue of Abel’s theorem, it is equal to L′(1)/L(1). But it is rather diﬃcult to prove. One may proceed as follows: d Let π+(X) = p : p prime X and = 1 and π (X) = p : p prime { ≤ p } − { d X and = 1 . Then π+(X) π (X) tends to less rapidly than π(X) ≤ p − } − − ∞ as X . If one could show that this function has the estimate 0(X/(log X)′) where→r ∞> 2, then the convergence of the series on the right side of (72) at s = 1, can be proved. For this estimate, one requires a generalization of the proof of prime number theorem for arithmetical series. We shall now compute L′(1)/L(1) for some values of d. Examples. d = 4. The quadratic ﬁeld Q( √d) = Q( √ 1) with discriminant 4 has class number− 1. Also, − − d = 1 if n 1 (mod 4) and = 1 if n 3 (mod 4). n ≡ − ≡ Therefore, s s s L(s) = 1− 3− + 5− − − Applications to Algebraic Number Theory 69 and 1 1 1 L(1) = 1− 3− + 5− − − the so-called Leibnitz series. From Dirichlet’s class number formula, we have

2π π 1 1 1 L(1) = = = 1− 3− + 5− . 4.2 4 − −

More interesting is the series for L′(1)/L(1): L (1) log 3 log 5 log 7 log 11 log 13 ′ = + + L(1) 4 − 4 8 12 − 12 − = (C log 4 4 log η(i)). − − Now π/12 ∞ 2πn η(i) = e− (1 e− ). − n=1

2π ∞ nπ We have 6− < 1/400 and hence e− is rapidly convergent. In other words 76 n=1 ∞ 2nπ 2 4 (1 e− ) is absolutely convergent and upto an error of 10− , it can be n=1 − replaced by 1. We obtain consequently

L (1) π ′ C log 4 + . L(1) ∼ − 3 2 Class number of the absolute class ﬁeld of Q( √d)(d < 0).

We shall now apply the method outlined in 57 to determine the class number § of the absolute class field of K0 = Q( √d), with d < 0. But, for the time being, however, let K0 be an arbitrary algebraic number field. The absolute class field K/K0 is, by definition, the largest field K containing K0, which is both abelian and unramified over K0 (i.e. with relative discriminant over K0 equal to (1)). The absolute class field, first defined by Hilbert, is also known as the Hilbert class field; its existence and uniqueness were proved by Furtwangler.¨ It can be shown that there are only finitely many abelian unramified extensions of K0 and all these are contained in one abelian extension and this is the maximal one. It has further the property that the Galois group G(K/K0) is isomorphic to the narrow ideal class group of K0. Applications to Algebraic Number Theory 70

Consider now any abelian unramiﬁed extension K1 of K0. Then G(K1/K0) is isomorphic to a factor group of G(K/K0), i.e. to a factor group of the narrow ideal class group of K0. The character group of G(K1/K0) is a subgroup of the group of h0 ideal class characters in the narrow sense. K

We have then for the zeta function, 77

ζK1 (s) = L(s, χ), χ the product on the right running over all characters in this subgroup. We shall now derive an expression for the quotient H/h of the class numbers H of K and h of K0 in the case K0 = Q( √d) with d < 0, by using the above product formula for K1 = K and comparing the residue at s = 1 on both sides. We have indeed ζ (s) = ′ L(s, χ) ζ (s), K1 d χ the product ′ running over all characters χ , 1. On comparing the residues at s = 1 on both sides, we obtain

2r1 (2π)r2 R 2πh ′ H = L(1, χ). (73) W √ D w √ d | | | | Here r1 and r2 denote the number of real and distinct complex conjugates of the field K1. The regulator of K1 is denoted by R and is defined as follows: By g0 g1 gr a theorem of Dirichlet, every unit ǫ of K1 is of the form ǫ = ǫ0 ǫ1 ...ǫr with rational integers gi, ǫ0 being a root of unity. Then ǫ1,...,ǫr (with r = r1 +r2 1) (1) (−r1 are called fundamental units. Define integers ek as follows. If K1 ,..., K )1 (r1+1) (r1+r2) are the real conjugates of K1 and K1 ,..., K1 the complex ones, then we define 1 if k r1 ek = ≤ 2 if r1 < K r.  ≤  Applications to Algebraic Number Theory 71

(k) (k) Then R = det(ek log el )(l, k = 1 to r); here ǫl denoting the conjugates of ǫl in the usual order. | | The regulator R is then a real number , 0 and without loss of generality, one may assume R to be positive. It can then be shown that R is independent of the choice of the fundamental units. In the case of the rational number field and an imaginary quadratic field, R is by definition, equal to 1. Further in (73), D denotes the absolute discriminant of K1 and W, the number of roots of unity in K1. We shall now take K1 = K, i.e. the absolute class field. Then K is totally 78 imaginary of degree 2h, i.e. r1 = 0, r2 = h; since K is unramified over K0, D = dh. We shall further assume that d < 4. Then (73) may be rewritten as − (2π)hRH πh 2π = ′ χ(A) log F(A) W d h/2 √ d √ d −  | | | | χ | |  A    or   2RH = h ′ χ(A) log F(A) . (74) W −  χ  A    With every element A of a finite group A ,..., A , we associate an inde- k { 1 h} terminate uA (k = 1,..., h). Then the determinant u 1 is called the group k Ak− Al determinant and was first introduced by Dedekind| and| extensively used by Frobenius. In the case of an abelian group we have then a decomposition of this group determinant, as follows:

uA 1 A = χ(A)uA = uA χ(A)uA . k− l       χ A  A  χ,1 A              Let us suppose Ah = E (the identity); we can show by an elementary transformation that

uA 1 A = uA uA 1 A uA 1 (k, l = 1 to h 1). k− l   k− t − k− − A      Then, from the above, we deduce that

uA 1 A uA 1 = χ(A)uA . k− t − k−   χ,1 A      Taking u = log F(A), we obtain A − Applications to Algebraic Number Theory 72

1 F(A− ) χ(A) log F(A) = det log k −   F(A 1A ) χ,1  A  k− l         2 N(bA ) η(zA ) = det log k | k |  =∆(say).  2   N(b 1 ) η(Z 1 )   Ak At− Ak At−   | |    From 74, we gather that 79 2H ∆ = . (75) W h R In other words, ∆/R is a rational number. We shall now discuss the (h 1)- rowed determinant ∆. − Now, bh is a principal ideal = (β ) (say) with β K . Then N(bh = β 2 A 1 l l 0 A 1 l l− ∈ l− | | h/2 so that N(bA 1 ) = βl . Define l− | | 24h η (zE) ρl = (l = 1 to h 1). 12 24h β η z 1 − l Al− We shall now prove that ρl are all units in the absolute class field K. For 1 the same, we first see that ρl depends only upon the class Al− . If b is replaced by b(λ) with λ K , then ∈ 0 24h η (1, zE) ρl = ρl 12h 12h 24h → (β1λ )λ η 1, z 1 − Al− 24 12 since, in the “homogeneous” notation, η (λω1,λω2) = λ− η(ω1,ω2). 1 Now, we pick out a prime ideal p in the class Al− with the property that pp′ = (p), p , p′; such prime ideals always exist in each class (, E) as a consequence of Dirichlet’s theorem. We shall show that the ρl defined with respect to p are units in the absolute class field K. p = B ,..., B p B = Let ( ) 1 r be the decomposition of in K. If we denote degK0 i ν, then r v = h. The ideal pv is principal and equal to (β) (say). Let [ω ,ω ] 1 2 be an integral base of K0 and [ω1∗,ω2∗] an integral base of p. Then we have (ω1∗ω2∗) = (ω1ω2)Pv where Pv is a 2-rowed square matrix of rational integers and Pp = p. Denoting| | by 24 η ((ω ,ω )Pp) ϕ ω = 12 1 2 Pp ( ) p 24 η ((ω1,ω2))

(with ω = ω2/ω1), we know from the theory of complex multiplication that 80 Applications to Algebraic Number Theory 73

ϕ (ω) K and the principal ideal Pp ∈ 12 ϕPp (ω) = (p′ ) (meaning the extended ideal in K). On taking the vth power on both sides,

v 12v v 12 12 ϕPp (ω) = (p′ ) = (p′ ) = (β ) or in other words,

v 12 ϕPp (ω) = β ǫ with ǫ, a unit in K. Writing out explicitly, we have, ﬁnally, since β 2 = Npv = pv, | | 12 24v β η ((ω ω )Pp) 1 2 = ǫ, . 24v a unit in K η ((ω1ω2)) The same also holds for a suitable unit ǫ, also with h instead of v since v h. (See references (2) Deuring, specially pp. 32-33, (4) Fricke, (5) Fueter). | If σ : α(h) α(k)(k = 1 to h) denotes the automorphism of K/K cor- → 0 responding to the class Ak under the isomorphism between G(K/K0) and the ideal class group of K0, it can be shown that

(k) h 1 h 2 h det(log ρl ) 12 − h − 2 H = W | | (k, l = 1 to h 1), (76) det(log ǫ(k) ) − | l | (k) (k) where ǫ1,...,ǫh 1 are a system of fundamental units and ǫ , ρ denote the − l l conjugates of ǫl and ρl(l = 1 to h 1) respectively, the conjugates being chosen in the manner indicated above. Since− the number on the left is strictly positive, the (h 1) units ρ1,...,ρh 1 are independent, i.e. they have no relation between − − them. Hence the group generated by these units is of ﬁnite index in the whole 81 unit group and the index is precisely given by the integer on the left side of (76). We have thus proved Applications to Algebraic Number Theory 74

Theorem 7. Let K be the absolute class field of an imaginary quadratic field K0 = Q( √d), d < 4, H, h the class numbers of K and K0 respectively, R the regulator of K and− W the number of roots of unity in K. Then for H we have the formula H W ∆ = h 2 R where ∆ = (log ρ(k) 1/12h) ,l,k = 1, 2,..., h 1 and ρ(k) are conjugates of h 1 | | l | | − l − independent units ρ1,...,ρh 1 in K, as found above. − Example. We shall take d = 5, i.e. K = Q( √ 5). − 0 − The class number h of K0 is 2. The absolute class field K is then a biquadratic field and is given by K ( √ 1) = Q( √ 5, √ 1). Further, 0 − − − Q( √5, √ 1) = K −

2 Q( √ 5) = K − 0

2 Q

W = 4 and the fundamental unit is ǫ = (1 + √5)/2. We take A1, A2(= E) 1 to be the two ideal classes of K0 and the unit ǫ1 = ǫ− . The bases for the two √ √ ideal representatives are given by bA1 = [1, (1 + 5)/2] and bA2 = [1, 5]. (Refer to the example on page 71. Then from the above,− we have −

12 2 η( √ 5) 4 ρ =  | − |  1  4  | |  1+ √ 5   η −   2      and from (69), we have then ρ = ǫ 12. From (76), we gather then that | 1| | 1| log( ρ ) 12.22 H = 4 | 1| = 4.12 or H = 1. log( ǫ ) | 1| 82 We shall consider, later, class numbers of more general relative abelian extensions (which are not necessarily unramiﬁed) of imaginary quadratic ﬁelds by using Kronecker’s second limit formula. Applications to Algebraic Number Theory 75

3 The Kronecker Limit Formula for real quadratic ﬁelds and its applications

In the last section, we applied Kronecker’s first limit formula to determine the constant term in the expansion at s = 1 of the zeta-function ζ(s, A) associated with an ideal class A of an imaginary quadratic field over Q. In what follows, we shall consider first, a similar problem for a real quadratic field K0 = Q( √d), d > 0. This problem which is more difficult, was solved by Hecke in his paper, “Uber¨ die Kroneckersche Grenzformel fur¨ reelle quadratische Korper¨ und die Klassenzahl relativ-abelscher Korper”.¨ We shall, however, follow a method slightly different from Hecke’s. We start from the series

∞ s 2s f (z, s) = y ′ m + nz − , −| | m,n= −∞ with z = x + iy, y > 0 and s = σ + it, σ > 1. It is easy to verify that f (z, s) is a ‘non-analytic modular function’, i.e. for a modular transformation z z∗ = (αz + β)/(γz + δ), we have f (z∗, s) = f (z, s). →We now make a remark which will not be used later, but which is interesting in itself, namely, f (z, s) satisﬁes the partial diﬀerential equation

y2∆ f = s(s 1) f (77) − where ∆ = (∂2/∂x2) + (∂2/∂y2) is the Laplace operator. For proving this, let us ﬁrst observe that if F(z, w) is a complex-valued function, twice diﬀerentiable in z and w and if (z, w) (z∗, w∗) where z∗(αz + 1 1 → β)(γz + δ)− , w∗ = (αw + β)(γw + δ)− with α, β, γ, δ real and αδ βγ = 0, then 83 it can be shown directly by computation that −

2 2 2 ∂ (F(z, w)) 2 ∂ F(z∗, w∗) (z w) = (z∗ w∗) . − ∂z∂w − ∂z∗∂w∗ Setting w = z and αδ βγ > 0, we see that y2∆ is an operator invariant under the transformation z z−. Now consider the function ys. It satisﬁes the equation → ∗ y2∆(ys) = s(s 1)ys. − If we use the invariance property of y2∆, then we see that

2 s s y ∆(y∗ ) = s(s 1)y∗ . − Applications to Algebraic Number Theory 76

Since y = (αδ βγ)y γz + δ 2, we have ∗ − | |− 2 s 2s s 2s y ∆(y γz + δ − ) = s(s 1)y γz + δ − | | − | | where γ, δ are real numbers not both zero. It is now an immediate consequence that f (z, s) satisfies the equation (77). s 2s 2πi(mu+nv) In fact, it also follows that the series y ′ m + nz − e satisfies the (m,n) | | differential equation (77). We shall now derive some properties of a function F which satisfies the equation (77). Suppose the function F itself can be written in the form,

F = (F 1)/(s 1) + F0 + F1(s 1) + ; − − − then on setting s(s 1)F = (s 1)2F + (s 1)F in (77) = and by comparing coeﬃcients, we obtain− the recurrence− − relation−

2 y ∆Fn = Fn 1 + Fn 2, n = 0, 1, 2,..., − − where F 2 = 0. Now,− consider the function f (z, s). From Kronecker’s ﬁrst limit formula, 2 F 1 = π and from the above recurrence relation, we obtain y ∆F0 = π. If we − 2 write F0 = π log y + G, then y ∆G = 0. In other words, G = F0 + π log y is a potential function.− This is also explained by the fact that

G = 2π(C log 2 log η(z) 2). − − | | 84 s 2s 2πi(mu+nv) For the function F(z, s) = y ′m,n m + nz − e , from Kronecker’s 2| | second limit formula, F 1 = 0 so that y ∆F0 = 0, i.e. F0 is a potential function, − which can also be seen directly from the expression for F0. These remarks stand in connection with the work of Maass and Selberg on Harmonic analysis. α β Now, let γ δ be the matrix of a hyperbolic substitution having two real fixed points ω, ω′(ω , ω′), both being finite. We shall assume without loss of generality, that ω <ω. Set u = (z ω)(z ω ) 1. Then u = (z ω)(z ω ) 1. ′ − − ′ − ∗ ∗ − ∗ − ′ − The transformation z z∗ corresponds to the transformation u u∗ = λu with λ, a positive real constant→ , 1. We can also assume without loss→ of generality that λ > 1 (otherwise, we may take the inverse substitution). If we introduce a v new variable v,by defining u = λ , then the transformation u u∗ = λu goes over to v v = v + 1. → → ∗ Applications to Algebraic Number Theory 77

Consider a non-analytic function f of z, invariant under the substitution z z = (αz + β)(γz + δ) 1. Then f considered as a function of v, has period → ∗ − 1. If v = v1 + iv2, then f (z) = g(v1, v2) has period 1 in v1. log u Now u = λv1 or v = | | so that e2πiv1 = u 2πi/ log λ. If g(v , v ) possesses | | 1 log λ | | 1 2 a valid Fourier expansion with respect to v1, it is of the form

∞ 2πinv1 f (z) = g(v1, v2) = cn(v2)e n= −∞ ∞ u 2πin/ log λ = c∗ u (78) n u | | n= −∞ log u/u since v = . In general, the Fourier coefficients c in (78), are not 2 2i log λ n∗ constants but if u has a constant argument, i.e. if u/u is a constant, then cn∗ are constants. We shall compute the Fourier coefficients cn∗ of f (z, s) in the case when ω, ω′ come from a real quadratic field K0 = Q( √d) with discriminant d > 0. It is interesting to see that upto certain factors, the Fourier coefficients are just 85 Hecke’s “zeta-functions with Grössencharacters” ssociated with K0. Suppose ǫ is a non-trivual (, 1) unit in K . Let [ω, 1] be an integral basis ± 0 of an ideal b in K0. Without loss of generality, we may suppose that ω>ω′ (otherwise ω has this property!). Now (ǫ)−b = b so that ǫω = αω + β, ǫ = γω + δ with rational integers α, β, γ, δ. We may also write

α β ω ω ω ω ǫ 0 ′ = ′ γ δ1 1 1 1 0 ǫ′ It follows then that αδ βγ = ǫǫ = 1. In the case αδ βγ = 1, deﬁne u − ′ ± − ωu + ω′ αz + β ωu∗ + ω′ such that z = or if z∗ = , then z∗ = ; in other words, u + 1 γz + δ u∗ + 1 z ω z ω u = − ′ and u = ǫ2u. If αδ βγ = N(ǫ) = 1, deﬁne u = − ′ and ω z ∗ − − ω z −αz + β − if z = , u = ǫ2u. Since z z is a hyperbolic transformation, the ∗ γz + δ ∗ − → ∗ points ω′, z, z∗, ω lie on the same segment of a circle in the case N(ǫ) = 1. We may assume without loss of generality, that z∗ lies to the right of z, or otherwise, 2 we can take the reciprocal substitution. We have then ǫ > 1, since ǫ>ǫ′, as a consequence of our assumption that z∗ lies to the right of z ǫ itself might be positive or negative. If ǫ is negative, we take instead of α, β,γ, δ, the integers Applications to Algebraic Number Theory 78

α, β, γ, δ so that we secure finally that ǫ > 1. In the second case, when −N(ǫ)−= −1, if− z lies on the small segment ω λω (see figure), z would lie on − ′ ∗ the big segment ω′ω which is the complement of the small segment obtained by reflecting the original one on ω′ω. If we take z on the semicricle ω′vω on ω′ω as diameter, then z and z∗ would lie on the same segment. So, in this case, under the same argument, ǫ > 1.

In either case, ǫ > 1 and consequently ǫ′ < 1 so that ǫ ǫ′ > 0. But 86 ǫ ǫ = γ(ω ω ) implies that γ > 0. − − ′ − ′ Now, if z lies on ω′vω, u is purely imaginary with arg u = π/2. On setting 2 u = u′i, we obtain the transformation u′ u′∗ = ǫ u′; u′ is real and positive. Hereafter we need not distinguish between→ the two cases. 2v If we deﬁne v such that u = ǫ (writing u instead of u′) then u u∗ s 2s → becomes v v + 1. The function f (z, s) = y ′m,n m + nz − , is invariant → 1 | | under the modular substitution z (αz + β)(γz +δ)− and hence as a function of v, it has period 1. It has a valid→ Fourier expansion in v of the form f (z, s) = ∞ 2πikv ake . Now k= −∞ 1 2πikv ak = f (z, s)e− dv 0 ǫ2 1 πik/ log ǫ du = f (z, s)u− . 2 log ǫ 1 u On changing the variable from u to z, the series f (z, s) being uniformly convergent on the corresponding segment of ω′vω, we may interchange integration and summation and obtain

ǫ2 s 1 y πik/ log ǫ du ak = u− . 2 log ǫ m + nz 2s u m,n 1 | | Applications to Algebraic Number Theory 79

ωui + ω From the relation z = ′ it follows that ui + 1 u(ω ω ) (m + nω)ui + (m + nω ) y = − ′ and m + nz = ′ u2 + 1 ui + 1 Setting β = m + nω then β b since m and n are both rational integers and ω ω = N(b) √d. Then our∈ integral becomes − ′ ǫ2 s 1 s s/2 u πik/ log ǫ du ak = (N(b)) d u− . 2 log ǫ 2 2 + 2 u bβ,0 1 β u β′ | To bring the right hand-side to proper shape again, we use an idea of Hecke. We shall now get rid of β in the integrand by introducing U = u β/β′ . The 87 integral then reduces to | |

β πik/ log ǫ (βǫ/β′ǫ′) us (πik/ log ǫ) du β s | | − , N( ) − 2 s β′ | | β/β (u + 1) u | ′| on writing u instead of U. For two elements β, γ, , 0, (β) = (γ) implies that γ = βǫn, n = 0, 1, 2,... with ǫ > 1, being the fundamental unit in Q( √d). Then± ± ± n n n γ βǫ γ = βǫ , γ′ = β′ǫ′ and = n . | | | | | | | | γ′ β′ǫ′ We have therefore, on taking into account γ =βǫn ofγ = βǫn, − (βǫ/β′ǫ′) s (πik/ log ǫ) | | u − du (u2 + 1)s u bβ,0 β/β′ | | | (βǫn+1/β ǫ n+1) ∞ ′ ′ us (πik/ log ǫ) du = | | − . 2 2 s n n (u + 1) u b(β),0 n= (βǫ /β′ǫ′ ) | −∞ | | 2n 2(n+1) Now since ǫ > 1, the intervals ( β/β′) ǫ , β/β′ ǫ ) ﬁll out the half- line (0, ) exactly once, without gaps| and overlaps,| | as n tends to on one side and∞+ on the other side. Hence we have −∞ ∞ (βǫn+1/β ǫ n+1) ∞ ′ ′ us (πik/ log ǫ) du us (πik/ log ǫ) du | | − = ∞ − , 2 2 s 2 2 s n n (u + 1) u (u + 1) u n= (βǫ /β′ǫ′ ) 0 −∞ | | and s πik s πik Γ Γ + us (πik/ log ǫ) du 2 − 2 log ǫ 2 2 log ǫ ∞ − = 2 2 s 0 (u + 1) u Γ(s) Applications to Algebraic Number Theory 80

Therefore, we have

1 2πikv ak = f (z, s)e− dv 0 s πik s πik Γ Γ + 1 2 − 2 log ǫ 2 2 log ǫ = (N(b))sds/2 2 log ǫ Γ(s) × πik/ log ǫ β s N(β) − , × β | | b(β),0 ′ | since the expression under the summation Σ does not change for associated 88 elements β and γ. Upto a product of Γ-factors, the Fourier coeﬃcients ak are Dirichlet series. For k = 0,

2 s 1 Γ 1 2 s/2 s a = f (z, s)dv = d (N(a))− 0 2 log ǫ Γ(s) 0 a A s ∈ Γ2 1 = 2ds/2ζ(s, A), 2 log ǫ Γ(s)

1 where A denotes the ideal class of b− in the wide sense. If k , 0, we deﬁne for σ > 1,

s ζ(s, χ, A) = χ(a)(N(a))− a A ∈ 1 s = (χ(b))− χ((β))(N(a))− ; a A ab=∈(β) The function ζ(s, χ, A) is called the zeta-function of the class A and associated with the Grossencharacter¨ χ, which is defined as follows: The characterχ is defined on principal ideals (β) of K0 as β πik/ log ǫ χ((β)) = β′ (χ((β)) is independent of the generator β by definition). Then χ is extended to all ideals i as follows: If ik = (v), define χ(i) as a hth root of χ((v)) so that v πik/ log ǫ χ(ih) = χ((v)) = v′ Applications to Algebraic Number Theory 81

It is clear that χ is multiplicative. 89 We have now s χ(b)ζ(s, χ, A) = χ((β))(N(a))− a A ab=(β∈),(0) s s = (N(b)) χ((β)) N(β) − , | | b (β),(0) | so that we may rewrite the expression for ak as follows:

s πik s πik Γ Γ + 2 log ǫ s/2 2 − 2 log ǫ 2 2 log ǫ a = χ(b)ζ(s, χ, A). k d Γ(s) One can make some applications from the nature of the Fourier coefficients ak. We know that the function f (z, s) satisfies the following functional equation, viz. s (1 s) π− Γ(s) f (z, s) = π− − Γ(1 s) f (z, 1 s). − − One can then show that from the analytic continuation of f (z, s) it follows that the Fourier coefficients ak as functions of s have also analytic continua- tions into the whole s-plane and satisfy a functional equation similar to the above. In the particular case, when k = 0, it follows that

s s/2 2 s (1 s) (1 s)/2 2 1 s π− d Γ ζ(s, A) = π− − d − Γ − ζ(1 s, A). 2 2 −

Hence, for the zeta-function ζk0 (s) = ζ(s, A), we have A

s s/2 2 s (1 s) (1 s)/2 2 1 s π− d Γ ζK0 (s) = π− − d − Γ − ζK0 (1 s). 2 2 − This was discovered first, by Hecke, who also introduced the zeta-functions with Grossencharacters¨ and derived a functional equation for the same. We shall now use Kronecker’s first limit formula to study the behaviour of 90 ζ(s, A) at s = 1. From Kronecker’s first limit formula, we have π f (z, s) = + 2π(C log 2 log √y η(z) 2 + ). s 1 − − | | − Applications to Algebraic Number Theory 82

It can be shown that the series on the right can be integrated term by term with respect to v (after transforming z into v) in the interval (0, 1), i.e. s Γ2 1 1 f (z, s)dv = 2ds/2ζ(s, A) 0 2 log ǫ Γ(s) π 1 = + 2π(C log 2) 2π log( √y η(z) 2)dv + . s 1 − − | | − 0 It follows therefore that the function on the left has a pole at s = 1 with residue π and the constant term in the expansion is provided by the integral which cannot in general be computed. It is to be noted here that in the case of the imaginary quadratic field, we had only the integrand on the right side and the argument z was an element of the field, but here z is a (complex) variable. One cannot get rid of the integral even if one uses the series expansion for the integrand. The above formula was found by Hecke and is the Kronecker limit formula for a real quadratic field. One can also consider ak for k , 0 and obtain a similar formula. Changing this integral to a contour integral, we shall later obtain an analogue of Kronecker’s solution of Pell’s equation in terms of elliptic functions. We have now, for real v, 1 v+1 a0 = f (z, s)dv = f (z, s)dv 0 v s Γ2 1 = 2ds/2ζ(s, A). 2 log ǫ Γ(s) Using the Legendre formula 91

s s + 1 1 s Γ Γ = √π2 − Γ(s), 2 2 we obtain s + 1 s + 1 Γ2 Γ2 Γ(s) Γ(s) 2 2 = = . s s s + 1 π22(1 s)Γ(s) Γ2 Γ2 Γ2 − 2 2 2 But on the other hand, s + 1 Γ2 2 = 1 + terms in (s 1)2, Γ(s) − Applications to Algebraic Number Theory 83 so that we have Γ(s) v+1 = s/2 ζ(s, A) 2 log ǫd− s f (z, s)dv Γ2 v 2 v+1 2 log ǫ 2 1 (s 1) 2 = (2 d− 2 ) − (1 + terms in (s 1) ) f (z, s)dv. π √d − v

2 1 (s 1) (s 1)(2 log 2 2 log √4 d) By setting (2 d− 2 ) − = e − − and expanding it in powers of (s 1), and applying Kronecker’s ﬁrst limit formula for f (z, s), we obtain ﬁnally,−

2 log ǫ 4 ζ(s, A) = (1 + (s 1)(2 log 2 2 log √d) + ) π √d − − ×

1 4 + 2C (2 log 2 2 log √d) × s 1 − − − −v+1 4 2 log( √y√d η(z) 2)dv + , − v | | i.e. v+1 2 log ǫ 1 4 ζ(s, A) = + 2C 2 log( √y√d η(z) 2)dv + . (79) √d s 1 − v | | − 2 log ǫ From this, we deduce that ζ(s, A) has a pole at s = 1 with residue which 92 √d is independent of the ideal class A. This result was ﬁrst discovered by Dirichlet. It would be nice if one could compute the integral and express it in terms of an analytic function, but it looks impossible. We shall only simplify it to a certain extent by converting it into a contour integral. z ω From the substitution ui = − ′ follows that ω z − z ω y(ω ω ) u = Im − ′ = − ′ ω z (z ω)(z ω) − − − and similarly 1 y(ω ω′) u− = − . (z ω )(z ω ) − ′ − ′ On multiplying the two, we have y2 1 = . (80) (z ω)(z ω ) (z ω)(z ω ) − − ′ − − ′ ω ω ω ω − ′ − ′ Applications to Algebraic Number Theory 84

Consider now the expression

(z ω)(z ω ) F(z) = − − ′ = az2 + bz + c. (81) N(b)

We then claim that a, b, c are rational with (a, b, c) = 1 and b2 4ac = d. From Kronecker’s generalization of Gauss’s theorem on the content− of a product of polynomials with algebraic numbers as coeﬃcients, we have for the product (ξ ωη)(ξ ω η), (1, ω)(1, ω ) = (1, (ω + ω ),ωω ) or in other words, − − ′ − − ′ − ′ ′ bb′ = (1,λ,) where λ and are rational, i.e.

1 λ , , = 1. N(b) N(b) N(b) 1 λ But, from our deﬁnition of F(z), a = , b = , c = so that N(b) N(b) N(b) (a, b, c) = 1. The discriminant of F(z) is

(ω + ω )2 4ωω (ω ω )2 b2 4ac = ′ − ′ = − ′ = d. − N(b)2 N(b)2 1 Further a = > 0. We can then show that the class of the quadratic form 93 N(b) F(z) is uniquely determined by the ideal class A. 4 From (80) it follows that y2d = F(z) F(z) or equivalently, √y√d = 4 F(z) F(z). Now, for the integral in (79), we have

v+1 z∗ 4 log ǫ 4 dz − log( √u√d η(z) 2)dv = 2 log( 4 F(z)η(z) 2) , | | | | F(z) √d v z since, by deﬁnition of v, √d dz dv = − 2 log ǫ F(z) and the transformation v v + 1 corresponds to z z∗ on the segment of the orthogonal circle. On the→ right side, z may be taken→ arbitrarily on the segment. Thus, let A be an ideal class, in the wide sense, of K0 = Q( √d), d > 0 and b an ideal in A with an integral basis [1,ω], ω>ω′. Corresponding to b, let α β be deﬁned as on p. 85 and let for z h, z = (αz + β)(γz + δ) 1. Let, γ δ ∈ ∗ − further, ǫ > 1 be the fundamental unit in K0. Then we have Applications to Algebraic Number Theory 85

Theorem 8 (Hecke). For the zeta-function ζ(s, A) associated with the class A, we have the limit formula

2 log ǫ 1 4C log ǫ z∗ dz lim ζ(s, A) = + 2 log( 4 F(z)η(z) 2) , s 1 − √d s 1 √d z | | F(z) → − where the integration is over a segment zz∗ of the semicircle on ω′ω as diameter and F(z) is deﬁned by (81).

Now, the question arises whether one could transform the integral in such a 94 way that it is independent of the choice of z and also the path of integration. It is possible to do that, by the method of Herglotz who reduced this to an integral involving simpler functins than η(z). If N(ǫ) = 1, then the transformation v v + 1 is equivalent to z z∗ = αz + β → → with α β , a modular matrix. γz + δ γ δ If N(ǫ) = 1, on replacing ǫ by ǫ2, N(ǫ2) = 1 and the transformation − α z + β v v + 1 goes over to the transformation z ′ ′ with α′ β′ a modular γ′ δ′ → → γ′z + δ′ matrix. Further, because of the periodicity of f (z, s),

v+2 v+1 f (z, s)dv = 2 f (z, s)dv v v so that we may assume without loss of generality that N(ǫ) = 1. Now, the behaviour of √4 F(z) η(z) under a modular substitution can be studied. We know that η(z∗) = ρ √γz + δη(z) where ρ is a 24th root of unity. Further, (z ω)(z ω′) (z∗ ω)(z∗ ω′) = − − − − (γz + δ)2 if one uses the fact that ω = ω∗ and γω + δ = ǫ. From these two, it follows that

√4 F(z) 4 F(z ) = κ ∗ + √γz δ with κ, a 4th root of unity. Therefore

4 4 F(z∗)η(z∗) = ρκ F(z)η(z). (82) (We emphasize here that (82) holds only for a hyperbolic substitution and a power of the same, since we have made essential use of the fact that ω and ω′ Applications to Algebraic Number Theory 86 are ﬁxed points of the substitution in obtaining the transformation formula for F(z).) Consider the function √4 F(z)η(z). It is an analytic function having no zeros in the upper half plane so that on choosing a single valued branch of 95 log(√4 F(z)η(z)), we have a regular function in the upper z-half plane. Let us denote log √4 F(z)η(z) by g(v). Then (82) implies that g(v+1) g(v) = 2πiλ (say) with λ rational. One can then express λ by means of the so-called− Dedekind sums. Setting g(v) 2πiλv = h(v), we have h(v + 1) = h(v) possesses − 2πiv ∞ 2πinv a Fourier development in e , or in other words, h(v) = cne , where n= −∞ v+1 = cn are constants, since h(v) is an analytic function of v and c0 v h(v)dv. Here v lies in a certain strip enclosing the real axis. + = v 1 Now, consider the complex integral c0 v h(v)dv. Here v is, in general, complex and the path of integration may be any curve between v and v+1 lying in the strip in which the Fourier expansion is vaid. We have

√d z∗ dz v+1 log( 4 F(z)η(z) 2) = 2 Re (g(v))dv (v real) −2 log ǫ | | F(z) z v v+1 = 2 Re h(v)dv v

= 2 Re(c0).

The computation of the integral on the left side therefore reduces to the determination of c0, which in turn is independent of v and the path of integration. The trick for computing the integral is to convert it into an infinite integral by letting z and z∗ α/γ. Expanding g(v) in an infinite series, one can express this→ ∞infinite integral→ as a definite integral of an elementary function. (See G. Herglotz.) We shall now obtain an analogue of Kronecker’s solution of Pell’s equation in this case. Let us define

z∗ dz g(A) = 2 log( 4 F(z)η(z) 2) . (83) | | F(z) z Then, associated with a character χ of the ideal class group (in the wide sense), we have L(s, χ) = χ(A)ζ(s, A), A the summation running over all wide classes A. 96 Applications to Algebraic Number Theory 87

From (79), we obtain, for χ , 1, the expansion,

L(s, χ) = χ(A)g(A) + terms involving (s 1). (84) − A Before proceeding to derive the analogue of Kronecker’s solution of Pell’s equation from the above, we shall examine under what conditions, a genus character, which is a character of the narrow class group of Q( √d), is also a character of the wide class group. Let χ be a genus character, defined as follows: If d = d1d2, then for all ideals a with (a, d1) = (1), χ is defined by d1 χ(a) = . (We may suppose without loss of generality that d1 is odd). N(a) Now, in general, χ is not a character of the wide class group. It will be so, if χ((α)) = 1 for all integers α with (α, d1) = (1). For elements α with N(α) > 0, this is true by the definition of χ. If N(α) < 0, then d d χ((α)) = 1 = 1 = 1 N((α)) N(α) − for all α, if it is true for one such α. d Take α = 1 + √d so that N(α) = 1 d < 0. We need examine only 1 , − d 1 − d1 being odd. Then

d d 1 1 +1 if d1 > 0 1 = − = − = d 1 d1 d1  1 if d1 < 0. − −  Since d is positive, either both d1, d2 are positive or both are negative. In the former case, χ continues to be a character of the wide class group and in the latter case, it is no longer a character of the wide class group. Case (i). Let us assume, that both d1 and d2 are positive. Then the genus character χ as deﬁned above, is also a wide class character. We have a decomposition of L(s, χ) due to Kronecker, from (66) as follows:

L(s, χ) = Ld1 (s)Ld2 (s) and on taking the values on both sides at s = 1, we obtain 97

L(1, χ) = Ld1 (1)Ld2 (1). From (84), we have now L(1, χ) = χ(A)g(A), summation running over A all wide classes A. Further 2 log ǫ1h1 Ld1 (1) = √d1 Applications to Algebraic Number Theory 88

if ǫ1 denotes the fundamental unit of Q( √d1) and h1, the wide class number of

Q( √d1) and similarly for Ld2 (1). Thus as an analogue of Kronecker’s solution of Pell’s equation as derived in (68), we have the following Proposition 13. Let χ be a genus character of Q( √d), d > 0, corresponding to the decomposition d = d1d2(d1 > 0, d2 > 0) of d. Let h1, h2 be the class numbers and ǫ1, ǫ2 the fundamental units of Q( √d1), Q( √d2) respectively. Then 4h1h2 log ǫ1 log ǫ2 = √d χ(A)g(A), (85) A where A runs over all the ideal classes of Q( √d) in the wide sense and g(A) is deﬁned by (83). The expression on the right side of (85) is not very simple. It is not known whether the number on the left side is rational or irrational. It is probable that the number on the left side is a complicated transcendental number, so that one cannot expect a simple value on the right side. Case (ii). Suppose d1 and d2 are both negative. Then again from the decomposition formula (66), we have

2 2πh1 2πh2 4π h1h2 L(1, χ) = Ld (1)Ld (1) = = 1 2 w √ d w √ d w w √d 1 | 1| 2 | 2| 1 2 where h1, h2 denote the wide class numbers of Q( √d1) and Q( √d2), and w1, w2 the number of roots of unity in Q( √d1) and Q( √d2) respectively. We shall see in 5 that 98 § π2 L(1, χ) = χ(B)G(B), √d B

B running over all narrow classes of Q( √d) and G(B) being numbers depending only on B. We shall further prove that G(B) are rational numbers which can be realized in terms of periods of certain abelian integrals of the third kind. We then have, as an analogue of Kronecker’s solution of Pell’s equation, the following: 4h h 1 2 = χ(B)G(B). w w 1 2 B

Example. Consider the ﬁeld Q( √10) with discriminant d = 40 = 5.8; d1 = 5 and d2 = 8. Applications to Algebraic Number Theory 89

The class number of Q( √10) is 2. Since the fundamental unit ǫ = 3 + √10 has norm 1, the narrow classes are the same as wide classes. We can now choose the− two ideal class representatives as follows

1 1 (1) = b1 = [1, √10] and b2 = 1, √10 with N(b2) = . 2 2

The class numbers h1 and h2 of Q( √5) and Q( √8) are both 1 and the funda- 1 + √5 mental units are respectively and 1 + √2. The only character χ , 1 2 has the property that χ(b1) = 1 and χ(b2) = 1, so that we have from (85), the following: −

1 + √5 2 log log(1 + √2) = √10(g(E) g(A)).  2  −     We shall make a remark for more general applications. Consider the absolute class field of Q( √d) with d > 0. For computing the class number of this field, one can proceed in the same way, as in the case of an imaginary quadratic field. For the same, one requires the computation of L(1, χ) for arbitrary narrow class characters. This will be done in 5, for a special type of characters. § 4 Ray class fields over Q( √d), d < 0 99 Let K be an algebraic number field of degree n over Q, the field of rational numbers. Let r1 and 2r2 be the number of real and complex conjugates of K (1) (r1) respectively, so that r1 + 2r2 = n. Let K ,..., K be the real conjugates of K and K(r1+1),..., K(n) be the complex conjugates of K. Let, for α K, α(i) K(i), i = 1, 2,..., n, denote the conjugates of α. Further let f be a given∈ ∈ α1 α2 integral ideal in K. Two numbers γ1 = , γ2 = with α1, β1, α2, β2 integral β1 β2 in K and with β1β2 coprime to f are (multiplicatively) congruent modulo f (in symbols, γ1 γ2( mod ∗f) if α1β2 α2β1( mod f)). If γ1, γ2 are integers in K, this is the≡ usual congruence modulo≡ f. b Let us consider non-zero fractional ideals a of the form a = where b and c c are integral ideals in K coprime to f. These fractional ideals a form, under the usual multiplication of ideals, an abelian group which we shall denote by Gf. Let Gf be the subgroup of Gf consisting of all principal ideals (α) for which α > 0 (i.e. α(i) > 0, for i = 1, 2,..., r ) and α 1( mod f). It is clear that if 1 ≡ ∗ (γ) Gf and γ 1( mod f), then γ might be written as α/β where α and β ∈ ≡ ∗ Applications to Algebraic Number Theory 90 are integers in K satisfying the conditions, α > 0, β > 0, α 1( mod f) and β 1( mod f). ≡ ≡ The quotient group Gf/Gf is a finite abelian group called the ray class group modulo f; its elements are called ray classes modulo f. Two ideals a and b in Gf are equivalent modulo f(a b mod f) if they lie in the same ray class modulo ∼ f, i.e. a = (γ)b with (γ) Gf. ∈ If r1 = 0 and f = (1), equivalence modulo f is the usual equivalence in the wide sense and the ray class group modulo f is the class group of K in the wide sense. If r1 = 2, r2 = 0 (i.e. K is a real quadratic field) and f = (1), then the equivalence modulo f is the equivalence in the narrow sense and the ray class group is the class group of K in the narrow sense. Let χ be a character of the group of ray classes modulo f. Then associated with χ, we define for σ > 1, the L-series s L(s, χ) = χ(a)(N(a))− a,0 where N(a) is the norm of a in K and the summation is extended over all integral ideals a coprime to f. It is clear that L(s, χ) is a regular function of s for 100 σ > 1. Moreover, due to the multiplicative character of χ, we have for σ > 1, an Euler-product decomposition for L(s, χ), namely

s 1 L(s, χ) = (1 χ(p)(N(p))− )− − p f | where the infinite product is extended over all prime ideals p coprime to f. Hecke has shown that L(s, χ) can be continued analytically as a meromorphic function of s in the whole s-plane and that when χ is a “proper” character, there is a functional equation relating L(s, χ) with L(1 s, χ), where χ is the conjugate character. If χ , 1 (the principal character), then− L(s, χ) is an entire function of s. If χ = 1 and f = (1), then L(s, χ) is the Dedekind zeta function of K. We are interested in determining the value at s = 1 of L(s, χ), in the case when K is a real or imaginary quadratic number field and χ is not the principal character. For this purpose, we need to apply Kronecker’s second limit formula. Later, we shall use this for the determination of the class number of the “ray class field” of K, corresponding to the ideal f. First we shall investigate the structure of a ray class character χ. Let α and α2 β(, 0) be integers coprime to f such that α = β( mod f). Then γ = satisfies β2 2 2 γ = 1( mod ∗f) and γ > 0 so that χ((γ)) = 1 i.e. χ((α )) = χ((β )). In other words, χ((α)) = χ((β)) or χ((α/β)) = 1. ± ± Applications to Algebraic Number Theory 91

Let L be the multiplicative group of λ , 0 in K. Corresponding to a ray class character χ modulo f, we define a character v of L as follows. For λ L, we find an integer α K such that α 1( mod f) and αλ > 0 and define∈ v(λ) = χ((α)). It is first∈ clear that v(λ)≡ is well-defined; for, if v(λ) = χ((β)) for another integer β K satisfying the same conditions, then we see at once α ∈ α that 1( mod f) and > 0 and so χ((α)) = χ((β)). It is easily verified β ≡ ∗ β that v(λ) = v(λ)v(). Moreover, v(λ) = 1; for, λ2 > 0 and by definition, ± v(λ2) = χ((α)) for α satisfying α 1( mod f) and α > 0 so that (v(λ))2 = 101 v(λ2) = χ((α)) = +1. We shall call v≡(λ), a character of signature. Consider the subgroup B of L consisting of λ > 0. Clearly v(λ) = 1 for all λ B. Thus v(λ) may be regarded as a character of the quotient group L/B. ∈ Now L/B is an abelian group of order 2r1 exactly, since one can find λ L for (i) ∈ which the real conjugates λ , i = 1, 2,..., r1, have arbitrarily prescribed signs. Also, there are 2r1 distinct characters u of L/B defined by

r1 λ(i) gi u(λ) = , gi = 0 or 1,λ L. λ(i) ∈ i=1 | | There can indeed be no more that 2r1 characters of L/B and hence v(λ) coincides with one of these characters u, of L/B. We do not however assert that every character u of L/B is realizable from a ray class character modulo f, in the manner described above. Let now γ K such that γ 1( mod ∗f) and (γ) Gf. By deﬁnition, v(γ) = χ((δ)) for∈ an integer δ such≡ that δ 1( mod ∈f) and δγ > 0. But χ((δ)) = χ((γ)) and hence v(γ) = χ((γ)). Let≡ α and β(, 0) be two integers coprime to f such that α = β( mod f). Then γ = α/β 1( mod f) and ≡ ∗ χ((α)) v(α) = χ((γ)) = v(γ) = . χ((β)) v(β) Thus for integral α(, 0) coprime to f, the ratio χ((α))/v(α) depends only on the residue class of α modulo f and is in fact, a character of the group G(f) of prime residue classes modulo f. We may denote it by χ(α). Thus

Proposition 14. Any character χ((α)) of the ray class group modulo f may be written in the form χ((α)) = v(α)χ(α) where v(α) is a character of signature and χ(α) is a character of the group G(f). Applications to Algebraic Number Theory 92

Before proceeding further, we give a simple illustration of the above. Let us take K to be Q, the field of rational numbers and f to be the principal ideal 102 ( d ) in the ring of rational integers, d being the discriminant of a quadratic field over| | Q. For an integer m coprime to d, we define d m for d > 0 m m χ((m)) =   |d| | |  for d < 0  m  | |  d  where is the Legendre-Jacobi-Kronecker symbol. The group Gf now m consists| of| fractional ideals (m/n) where m and n are rational integers coprime to d. We extend χ to the ideals (m/n) in Gf by setting χ((m/n)) = χ((m))/χ((n)). Now it is known that if m n( mod d) and mn is coprime to d, then ≡ d d = if d > 0 m n | | | | and d m d n = if d < 0 m m n n | | | | | | | | Thus d χ(m) = for d > 0 m | | and d m χ(m) = for d < 0 m m | | | | m are characters of G( d ). Moreover v(m) = is clearly a character of signa- | | m ture. It is easily verified that χ((m/n)) is a ray| | class character modulo ( d ) and that χ((m)) = v(m)χ(m). | | s Now L(s, χ) = a(a)(N(a))− , where A runs over all the ideal classes A in the wide sense and a over all the non-zero integral ideals in A, which are 1 coprime to f. In the class A− , we can choose an integral ideal bA coprime to f and abA = (β) where β is an integer divisible by bA and coprime to f. 103 Conversely, and principal ideal (β) divisible by bA and coprime to f is of the form abA, where a is an integral ideal in A coprime to f. Moreover, χ(bA)χ(a) = χ((β)) and N(b ) N(a) = N(β) , where N(β) is the norm of β. Thus A | | s s L(s, χ) = χ(b )(N(b )) χ((β)) N(β) − A A | | A b(β) A| Applications to Algebraic Number Theory 93

where the inner summation is over all principal ideals (β) divisible by bA and 1 coprime to f. Since A− runs over all classes in the wide sense when A does so, we may assume that bA is an integral ideal in A coprime to f. Moreover, in view of Proposition 14, we have for σ > 1,

s s L(s, χ) = χ(b )(N(b )) v(β)χ(β) N(β) − (86) A A | | A b(β) A| where now A runs over all the ideal classes of K in the wide sense, bA is a fixed integral ideal in A coprime to f and the inner sum is over all principal ideals (β) divisible by bA and coprime to f. We may extend χ(β) to all residue classes modulo f by setting χ(α) = 0 for α not coprime to f. Thus we may regard the inner sum in (86) as extended over all principal ideals (β) divisible by bA. In order to render the series in (86) suitable for the application of Kronecker’s limit formula, we have to replace χ(β) by an exponential of the form e2πi(mu+nv) occurring in the limit formula. We shall, in the sequel, express χ(β) as an exponential sum by using an idea due to Lagrange, which is as follows. Let x1,..., xn be n distinct roots of a polynomial f (x) of degree n, with th coefficients in a field K0 containing all the n roots of unity. Let, further, the field M = K0(x1,..., xn) be an abelian extension of K0, with galois group H. σi Let us assume moreover that if σ1,...,σn H, then xi = x1 i = 1, 2,..., n. ∈ n Now, if χ is a character of H, then let us define yχ = χ(σi)xi. It is clear that i=1 y M and χ ∈ n n σ j σiσ j σiσ j yχ = χ(σi)x1 = χ(σ j) χ(σiσ j)x1 = χ(σ j)yχ. i=1 i=1 Hence yn K . Now, if χ denotes the conjugate character of χ and σ H, then χ ∈ 0 ∈ n n σiσ σ yχ = χ(σiσ)x1 = χ(σ) χ(σi)xi . i=1 i=1

n , = 1 σ If yχ 0, then χ(σ) y−χ χ(σi)xi . We shall use a similar method to express 104 i=1 χ(β) as an exponential sum whose terms involve β in the exponent. First we need the following facts concerning the different of an algebraic number field M of finite degree over Q. Let for α M, S (α) denote the ∈ trace of α. Let a be an ideal (not necessarily integral) in M and let a∗ be the “complementary” ideal to a, namely the set of λ M, for which S (λα) is a rational integer for all α a. It is known that aa is∈ independent of a and in fact ∈ ∗ Applications to Algebraic Number Theory 94

1 aa∗ = (1)∗ = ϑ− , where ϑ is the different of M. Further clearly (a∗)∗ = a and N(ϑ) = d , where d is the discriminant of M. These facts can be verified in a | | simple way if M is a quadratic field over Q, with discriminant d. Let [α1, α2] be an integral basis of a. The ideal a is the set of λ M for which S (λα ) ∗ ∈ 1 and S (λα2) are both rational integers. If α1′ , α2′ denote the conjugates of α1 α2′ α1′ and α2 respectively, then it is easliy verifed that , − α α α α α α α α 1 2′ − 2 1′ 1 2′ − 2 1′ √ is an integral basis of a∗. Now α1α2′ α2α1′ = N(a) d and this means that − ± 1 a∗ = (1/N(a) √d)a′. Since (N(a)) = aa′, we see that a∗ = a− (1/ √d) i.e. aa∗ = 1 (1/ √d) = ϑ− . Moreover N(ϑ) = d . | | 1 1 Let us consider now the ideal a = f− ϑ− in K; clearly a∗ = f. Let us choose 1 1 in the class of fϑ, an integral ideal q coprime to f. Then qf− ϑ− = (γ) for γ K 1 ∈ i.e. (γ)ϑ = qf− has exact denominator f. If K were a quadratic field and f, a principal ideal, then fϑ is principal and we may take q = (1). We shall consider γ fixed this way once for all, in the sequel. Let us observe that if λ f, S (λγ) is a rational integer. ∈ With a view to express χ(β) as an exponential sum, we now define the sum

T = χ(λ)e2πiS (λγ) (87) λ mod f where λ runs over a full system of representatives of residue classes modulo f; for f = (1), clearly T = 1. We see that T is defined independently of the 105 choice of representatives λ, for, if runs over another system of representatives modulo f, then λ ( mod f) in some order and in this case, χ(λ) = χ() and e2πiS (λγ) = e2πiS (γ≡) since S ((λ )γ) is a rational integer. Moreover, in this sum, λ may be supposed to run only− over representatives of elements of G(f), since χ(α) = 0, for α not coprime to f. The sums of this type were first investigated by Hecke; similar sums for the case of the rational number field have been studied by Gauss, Dirichlet and for complex characters χ, by Hasse. Let α be an integer in K coprime to f. Then αλ runs over a complete set of prime residue classes modulo f when λλ does so. Thus

T = χ(αλ)e2πiS (αλγ) λ mod f = χ(α) χ(λ)e2πiS (αλγ), λ mod f i.e. Tχ(α) = χ(λ)e2πiS (αλγ). (88) λ mod f Applications to Algebraic Number Theory 95

We shall presently see that (88) is true even for α not coprime to f and that T , 0, when χ is a so-called proper character of G(f). Let g be a proper divisor of f and ψ, a character of G(g). We may extend ψ into a character χ of G(f) by setting χ(λ) = ψ(λ) for λ coprime to f and χ(λ) = 0, otherwise. We say that a character χ of G(f) is proper, if it is not derivable in this way from a character of G(g) for a proper divisor g of f. A ray class character χ modulo f is said to be proper if the associated character χ of G(f) is proper; χ is then said to have f as conductor. Let χ be a ray class character modulo f and let g be a proper divisor of f. Moreover let for integers α, β coprime to f such that α β( mod g) and ≡ αβ > 0, χ(α) = χ(β). We can associate with χ, a ray class character χ0 modulo g as follows. If p is a prime ideal coprime to f, define χ0(p) = χ(p). If p is a prime ideal coprime to g but not to f, we can find a number α such that α 1( ≡ mod ∗g), α > 0 and (α)p is coprime to f. We then set χ0(p) = χ((α)p). We see that χ0 is well-defined for all prime ideals coprime to g and we extend χ0 106 multiplicatively to all ideals in the ray classes modulo g. Clearly, χ0 is a ray class character modulo G and for integral α coprime to g, χ0((α)) = χ0(α)v(α), where χ0(α) is the associated character of G(g) and v(α) is the same signature character as the one associated with χ. We now say the ray class character χ modulo f having the property described above with respect to g, has g as conductor, if the associated χ0 has g as conductor. Let from now on, χ be a proper ray class character modulo f. Since χ(α) = 0 for α not coprime to f, all we need to prove (88) for proper χ and for α not coprime to f is to show that the right hand side of (88) is zero. Let b be the greatest common divisor of (α) and 1 f and let g = fb− . Since χ is proper, it is not derivable from any character of G(g). In other words, there exist integers λ and coprime to f such that λ ( mod g) and χ(λ) , χ(). Otherwise, if for all integers λ and coprime≡ to f and for which λ ( mod g) it is true that χ(λ) = χ(), then we can define ≡ a character χ0 of G(g) such that for λ coprime to f, χ(λ) = χ0(λ) which is a contradiction. Thus we can find integers and ν coprime to f such that v( mod g) and χ() , χ(v). Now clearly ≡

χ() χ(λ)e2πiS (αλγ) 2πiS (αλγ) λ mod f χ(λ)e =  (89) χ(v) χ(λ)e2πiS (αλγν). λ mod f   λ mod f   Further since α αν( mod f), S (αλγ ( ν)) is a rational integer and hence ≡ − χ(λ)e2πiS (αλγ) = χ(λ)e2πiS (αλγν). λ mod f λ mod f Applications to Algebraic Number Theory 96

But since χ() , χ(ν), we see from (89), that χ(λ)e2πiS (αλγ) = 0. λ mod f Hence (88) is true for all integral α. We now proceed to prove the T , 0. In fact, using (89), we have

2πiS (αγ) TT = T χ(α)e− α mod f 2πiS (α(λ 1)γ) = χ(λ) e − (90) λ mod f α mod f Now if α runs over a complete system of representatives of residue classes 107 modulo f, so does α + ν for every integer ν and hence

e2πiS (αγ) = e2πiS (γν) e2πiS (αγ) α mod f α mod f Thus, if there exists at least one integer ν such that S (γν) is not a rational integer, e2πiS (αγ) = 0. Now S (γν) is a rational integer for all integers ν α mod f 1 1 if and only if γ ϑ i.e. if and only if ()qf− is integral i.e. f. Thus ∈ − ∈ 0, if < f, e2πiS (αγ) = N(f), if f. α mod f  ∈  From this and from (90), we have thenT 2 = N(f), i.e. T = √N(f). The determination of the exact value of T/ T | is| of the same order| | of diﬃculty as the corresponding problem for “generalized| | Gauss sums” considered by Hasse. We have, ﬁnally, for proper χ modulo f, as a consequence of (88),

1 2πiS (λβγ) χ(β) = T − χ(λ)e (91) λ mod f Let us notice that β appears in the exponent in the sum on the right-hand side of (??). We now insert the value of χ(β) as given by (??) in the series on the right hand side of (86). In view of the absolute convergence of the series for σ > 1 and in view of the fact that the sum in (93) is a ﬁnite sum, we are allowed to rearrange the terms as we like. We then obtain for σ > 1 and for a ray class character χ modulo f with f as conductor, 1 L(s, χ) = χ(λ) χ(b )(N(b ))s T A A × λ mod f A Applications to Algebraic Number Theory 97

2πiS (λβγ) s v(β)e N(β) − . (92) × | | b(β) A| In (92), λ runs over a full system of representatives of prime residue classes 108 modulo f, A over representatives of the ideal classes of K in the wide sense and the inner sum is over all principal non-zero ideals (β) divisible by bA. We shall use (92) to determine the value of L(s, χ) at s = 1 when K is a real or imaginary quadratic ﬁeld over Q. In this section, we shall consider only the case when K is an imaginary quadratic ﬁeld over Q, with discriminant D < 0. Here v(β) = 1 identically, since there is no nontrivial character of signature. Moreover, we could assume f , (1), for otherwise, L(s, χ) precisely the Dedekind zeta function of K. Let, therefore, f , (1) and let w and wf denote respectively the number of all roots of unity and of roots of unity ǫ satisfying ǫ 1( mod f). From (92), we have for σ > 1, ≡ 1 L(s, χ) = χ(λ) χ(b )(N(b ))s w T A A × λ mod f A 2πiS (λβγ) s e (N(β))− (93) × bβ,0 A|

where the inner summation is over all β , 0 in bA. We now contend that as λ runs over a full system of representatives of the prime residue classes modulo f and bA over a complete set of representatives (integral and coprime to f) of the classes in the wide sense, then (λ)bA covers exactly w/wf times, a complete system of representatives of the ray classes modulo f. In fact, let ǫ1,...,ǫv(v = w/wf) be a complete system of roots of unity in K, incongruent modulo f. The corresponding residue classes modulo f form a subgroup E(f) of G(f). Let λi, i = 1,..., r be a set of integers whose residue classes modulo f constitute a full system of representatives of the cosets of G(f) modulo E(f). Then it is easily veriﬁed that when λ runs over the elements λ1,...,λr and bA over the representatives of the classes in the wide sense, (λ)bA covers exactly once a full system of representatives of the ray classes modulo f. Our assertion above is an immediate consequence. Thus, we have from (93), for σ > 1,

1 s 2πiS (βγ) s L(s, χ) = χ(bB)(N(bB)) e (N(β))− , (94) Twf B bβ,0 B|

where B runs over the ray classes modulo f and bB is a ﬁxed integral ideal in B 109 coprime to f. Applications to Algebraic Number Theory 98

Let [β1, β2] be an integral basis of bB; we can assume, without loss of gen- β2 erality that = zB = xB + iyB with yB > 0. Then if β bB, β = mβ1 + nβ2 for β1 ∈ rational integers m, n and N(β) = N(β ) m + nz 2. Moreover, 1 × | B| N(b ) D = β β β β = 2y N(β ). B | | | 1 2 − 2 1| B 1 Thus

s 2πiS (βγ) s (N(bB)) e (N(β))− bβ,0 B| 2y s B ′ 2πiS ((mβ1+nβ2)γ) 2s = e m + nzB − , √ D | | | | m,n where, on the right hand side, the summation is over all ordered pairs of rational integers (m, n) not equal to (0, 0). Let us set now uB = S (β1γ) and vB = S (β2γ) and let f be the smallest positive rational integer divisible by f. Then in view of the fact that (γ)ϑ has exact denominator f and bB is coprime to f, it follows that uB and vB are rational numbers with the reduced common denominator f . Since f , (1), uB and vB are not simultaneously integral. We then have

s 2πiS (βγ) s (N(bB)) e (N(β))− bβ,0 B| 2 s s ′ 2πi(muB+nvB) 2s = y e m + nzB − . (95) √ D B | | | | m,n We know that the function of s deﬁned for σ > 1 by the inﬁnite series on the right hand side of (95) has an analytic continuation which is an entire function of s. Its value at s = 1 is given precisely by Kronecker’s second limit formula. Indeed, by (39), we have

2 ′ ϑ (v u Z , z ) 2 2πi(muB+nvB) 2 1 B B B B πiu zB yB e m + nzB − = π = π log − e B . m,n | | − − η(z ) B πiuBvB Inserting the factor e− of absolute value 1 on the right hand side, we have 110

2πi(mu +nv ) 2 y ′ e B B m + nz − B | B| m,n 2 ϑ1(vB uBzB, zB) πiu (u z v ) = π log − e B B B− B . (96) − η(zB) Applications to Algebraic Number Theory 99

Now we deﬁne for real numbers u, v not simultaneously integral and z H, the function ∈ πiu(uz v) ϑ1(v uz, z) ϕ(v, u, z) = e − − . η(z)

In each ray class B modulo f, we had chosen a ﬁxed integral ideal bB with β2 integral basis [β1, β2] and deﬁned uB = S (β1γ), vB = S (β2γ) and zB = . β1 Now, the left hand side of (95) depends only on the ray class B. Hence from (95) and (96), it is clear that log ϕ(v , u , z ) 2 depends only on B and not on | B B B | the special choice of bB or β1, β2. From (94), (95) and (96) we can now deduce Theorem 9. If χ is a proper ray class character modulo an integral ideal f , (1) of Q( √D), D < 0, the associated L(s, χ) can be continued analytically into an entire function of s and its value at s = 1 is given by

2π 2 L(1, χ) = χ(bB) log ϕ(vB, uB, zB) , (97) −Twf √ D | | | | B where B runs over all the ray classes modulo f and T is the sum defined by (87). We shall need (97) later for the determination of the class number of the ‘ray class field’ of Q( √D). By the ray class field (modulo f) of an algebraic number field k, we mean the relative abelina extension K0 of k, with Galois group isomorphic to the group of ray classes (modulo f) in k such that the prime divisors of f are the only prime ideals which are ramified in K0. We are now interested first in determining the nature of the numbers ϕ(vB, uB, zB). For this purpose, we observe that ϕ(v, u, z) is a regular function of 111 z in H and we shall study its behaviour when z is subjected to modular transformations and the real variables v and u undergo certain linear transformations. In fact, using the transformation formula for ϑ1(w, z) and η(z) proved earlier and the definition of ϕ(v, u, z), one easily verifies the following formulae, viz.

πiu ϕ(v + i, u, z) = e− ϕ(v, u, z), − ϕ(v, u + 1, z) = eπivϕ(v, u, z), − (98) ϕ(v + u, u, z + 1) = eπi/6ϕ(v, u, z), 1 πi/2 ϕ( u, v, z− ) = e− ϕ(v, u, z). − − Now, corresponding to a modular transformation z z = (az + b) (cz + d) 1 → ∗ − we deﬁne v∗ = av + bu and u∗ = cv + du. The last two transformation formulae Applications to Algebraic Number Theory 100 for ϕ(v, u, z) above, merely mean that corresponding to the elementary modular transformations z z = z + 1 or z z = z 1, we have → ∗ → ∗ − −

ϕ(v∗, u∗, z∗) = ρϕ(v, u, z), where ρ is a 12th root of unity. Since these two transformations generate the modular group, we see that for any modular transformation z z , → ∗

ϕ(v∗, u∗, z∗) = ρϕ(v, u, z) (98)′ where ρ = ρ(a, b, c, d) is a 12th root of unity which can be determined explicitly. Let (v, u) be a pair of rational numbers with reduced common denominator f > 1. Let us deﬁne for z H, ∈ Φ(v, u, z) = ϕ12 f (v, u, z).

Then as a consequence of the above formulae for ϕ(v, u, z), we see that Φ(v + 1, u, z) = Φ(v, u, z), Φ(v, u + 1, z) = Φ(v, u, z) and Φ(v∗, u∗, z∗) = Φ(v, u, z). If z z = (az + b)(cz + d) 1 is a modular transformation of level f , then v v → ∗ − ∗ − and u∗ u are rational integers and in view of the periodicity of Φ(v, u, z) in v and u, we− see that

Φ(v, u, z) = Φ(v∗, u∗, z∗) = Φ(v, u, z∗).

112 Let (vi, ui)i = 1, 2,..., q run over all the paits of rational numbers lying between 0 and 1 and having reduced common denominator f . Then corresponding to each pair (vi, ui), we have a function Φi(z) = Φ(vi, ui, z) which, as seen above, is invariant under modular transformations of level f . Moreover, 1 if z z∗ = (az + b)(cz + d)− is an arbitrary modular transformation, then for some→i, v v ( mod 1) and u u ( mod 1) so that in view of the periodicity ∗j ≡ i ∗j ≡ i of Φ(vi, ui, z) in vi and ui we see that

Φi(z∗) = Φ(vi, ui, z∗) = Φ(v∗j, u∗j, z∗) = Φ(v j, u j, z) = Φ j(z).

In other words, the functions Φi(z) are permuted among themselves by an arbitrary modular transformation. Now, Φi(z) is regular in H and invariant under modular transformations 2πiz/ f of level f . Moreover, Φi(z) has in the local uniformizer e at inﬁnity, a power-series expansion with at most a ﬁnite number of negative powers. Since Φi(z∗) = Φ j(z) for some j, we see that Φi(z) has at most a pole in the local uniformizers at the ‘parabolic cusps’ of the corresponding fundamental domain. Applications to Algebraic Number Theory 101

Thus Φi(z) is a modular function of level f . Since a modular transformation merely permutes the functions Φi(z), we see that any elementary symmetric function of the functions Φi(z) is a modular function. From the theory of complex multiplications, it can be shown by considering the expansions ‘at infinity’ of the functions Φi(z), that Φi(z) satisfies a polynomial equation whose coefficients are polynomials, with rational integral coefficients, in j(z) (the elliptic modular invariant). If z lies in an imaginary quadratic field K over Q, it is known that for z < Q, j(z) is an algebraic integer. Since Φi(z) depends inte- grally on j(z), it follows that for such z, Φi(z) is an algebraic integer. Actually, from the theory of complex multiplication, one may show that for z K(z < Q), ∈ Φi(z) is an algebraic integer in the ray class field modulo f over K. In particular, 12 f the numbers ϕ (vB, uB, zB) corresponding to the ray classes B modulo f in K, are algebraic integers in the ray class field modulo f, over K. Let now K0 be the ray class field modulo f over K(= Q( √D), D < 0). Let ∆, R, W and g be respectively the discriminant of K0 relative to Q, the regulator 113 of K0, the number of roots of unity in K0 and the order of the Galois group of K0 over K. Let H and h denote respectively the class numbers of K0 and K. Itis clear that K0 has no real conjugates over Q and has 2g complex conjugates over Q. Moreover ∆ = D gN(ϑ), where N(ϑ) is the norm in K/Q of the relative | | | | discriminant ϑ of K0 over K. From class field theory, we know that

ζ (s) = ζ (s) L(s, χ ) ( ) K0 K 0 ∗ χ0,1 where χ runs over all the non-principal ray class characters modulo f and if fχ is the conductor of χ, then χ0 is the proper ray class character modulo fχ associated with χ. It is known that ϑ = fχ. Multiplying both sides of ( ) χ ∗ above by s 1 and letting s tend to 1, we have − (2π)g H R 2πh = L(1, χ0). W √ ∆ w √ D | | | | χ,1 But, from (97), we have

2π 2 L(1, χ0) = χ0(bB0 ) log ϕ vB0 , uB0 , zB0 −T0wf √ D χ | | B0 where 2πiS (λγ0) T0 = T0(χ0) = χ0(λ)e , λ mod fχ Applications to Algebraic Number Theory 102

γ K is chosen such that (γ √d) has exact denominator f , B runs over 0 ∈ 0 χ 0 all the ray classes modulo fχ, bB0 is a ﬁxed integral ideal in B0 coprime to fχ and wfχ is the number of roots of unity congruent to 1 modulo fχ. Witn N(fχ) g denoting the norm in K over Q, of the ideal fχ, we have ( N(fχ)) D = ∆ . It χ | | | | is now easy to deduce Theorem 10. For the class number H of the ray class ﬁeld modulo f over K = Q( √D), D < 0, we have the formula

H W N(Fχ) 2 = χ (b ) log ϕ v , u , z  0 B0 B0 B0 B0  h w R − T0wfχ χ,1  B0      When f is a rational integral ideal in K, Fueter has derived a ‘similar’ for- 114 mula for the class number of the ray class field modulo f or the ‘ring class field’ modulo f over K, by using a “generalization” of Kronecker’s first limit formula. Unlike in the case of the absolute class field over K, it is not possible, in general, (for f , (1)) to write the product occurring in the formula for H/h χ,1 above, as a (g 1)-rowed determinant whose elements are of the form log η(k) − | i | where η(k), i = 1, 2,..., g 1 are conjugates of independent units η lying in i − i K0.

5 Ray class ﬁelds over Q( √D), D > 0.

Let K be a real quadratic field over Q, with discriminant D > 0. Let f be a given integral ideal in K and χ(, 1), a proper character of the group of ray classes modulo f in K. As in the case of the imaginary quadratic field, we shall first determine the value at s = 1 of the L-series L(s, χ) associated with χ and use this later to determine the class number of special abelian extensions of K. We start from formula (92) proved earlier, namely, for σ > 1,

1 L(s, χ) = T − χ(λ) χ(b ) A × λ mod f A s 2πiS (λβγ) s (N(b )) v(β)e N(β) − , × A | | b (β),(0) A| where A runs over all the ideal-classes of K in the wide sense, bA is a fixed integral ideal in A, chosen once for all and coprime to f and the inner sum is extended over all non-zero principal ideals (β) divisible by bA. Further, χ(λ) and v(λ) are respectively the character of G(f) and the character of signature Applications to Algebraic Number Theory 103 associated with the ray class character χ as in Proposition 14. Besides, γ is a q number in K such that (γ √C) has exact denominator f i.e. (γ) = where f( √D) (q, f) = (1); the number γ will be fixed throughout this section. Let, for α K, α′ denote its conjugate. We may, in the above, suppose ∈ β2 that [β1, β2] is a fixed integral basis of bA such that if ω = , then, without β1 loss of generality, ω>ω . We have, then, (ω ω) N(β ) = N(b ) √D. 115 ′ − | 1 | A Further, let uA = S (λβ1γ) and vA = S (λβ2γ); then, if β = mβ1 + nβ2, we have e2πiS (λβγ) = e2πi(muA+nvA). The character of signature v(λ) associated with the ray class character χ may, in general, be one of the following, namely, for λ , 0 in K, (i) v(λ) = 1 N(λ) (ii) v(λ) = N(λ) | | λ (iii) v(λ) = λ | | λ (iv) v(λ) = ′ . λ | ′| In what follows, we shall be concerned only with such ray class characters χ, for which the corresponding v(λ) is defined by (i) or (ii). We shall not deal with the characters χ, for which either (iii) or (iv) occurs, the determination of L(1, χ) being quite complicated in these cases. Let us first suppose that the character of signature v(λ) associated with χ is given by v(λ) = 1 for all λ , 0 in K. In other words, for an integer α in K, coprime to f, χ((α)) = χ(α). We may, moreover, suppose that f , (1), since otherwise, χ is a character of the wide class group and L(s, χ) is just the corresponding Dedekind zeta function of K. Let z = x + iy be a complex variable with y > 0. Corresponding to an ideal class A of K in the wide sense and an integer λ coprime to f, we define for σ > 1, the function

s 2πi(mu +nv ) 2s g(z, s,λ, u , v ) = y ′ e A A m + nz − A A | | m,n where the summation is over all pairs of rational integers m, n not simultaneously zero. As a function of z, g(z, s,λ, uA, vA) is not regular but as a function of s, it is clearly regular for σ > 1. In fact, since uA and vA are not both integral, we know that g(z, s,λ, uA, vA) has an analytic continuation which is an Applications to Algebraic Number Theory 104

entire function of s. For the sake of brevity, we shall denote it by gA(z,λ); this does not mean that it is a function of z and λ depending only on the class A. It does depend on the choice of the ideal bA, the integral basis [β1, β2], the residue 116 class of λ modulo f and further on γ. But we have taken bA, β1, β2 and γ to be fixed. In order to find L(1, χ), we shall employ the idea of Hecke already referred 2πiS (λβγ) s to on p. 86 ( 3, Chapter II) and express the series e N(β) − as § b (β),(0) | | A| an integral over a suitable path in H, with the integrand involving gA(z,λ). We shall denote the group of all units in K by Γ and the subgroup of units congruent to 1 modulo f, by Γf. Moreover, let Γf∗ be the subgroup of Γf consisting of ρ such that ρ > 0. The group Γf∗ is infinite cyclic and let ǫ be the generator of Γ , which is greater than 1. We see that ǫ 1( mod f), N(ǫ) = 1, f∗ ≡ ǫ>ǫ′ > 0. Since [ǫβ1, ǫβ2] is again an integral basis of bA, ǫβ2 = aβ2+bβ1, ǫβ1 = cβ2+ dβ1 where a, b, c, d are rational integers such that ad bc = N(ǫ) = 1. Further, − 1 if ω = β2/β1, then ǫω = aω + b, ǫ = cω + d so that ω = (aω + b)(cω + d)− . We also have the same thing true for ω′, namely, ǫω′ = aω′ + b, ǫ′ = cω′ + d, 1 so that we have ω′ = (aω′ + b) (cω′ + d)− . Thus the modular transformation z z = (az+b) (cz+d) 1 is hyperbolic, with ω, ω as fixed points and it leaves → ∗ − ′ fixed the semicircle on ω′ω as diameter. If now we introduce the substitution 1 1 z = (ωpi + ω′)(pi + 1)− (or equivalently, po = (z ω′)(ω z)− we see that when z( H) lies on this semicircle, p is real and positive.− As− a matter of fact, ∈ when z( H) describes the semicircle form ω′ to ω, p runs over all positive real ∈ 1 numbers from 0 to . Let p∗i = (z∗ ω′) (ω z∗)− ; it is easy to verify that 2 ∞ − − p∗ = ǫ p. This means that p∗ > p, since ǫ > 1. Consequently, whenever z lies on the semicircle on ω′ω as diameter, z∗ again lies on the semicircle and always to the right of z. Consider now

z 0∗ dp F(A,λ, s) = g(z, s,λ, uA, vA) , z0 p the integral being extended over the arc of this semicircle, from a ﬁxed point 1 z0 H to z0∗ = (az0 + b)(cz0 + d)− . In view of the uniform convergence of the ∈ s 2πi(mu +nv ) 2s series y ′ e A A m + nz on this zrc we have, for σ > 1, m,n | |− z0∗ z0∗ dp 2πiS (λβγ) s 2s dp g (z,λ) = ′ e y m + nz − , A P | | P z0 m,n z0 where β = mβ1 + nβ2. Setting = m + nω = β/β1, we verify easily that 117 Applications to Algebraic Number Theory 105 m + nz 2 = (2 p2 + 2)(p2 + 1) 1 and y = p(ω ω )(p2 + 1) 1. Thus | | ′ − − ′ −

β K, all the integers in K which are associated with β with respect to Γf∗ are ∈ k l of the form βǫ± , k = 0, 1, 2,.... Keeping β ﬁxed, if we replace β by βǫ in the integral on the right hand side of (99), then the integral goes over into

ﬁxed integer β with respect to Γf∗, is given by

e2πiS (λβγ) ps dp e2πiS (λβγ) Γ2(s/2) ∞ = N(β) s (p2 + 1)s p N(β) s 2Γ(s) | | 0 | | As a consequence, 118

s 2 (N(bA) √D) Γ (s/2) 2πiS (λβγ) s F(A,λ, s) = e N(β) − (100) 2Γ(s) | | b β(Γ ) A| f∗ where, on the right hand side, the summation is over a complete set of integers

β , 0 in bA, which are not associated with respect to Γf∗. Applications to Algebraic Number Theory 106

Let ρ denote a complete set of units incongruent modulo f and let ǫ be { i} { j} a complete set of representatives of the cosets of Γf modulo Γf∗. It is clear that ρ ǫ runs over a full set of representatives of the cosets of Γ modulo Γ . Thus { i j} f∗ 2 Γ (s/2) s 2πiS (λρ ǫ βγ) s F(A,λ, s) = (N(b ) √D) e i j N(β) − (101) 2Γ(s) A | | ρi,ǫ j b(β) A| where, on the right hand side, the inner sum is over all non-zero principal ideals (β) divisible by bA. By the signature of an element α , 0 in K, we mean the pair (α/ α , α / α ). | | ′ | ′| Now we can certainly ﬁnd a set l of integers l such that l 1( mod f) and the set ǫ consists precisely{ of} 4 elements with the 4 possible≡ diﬀerent { j l} signatures. Let λk be a set of integers λk constituting a complete system of representatives of{ the} cosets of G(f) modulo E(f). Here E(f) is the group of prime residue classes mod f, containing at least one unit in K. The set λkρi is seen to be a complete set of representatives of the prime residue classes{ } modulo f. It is easy to prove

Proposition 15. The set of ideals (λkl)bA serves as a full system of representatives of the ray classes modulo{ f. }

Proof. Obviously it suﬃces to show that (λkl) runs over a complete set of representatives of the ray classes modulo f{lying in} the principal wide class. In fact, if (α) Gf, then α ρiλk( mod ∗f) for some ρi and λk and moreover, 119 ∈ ≡ α we can ﬁnd ǫ jl such that > 0. Since ǫ jl 1( mod f), we have λkρiǫ jl ≡ (α) (λ ρ ǫ ) = (λ modulo f. It is easy to verify that no two elements of ∼ k l i j k l the set (λkl) are equivalent modulo f. From{ (101),} we then have for σ > 1

χ(λkl)χ(bA)F(A,λkl, s)

λk,l,A Γ2(s/2) = χ(λ )χ(b )(N(b ) √D)s 2Γ(s) k l A A × λk,l,A 2πiS (λ ρ ǫ βγ) s e k l i j N(β) − × | | bA (β) ρi|,ǫ j Γ2(s/2) = χ(λ )χ(b )(N(β ) √D)s 2Γ(s) k A A × λk,ρi,A 2πiS (λ ρ ǫ βγ) s χ( ) e k l i j N(β) − . (102) × l | | ǫ j,l b(β) A| Applications to Algebraic Number Theory 107

1( mod f), e2πiS (λklρiǫ jβγ) = e2πiS (λkρiβγ) and since 1( mod f), χ( ) = 1. ≡ l ≡ l Hence the sum in (102) is independent of ǫ j, l and we have

χ(λkl)χ(bA)F(A,λkl, s)

λk,l,A Γ2(s/2) = 4 Ds/2 χ(λ ρ )χ(b )(N(b ))s 2Γ(s) k i A A × λk,ρi,A 2πiS (λ ρ βγ) s e k i N(β) − . | | b (β),(0) A|

Let (λkl)bA lie in the ray class B modulo f. Denoting (λkl)bA by bB, we know that bB runs over a full system of representatives of the ray classes modulo f. Also, if α1 = λklβ1, α2 = λklβ2, then [α1, α2] is an integral basis 120 of bB and we may now denote the function

s 2πiS ((mα +nα )γ) 2s g (z,λ ) = y ′ e 1 2 m + nz − A k l | | m,n

z0∗ dp by gB(z, s) and gB(z, s) by F(B, s). z0 p The use of the notation F(B, s) is justiﬁed for, from (101), we see that F(B, s) depends only on the ray class B modulo f (and on γ) and not on the particular integral ideal bB chosen in B. For, if we replace bB by ()bB where > 0, 1( mod f) and ()b is an integral ideal coprime to f, then it is ≡ ∗ B easy to show that S (λklρiǫ jβγ) S (λklρiǫ jβγ) is a rational integer and in addition, (N(b ))s N(β) s again depends− only on the ideal class of b . A | |− A Now χ((λkl))χ(bA) = χ(bB). Moreover, λkρi runs over a complete set of representatives λ of the prime residue classes{ modulo} f. As a consequence, we have Γ2(s/2)Ds/2 χ(b )F(B, s) = 2 χ(λ) B Γ(s) × B λ mod f s s χ(b )(N(b )) (λβγ) N(β) − × A A ∗∗∗∗∗∗ | | A b(β) A| where B runs over all ray classes moudlo f ******** a complete system of prime residue classes modulo f and A ***** ideal classes in the wide sense. In other words, we have for σ **********,

1 s/2 χ(b )F(B, s) = (s/2)(Γ(s))− D T L(s, χ). (103) B ∗∗∗∗∗ B Applications to Algebraic Number Theory 108

The function gB(z, s) is an Epstein zeta-function of the type of ζ(s, u, v, z, 0) discussed earlier in 5, Chapter I. Using the simple functional equation for the Epstein zeta function§ ζ(s, u, v, z, 0), we shall now derive a functional equation for L(s, χ). Proposition 16. If χ is a proper ray class character modulo f(, (1)) whose associated character of signature v(λ) 1, then L(s, χ) is an entire function of s satisfying the following functional≡ equation, namely, if ξ(s, χ) = s 2 s/2 π− Γ (s/2)(DN(F)) L(s, χ), then χ(q) ξ(s, χ) = ξ(1 s, χ), T/ √N(f) − 121 Remark. The functional equation ( ) for ****** generalization to arbitrary algebraic number **** derived by Hecke∗ by using the generalized **** formula’ (for algebraic number ****** same, it is interesting to deduce the same **** by using the functional equation satisﬁed by ζ(s, u, v, z, 0). Proof. From (92), we have

s 2 s/2 s 2 s/2 s π− Γ (s/2)D T L(s, χ) = π− Γ (s/2)D χ(λ) χ(bA)(N(bA)) λ A 2πiS (λβγ) s e N(β) − × | | b (β),0 A| π sΓ2(s/2)Ds/2 = − χ(λ) χ(b )(N(b ))s e(f) A A × λ A

× b β(Γ ) A| f∗

vA where e(f) is the index of Γ∗ in Γ. Now, if we set u∗ = − , then, in the notation f uA of 5, Chapter I, we have g (z, s,λ, u , v ) = ζ(s, u ,0, z, 0) and by (100) § A A A ∗ s 2 s/2 2 π− Γ (s/2)D T L(s, χ) = χ(λ) χ(b ) e(f) A × λ A

z0∗ s dp π− Γ(s)ζ(s, u∗, 0, z, 0) . × z0 p

By the method of analytic continuation of ζ(s, u∗, 0, z, 0) discussed earlier in 5, Chapter I, we have §

z0∗ s dp π− Γ(s)ζ(s, u∗, 0, z, 0) p Applications to Algebraic Number Theory 109

z∗ 0 dp ∞ 1 2 dt = ′ πiy− m+nz +2πi(muA+nvA) s  e− | |  t . z p 0 t 0 m,n     1  ∞ ... + ∞ ... 122 The inner integral 0 may be split up as 0 1 . Now we may use πy 1t m+nz 2+2πi(mu +nv ) the theta-transformation formula for the series e− − | | A A and m,n further make use of the inequality y 1 m + nz 2 c(m2 + n2) uniformly on the − | | ≥ arc from z0 to z0∗, for a constant c independent of m and n. If, in addition, we keep in mind that χ is not the principal character, we can show that L(s, χ) is an entire function of s.

Using now the functional equation of ζ(s, u∗, 0, z, 0), we have

s 2 s/2 2 π− Γ (s/2)D T L(s, χ) = χ(λ) χ(b ) e(f) A × λ A

z0∗ (1 s) dp π− − Γ(1 s)ζ(1 s, 0, u∗, z, 0) . × z0 − − p

1 s 2(1 s) For σ < 0, ζ(1 s, 0, u∗, z, 0) = y − m vA + (n + uA)z − − and by the − m,n | − | same arguments as above, we can show that

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ 2 1 s 1 s (1 s) = Γ − (N(bA) √D) − N() − − 2 | | =β+ν β b ∈ ′A √ where, on the right hand side, = β1′ S (λβ2γ) + β2′ S (λβ1γ) = DλγN(bA) and runs over a complete set of numbers− of the form β + ν with−β b , such ∈ ′A that they are not mutually associated with respect to Γf∗. Let δ be an integer in K such that af = (δ) with an integral ideal a coprime to f. Then, for σ < 0, we obtain from above that

(1 s) s 2 s/2 π− − (1 s)/2 2 π− Γ (s/2)D T L(s, χ) = D − Γ ((1 s)/2) e(f) − × 1 s 1 s χ(b )(N(b )) − (N(af)) − χ(λ) × A A × A λ (1 s) N(β) − − , × | | β νδ( mod ab′ f) ≡ A β,0,ab′ β(Γ ) A| f∗ Applications to Algebraic Number Theory 110 where, in the inner sum on the right hand side, β runs over a complete set of 123 integers in ab′A which are not mutually associated with respect to Γf∗ and which satisfy β νδ( mod ab′ f). If β νδ mod ab′ f, then ≡ A ≡ A χ(β) = χ(νδ) = χ( γ √DλδN(b )) = χ((N(b )γ √Dλδ)) − A A √ = χ(bA)χ(b′A)χ((γ Dδ))χ(λ) = χ(bA)χ(b′A)χ(aq)χ(λ).

Hence χ(λ) = χ(β)χ(bAb′Aaq). Further if λ runs over representatives of prime residue classes modulo f and β runs independently over a complete set of integers in ab′ congruent modulo ab′ f to √DλγδN(b ) and not mutually associ- A A − A ated with respect to Γf∗, then β runs over a complete set of integers in ab′A not mutually associated with respect to Γf∗ and satisfying ((β), ab′Af) = ab′A, If we 1 note, in addition, that χ(β) = 0 for those β in ab′A for which (β)(ab′A)− is not coprime to f, we can show ﬁnally that

(1 s) χ(λ) N(β) − − = χ(b b′ aq) | | A A × ab′β(Γ ) A| 1∗ β νδ( mod ab′ F) ≡ A (1 s) χ(β) N(β) − − . × | | ab′β(Γ ) A| f∗ β,0 Thus, for σ < 0, π (1 s) π sΓ2 / s/2 , χ = − − (1 s)/2Γ2 / χ q − (s 2)D T L(s ) s 1 D − ((1 s) 2) ( ) (N(f)) − − × 1 s χ(ab′ )N((ab′ )) − × A A × A (1 s) χ(β) N(β) − − | | ab′ (β),(0) A| χ(q) = π (1 s) (1 s)/2Γ2 / , χ , − − D − ((1 s) 2) s 1 L(1 s ) − (N(f)) − − since ab′A again runs over a system of representatives of the ideal classes similar to bA. Since L(s, χ) is an entire function of s, the above functional relation is 124 s s s/2 clearly valid for all s. We have now only to set ξ(s, χ) = π− Γ (s/2)(DN(f)) L(s, χ), to see that the equation ( ) is true. ∗ Let now [α1, α2] be an integral basis for the ideal bB in the ray class B and let ω = α2/α1, uB = S (α1γ), vB = S (α2γ). By Kronecker’s second limit formula, we have g (z, s) = π log ϕ(v , u , z) 2 B − | B B | Applications to Algebraic Number Theory 111

+ terms involving higher powers of (s 1) .... − Letting s tend to 1, we have from (103),

z 1 0∗ dp L(1, χ) = χ(b ) log ϕ(v , u , z) 2 . − B | B B | p 2T √D B z0 Now it is easy to verify that

dp (ω ω ) √D = − ′ dz = dz, p −(z ω)(z ω ) − F (z) − − ′ B where √D 2 F (z) = (z ω)(z ω′) = a z + b z + c ( ) B ω ω − − 1 1 1 ∗ − ′ has the property that a1, b1, c1 are rational integers with the greatest common divisor 1, a > 0 and b2 4a c = D. Thus we have 1 1 − 1 1 Theorem 11. For a proper ray class character modulo f , (1) in Q( √D), D > 0, with the associated v(λ) 1, the value of L(s, χ) at s = 1 is given by ≡ z 1 0∗ dz L(1, χ) = χ(b ) log ϕ(v , u , z) 2 , 2T B | B B | F (z) B z0 B where B runs over all the ray classes modulo f and F (z) is given by ( ) above. B ∗ As remarked earlier, the terms of the sum on the right hand side depend only on γ (fixed!) and on the ray class B and not on the choice of the ideal bB in B. We observe further that it does not seem to be possible to simplify this 125 formula any further, in an elementary way. One might try to follow the method of Herglotz to deal with the integrals but then the invariance properties of the integrand are lost in the process. We proceed to discuss the case when the character of signature v(λ) associated with the given ray class character χ is of type (ii), i.e. for integral α(, 0) N(α) coprime to f, we have χ((α)) = χ(α) . In this case, we shall now see N(α) that the value at s = 1 of L(s, χ) can be| determined| in terms of elementary functions, using the first or second limit formula of Kronecker, according as f = (1) or f , (1). The fact that the character of signature v(λ) associated with χ is defined by N(λ) v(λ) = for λ , 0 (104) N(λ) | | Applications to Algebraic Number Theory 112 implies automatically a condition on f. For instance, if ρ is a unit congruent to N(ρ) 1 modulo, f, then = χ(ρ)v(d) = χ((ρ)) = 1. In other words, every unit N(ρ) congruent to 1 modulo| f| should necessarily have norm 1. To determine L(1, χ), we start as before from formula (92), namely, for σ > 1,

1 s L(s, χ) = T − χ(λ) χ(b )(N(b )) A A × λ mod f A

2πiS (λβγ) N(β) s e N(β) − . × N(β) | | b (β),(0) A| | | We shall try to express the inner sum on the right hand side as an integral, as before. We shall follow the same notation as in the case treated above. ∂ 1 ∂ i ∂ Let = ; then in view of the uniform convergence of the ∂z 2 ∂x − 2 ∂y series deﬁning the function gA(z, s,λ, uA, vA) (denoted by gA(z,λ) for brevity) for σ > 1, we have

∂gA(z,λ) 2πi(mu +nv ) ∂ s 2s = ′ e A A y m + nz − ∂z ∂ { | | } m,n z s s 1 2πi(mu +nv ) (2s 2) 2 = y − ′ e A A m + nz − − (m + nz)− . (105) 2i | | m,n 126 z ∂g (z,λ) Let us now consider the integral 0∗ A dz, extended over the same z0 ∂z arc of the semi-circle in H as considered earlier. Due to the uniform convergence of the series (105) on this arc, we have

z∗ z∗ s 1 0 ∂gA(z,λ) s 0 y − dz = ′ e2πi(muA+nvA) dz ∂z 2i m + nz 2s 2(m + nz)2 z0 m,n z0 | | − s = Ds/2(N(b ))s 2 A × p0∗ s 1 2πiS (λβγ) s p − dp e N(β1) − . × | | (2 p2 + 2)s 1(pi + )2 bβ,0 p0 ′ − ′ A| Eﬀecting the substitution p / p, we see that → | ′ | p 0∗ ps 1 − dp 2 2 2 s 1 2 p0 ( p + ′ ) − (pi + ′) Applications to Algebraic Number Theory 113

z0∗ ∂gA(z,λ) s s/2 s dz = D (N(bZ)) z0 ∂z 2 × (ββ /β β ) p ǫ2 1′ ′ 1 0 ps 1 2πiS (λβγ) β s | | − . e N( ) − 2 s 1 2 dp × | | (ββ /β β ) p (p + 1) (pi + τ) bβ,0 1′ ′ 1 0 − A| | | Now, since v(ǫ) = 1, the value of τ corresponding to β and βǫk(k = 1, 2,...) 127 is the same. Moreover e2πiS (λβγ) = e2πiS (λβǫkγ). If then we apply± the± same analysis as in the former case, we obtain

z0∗ ∂gA(z,λ) s s/2 s dz = D (N(bA)) z0 ∂z 2 × s 1 2πiS (λβγ) s ∞ p − dp e N(β) − (106) × | | (p2 + 1)s 1(pi + τ)2 b β(Γ ) 0 − A| f∗ where, on the right hand side, the summation is over a complete set of integers

β , 0 in bA, which are not associated with respect to Γf∗. Eﬀecting the transformation p p 1, we see that → − ps 1 ps 1 ∞ − dp = ∞ − dp. (p2 + 1)s 1(τ + ip)2 − (p2 + 1)s 1(τ ip)2 0 − 0 − − Taking the arithmetic mean, we have

s 1 ∞ p − 2 s 1 2 dp 0 (p + 1) − (τ + ip) 1 ps 1 1 1 = ∞ − dp 2 (p2 + 1)s 1 (τ + ip)2 − (τ ip)2 0 − − ps+1 dp = τ ∞ 2 i 2 s+1 − 0 (p + 1) p v(β)v(β ) Γ2((s + 1)/2) = 1 . (107) i Γ(s + 1) Applications to Algebraic Number Theory 114

From (106) and (107), we obtain for σ > 1,

z0∗ s/2 2 ∂gA(z,λ) D Γ ((s + 1)/2) s v(β1) dz = (N(bA)) z0 ∂z 2i Γ(s) × 2πiS (λβγ) s v(β)e N(β) − . × | | b β(Γ ) A| f∗

Since ρ ǫ is a system of representatives of Γ modulo Γ , we have 128 { i j} f∗

z0∗ s/2 2 ∂gA(z,λ) D Γ ((s + 1)/2) s v(β1) dz = (N(bA)) z0 ∂z 2i Γ(s) × 2πiS (λβρ ǫ γ) s v(βρ ǫ )e i j N(β) − , (108) × i j | | bA (β) ρi|,ǫ j where the summation on the right hand side is extended over all non-zero principal ideals divisible by b and over the ﬁnite sets of representatives ρ and A { i} ǫ j . Summing over all ideal classes A and the elements of the sets λk and l we{ } obtain from (108), { } { }

z∗ 0 ∂gA(z,λkl) χ(λkl)χ(bA)v(β1) dz z0 ∂z λk,l,A Ds/2 Γ2((s + 1)/2) = χ(λ )χ(b )(N(b ))s 2i Γ(s) k l A A × λk,l,A 2πiS (λ βρ ǫ γ) s v(βρ ǫ )e k 1 i j N(β) − . (109) × i j | | bA (β) ρi|,ǫ j

It is quite easy to verify that

v(ρiǫ j) = χ((ρiǫ j))χ(ρiǫ j) = χ(ρiǫ j) = χ(ρi), e2πiS (λklβρiǫ jγ) = e2πiS (λkρiβγ).

Using these facts and applying the same arguments as in the earlier situation, we see that the right hand side of (109) is precisely

Γ2((s + 1)/2) 2iDs/2 χ(λ) χ(b )(N(b ))s − Γ(s) A A × λ mod f A 2πiS (λβγ) s v(β)e N(β) − × | | b(β) A| Applications to Algebraic Number Theory 115

Γ2((s + 1)/2) = 2iDs/2 T L(s, χ). − Γ(s)

To simplify the expression on the left hand side of (109), we first observe 129 that χ(λk1) = χ((λkl))v(λkl). Moreover, if we denote, as before, the ideal (λkl)bA by bB where B is the ray class modulo f containing it, then, by Propo- sition 15, bB runs over a full system of representatives of ray classes modulo f. We may assume, without risk of confusion, that [β1, β2] is still an integral basis of bB and set uB = S (β1γ), vB = S (β2γ). Further we may denote s 2πi(muB+nvB) 2s gA(z, s,λkl, uA, vA) = y ′ e m + nz − by gB(z) = gB(z, s). The m,n | | function gB(z, s) depends not just on the ray class B but also on the choice of the ideal bB in B and on the integral basis [β1, β2] of bB. Since bB and [β1, β2] are fixed, the modular transformation z (az + b)(cz + d) 1 is uniquely deter- → − mined by ǫβ2 = aβ2 + bβ1, ǫβ1 = cβ2 + dβ1; ω = β2/β1, ω′ = β2′ /β1′ are the two 1 fixed points of this transformation and z0∗ = (az0 + b)(cz0 + d)− . From (109), we have now, for σ > 1,

/ z iD s 2 Γ(s) 0∗ ∂g (z) L(s, χ) = − χ(b )v(β ) B dz. (110) 2T Γ2((s + 1)/2) B 1 ∂z B z0

1 z0∗ ∂gB(z) Let us denote v(β1) dz by G(B, s). We see from (108) that, −2π2i z0 ∂z for σ > 1,

Ds/2Γ2((s + 1)/2) G(B, s) = (N(b ))s 4π2Γ(s) B × 2π √ 1S (βρ γ) s v(βρ )e − i N(β) − . × i | | ρi,ǫ j b (β),(0) B|

Obviously, the right hand side remains unchanged, when bB is replaces by an- 130 other integral ideal in B. Thus, G(B, s) clearly depends only on B (and on γ and s) but not on the special choice of bB or its integral basis. Writing χ(B) for χ(bB), we see that (110) goes over into

π2 Γ(s) L(s, χ) = χ(B)G(B, s). (111) T Ds/2 ((s + 1)/2) B ∂g (z, s) The function B is essentially an Epstein zeta-function of the type ζ(s, u, v, Q, P) ∂z z ∂g (z, s) discussed in 5, Chapter I and it can be shown as before that 0∗ B dz § z0 ∂z Applications to Algebraic Number Theory 116 is an entire function of s, by using the method of analytic continuation of the Epstein zeta-function. Thus formula (110) gives the analytic continuation of L(s, χ) into the entire s-plane, as an entire function of s. Moreover, if we use ∂g (z, s) the functional equation satisﬁed by B , we can show as in Proposition ∂z 16 that L(s, χ) satisﬁes the following functional equation, viz. if we set s 2 s/2 Λ(s, χ) = π− Γ ((s + 1)/2)(DN(f)) L(s, χ), then χ(q)v(γ √D) Λ(s, χ) = Λ(1 s, χ). (112) T/ √N(f) − Let us observe that the constant on the right hand side is of absolute value 1. In order to determine the value of L(1, χ), we distinguish between the two cases f , (1) and f = (1).

(i) Let first f , (1). Then the numbers uB and vB corresponding to a ray class B modulo f are rational numbers which are not both integral. For the function gB(z, s) we have the following power-series expansion at 131 s = 1, by Kronecker’s second limit formula, namely g (z, s) = π log ϕ(v , u , z) 2 B − | B B | (terms involving higher powers of (s 1)). − Applying ∂/∂z to gB(z, s) and noting that (∂/∂z) log ϕ(vB, uB, z) = 0, we ∂g (z, s) have for B the power-series expansion, ∂z ∂g (z, s) ∂ B = π log ϕ(v , u , z) ∂z − ∂z B B + (terms involving higher powers of (s 1)). − Now since ϕ(vB, uB, z) is regular and non-vanishing in H, we can choose a fixed branch of log ϕ(vB, uB, z) in H and (∂/∂z) log ϕ(vB, uB, z) is just (d/dz) log ϕ(vB, uB, z). The coefficients of the higher powers of (s 1) might involve z. Letting s tend to 1, we have −

v(β ) z G(B, 1) = 1 log ϕ(v , u , z) 0∗ 2πi B B z0 We shall denote G(B, 1) by G(B) and then we have, from (111) π2 L(1, χ) = χ(B)G(B). T √D B Applications to Algebraic Number Theory 117

(ii) Let now f = (1). Then, for all B, we see that g (z, s) = ys m + nz 2s B m,n | |− and by Kronecker’s ﬁrst limit formula, we have the following power- series expansion for gB(z, s) at s = 1, namely π g (z, s) = 2π(C log 2) π log(y η(z) 4) + . B − s 1 − − | | − The application of ∂/∂z does away with the terms π/(s 1) and 2π(C log 2). Noting that − − ∂ 1 1 ∂ log y = = and (η(z))2 = 0, ∂z 2iy z z ∂z − ∂gB(z, s) we have for , the following power-series expansion at s = 1, 132 ∂z namely

∂g (z, s) π d B = π log(η(z))2+ ∂z −z z − dz − + terms involving higher powers of (s 1). − We wish to replace 1/(z z) if possible by a regular function of z; but we − shall see now that we can do this when z lies on the semi-circle on ω′ω as diameter. In fact, the equation to the semi-circle on ω′ω as diameter is z ω z ω − + − = 0, z h. z ω z ω ∈ − ′ − ′ ω + ω This means that zz + p(z + z) + q = 0 with p = ′ , q = ωω , when − 2 ′ z lies on the semi-circle. But then 1 z + p = z z z2 + 2pz + q − 1 d = log(z2 + 2pz + q) 2 dz d = log (z ω)(z ω ). dz − − ′ Hence, for z on the semi-circle, we have the power-series expansion ∂g (z, s) d B = π log (z ω)(z ω )η2(z)+ ∂z − dz − − ′ + (higher powers of s 1). − Applications to Algebraic Number Theory 118

Thus letting s tend to 1,

z v(β1) 2 0∗ G(B, 1) = log (z ω)(z ω′)η (z) 2πi − − z0 2 for a ﬁxed branch of log( √(z ω)(z ω′)η (z)). Denoting G(B, 1) by G(B) again, we have from (111),− −

π2 L(1, χ) = χ(B)G(B). √D B

We are thus led to 133 Theorem 12. For a proper ray class character χ(, 1) modulo f in Q( √D)(D > 0), with associated v(λ) deﬁned by (104), we have

π2 L(1, χ) = χ(B)G(B) T √D B where the summation is over all ray classes B modulo f and

v(β ) z 1 log ϕ(v , u , z) 0∗ , for f , (1), 2πi B B z0 G(B) =  (113) v(β ) z∗  1 2 0  log( √(z ω)(z ω′)η (z)) , for f = (1). 2πi z0  − −  We are now interested in the explicit determination of the values of G(B) corresponding to the various ray classes B, in terms of elementary arithmetical functions. The expressions inside the square brackets in (113) represent analytic functions of z and we know that G(B) itself does not depend on the point z0 chosen on the semi-circle on ω′ω as diameter. Thus G(B) is, in both cases, the value of an analytic function z, which is a constant on the semi-circle on ω′ω as diameter; as a consequence, G(B) is a constant independent of z0 and in order to calculate G(B), we might replace z0 in (113) by any point z in H, not necessarily lying on the semi-circle on ω′ω as diameter. z∗ First we observe that (1/2πi)[log ϕ(vB, uB, z)]z is a rational number with at most 12 f in the denominator, where f is the smallest positive rational integer 1 divisible by f. This can be verified as follows. Let z∗ = (az + b)(cz + d)− where the rational integers a, b, c, d satisfying the condition ad bc = 1 are − uniquely determined by the unit ǫ and a fixed integral basis [β1, β2] of a fixed integral ideal bB in B, coprime to f. Corresponding to z∗, let v∗B = avB + buB and u∗B = cvB + duB. We know already from the properties of ϕ(vB, uB, z) that Applications to Algebraic Number Theory 119

th ϕ(vV∗ , u∗B, z∗) = ρϕ(vB, uB, z) where ρ is a 12 root of unity depending only on a, b, c, d. On the other hand v∗B = aS (β2γ) + bS (β1γ) = S (ǫβ2γ) and hence v∗B vB = S ((ǫ 1)β2γ) is a rational integer k, say; similarly, u∗B uB is again a rational− integer− l, say. But we know from (99) that −

ϕ(v∗B, u∗B, z∗) = ϕ(vB + k, uB + l, z∗) = τϕ(vB, uB, z∗), where τ isa2 f -th root of unity. Hence 134

ϕ(vB, uB, z∗) = θϕ(vB, uB, z), where θ is a 12 f -th root of unity. Choosing a fixed branch of log ϕ(vB, uB, z), 1 we see that [log ϕ(v , u , z)]z∗ is a rational number with at most 12 f in the 2πi B B z 12 f denominator. Now, we know that for z H, the function ϕ (vB, uB, z) is ∈ z∗ a modular function of level f and [log ϕ(vB, uB, z)]z is just a ‘period’ of the abelian integral log ϕ(vB, uB, z). The explicit computation of these ‘periods’ of the abelian integral log ϕ(vB, uB, z) has been essentially considered by Hecke, in connection with the determination of the class number of a biquadratic field obtained from a real quadratic field k over Q, by adjoining the square root of a “totally negative” number in k; employing the ideal of the well-known Riemann-Dedekind method, Hecke used for this purpose, the asymptotic behaviour of log ϑ11(w, z) as z tends to infinity and z∗ to the corresponding ‘rational point’ a/c on the real axis. It is to be remarked, however, that from Hecke’s considerations, one can at first sight conclude only that the quantities 1 [log ϕ(v , u , z)]z∗ are rational numbers with at most 24 f 2 in the denomina- 2πi B B z tor. 1 Regarding [log √(z ω)(z ω )η2(z)]z∗ , we can conclude again that 2πi − − ′ z they are rational numbers with at most 12 f in the denominator. For this purpose, we notice that

z ω z ω′ z∗ ω = − and z∗ ω′ = − − ǫ(cz + d) − ǫ′(cz + d) 2 2 th and η (z∗) = ρ(cz + d)η (z), ρ being a 12 root of unity depending only on a, b, c, d. Hence

2 2 (z ω)(z ω )η (z∗) = ρ (z ω)(z ω )η (z). ∗ − ∗ − ′ − − ′ 2 Choosing a ﬁxed branch of log( √(z ω)(z ω′)η (z)) in H, we see that our assertion is true. − − Applications to Algebraic Number Theory 120

We may now proceed to obtain the value of G(B), first for f , (1). We make a few preliminary simplifications. We can suppose without loss 135 of generality that 0 uB, vB < 1, since, in view of (98), ϕ(vB, uB, z) merely picks up a 2 f th root of≤ unity as a factor when v is replaced by v 1 or u by B B ± B u 1 and this gets cancelled, when we consider [log ϕ(v , u , z)]z∗ . Moreover, B ± B B z since we know already that G(B)v(β1) is real (in fact, even rational) it is enough 1 to find the real part of [log ϕ(v , u , z)]z∗ . Now 2πi B B z

πiu(uz v) πi/2+πiz/6 πi(v uz) πi(v uz) ϕ(v, u, z) = e − − (e − e− − ) − × ∞ 2πi(v uz)+2πimz ∞ 2πi(v uz)+2πimz (1 e − ) (1 e− − ) × − − m=1 m=1 πiu(uz v) πi/2+πiz/6+πi(v uz) = e − − − × ∞ m u ∞ 1 m+u (1 VQ − ) (1 V− Q ), × − − m=1 m=0

2πiv 2πiz u 2πiuz where we have set V = e , Q = e and Q− = e− . Moreover, if u = 0, then 0 < v < 1 and in the second inﬁnite product above we notice that m+u 0, in any case. We deﬁne the branch of (1/2πi) log ϕ(v, u, z) by ≥ 1 z 1 1 v log ϕ(v, u, z) = u2 u + + (1 u)+ 2πi 2 − 6 − 4 2 −

1 ∞ m u + log(1 VQ − )+ 2πi − m=1

1 ∞ 1 m+u + log(1 V− Q ), 2πi − m=0 where, on the right hand side, we take the principal branches of the logarithms. (V 1Qu)n Noting that the series ∞ − converges even if u = 0, since then 0 < v < n=1 n m u n 1 m+u n ∞ (VQ − ) ∞ ∞ (V− Q ) 1 and further the series n∞=1 and converge m=1 n m=1 n=1 n absolutely, we conclude that 1 1 1 1 v log ϕ(v, u, z) = u2 u + z + (1 u) 2πi 2 − 6 − 4 2 − − u n n 1 u n 1 ∞ 1 (VQ− ) Q + (V− Q ) − 2πi n 1 Qn n=1 − Applications to Algebraic Number Theory 121

Thus, for f , (1), 136

1 2 1 G(B)v(β1) = u uB + (z∗ z) 2 B − 6 − − 1 u n 1 1 +u n z∗ ∞ 1 (VQ 2 B ) + (V Q 2 B ) − − − (114) −  2πin Q n/2 Qn/2  n=1 −  − z   where we have used V to stand for e2πivB without risk of confusion. Now c = (ǫ ǫ )/(ω ω ) > 0; so, if we set z = (d/c) + (i/cr) with r > 0, − ′ − ′ − then z∗ = (a/c) + (ir/c). As r tends to zero, z∗ tends to the rational point a/c vertically and z tends to infinity. In view of our earlier remarks, we may let r tend to zero in (114), in order to find the value of G(B)v(β1); moreover, it suffices to find the real part of the right hand side of (114), since G(B)v(β1) is rational. Thus, we have for f , (1),

1 2 1 (a + d) G(B)v(β1) = u uB + + σ1 + σ2 2 B − 6 c where

uB n n 1 uB n 1 ∞ 1 (VQ1− ) Q1 + (V− Q1 ) σ1 = lim real part of , r 0  2πi n 1 Qn  → n=1 1  −   1 u 1 +u   2 − B 1 − 2 B n 1 ∞ 1 (VQ2 ) + (V− Q2 ) σ2 = lim real part of − r 0  2πi n n/2 n/2  →  n=1 Q−2 Q2   −  2πiz 2πiz∗  with Q1 = e  and Q2 = e . 

As r tends to zero, Q1 tends to zero exponentially and it is easy to see that σ1 = 0 unless uB = 0 and in this case

∞ V n Vn σ = − − 1 4πin n=1

So, if we deﬁne λ(uB) to be zero if 0 < uB < 1 and equal to 1 if uB = 0, then 137

∞ V n Vn σ = λ(u ) − − . 1 B 4πin n=1

In order to evaluate σ2, we need the identity

c c c c q q− c 2k+1 (c 2k+1) 1 − = q − = q− − , (q , q− ) q q 1 − − k=1 k=1 Applications to Algebraic Number Theory 122 i.e. 1 c qc 2k+1 c q (c 2k+1) = − = − − . (115) q q 1 qc q c qc q c − − k=1 − − k=1 − − n/2 Employing the identities (115) with q = Q−2 , we have

c k c/2 uB n k c/2 uB n ∞ 1 (VQ2− − ) + (VQ2− − )− σ2 = lim real part of . − r 0  2πin cn/2 +cn/2  → k=1 n=1 Q− Q  2 − 2    Let now tk = 1 (2/c)(k uB) for k = 1, 2,... c; clearly 1 tk < 1 for − − 2πi(vB+(a/c)(k uB−)) ≤ πr 0 uB < 1. Further, let for k = 1,..., c, Vk = e − and let P = e . ≤ c 2πia 2πr 2 It is clear that Q2 = e − = P− and P tends to 1 as r tends to zero. Now it can be verified that c ∞ 1 (V Ptk )n + (V Ptk ) n = k k − σ2 lim real part of n n − P 1  2πin P P−  →  k=1 n=1 −    i.e.   c n n tkn tkn ∞ 1 (Vk Vk− )(P P− ) σ2 = lim − − . (116) − P 1 4πin Pn P n → k=1 n=1 − − Ptkn P tkn − = We know that lim n − n tk and if we can establish the uniform P 1 P P− convergence of the→ inner series− in (116) with respect to P for 1 P < , then we can interchange the passage to the limit and the summations≤ in (116).∞ For this purpose, we remark that, for fixed k with 1 k c, the function 138 xtk x tk ≤ ≤ = − < fk(x) − 1 is a bounded monotone function of x for 1 x . In fact, x x− ≤ ∞ it is monotone− decreasing for 0 < tk < 1, identically zero for tk = 0, monotone increasing for 1 < t < 0 and identically 1 for t = 1. Moreover if t < 1, − k ± k ± | k| fk(x) tends to 0 as x tends to infinity and fk(x) tends to tk as x tends to 1. n Thus the sequence fk(P ) , n = 1, 2,... is a bounded monotone sequence for { } n n V V− 1 k c. Further the series ∞ k − k is convergent. By Abel’s criterion, ≤ ≤ n=1 4πin the inner series in (116) converges uniformly with respect to P for 1 P < . As a consequence, we have from (116), ≤ ∞

c n n k uB 1 ∞ Vk Vk− σ2 = − − c − 2 2πin k=1 n=1 Let now for real x, ∞ e2πinx e 2πinx P (x) = − − ; 1 − 2πin n=1 Applications to Algebraic Number Theory 123 as is well-known, we have 1 P x [x] for x not integral, 1 =  − − 2 0 for x integral,   where [x] denotes the integral part of x. Further, let for real x,

∞ e2πinx P (x) = ′ . 2 − (2πin)2 n= −∞ P 1 2 Clearly 2(x) = 2 (x [x]) (x [x]) + 1/12. With this notation then, we have for f , (1), { − − − } (a + d) 1 G(B)v(β ) = P (u ) + λ(u )P (v ) 1 2 B c 2 B 1 B − c k uB 1 k uB − P1 a − + vB . − c − 2 c k=1

Now, for 1 k c and 0 uB 1, 0 < (k uB)/c < 1 and (k uB)/c = 1 only when k = c≤and≤u = 0. Therefore,≤ ≤ for 1 −k < c, − B ≤ k uB k uB 1 P1 − = − c c − 2 and for k = c, 139 k uB k uB 1 1 P1 − = − λ(uB). c c − 2 − 2

Further when k = c and uB = 0,

k uB P1 a − + vB = P1(a + vB) = P1(vB). c Thus (a + d) 1 G(B)v(β ) = P (u ) + λ(u )P (v ) 1 2 B c 2 B 1 B − c k uB k uB 1 P1 − P1 a − + vB λ(uB)P1(vB) − c c − 2 k=1 c (a + d) k u k u ′ B B = P2(uB) P1 − P1 a − + vB . c − c c k=1 (117) Applications to Algebraic Number Theory 124

Since P1(x) is an odd function of x,

k uB k + uB P1 − = P1 − c − c and k uB k + uB P1 a − + vB = P1 a − vB . c − c − Moreover, in the sum in (117), we can lek k run over any complete set of residues modulo c, in view of the periodicity of P1(x). Thus

c (a + d) k + uB k + uB G(B)v(β1) = P2(uB) P1 − P1 a − vB c − c c − k=1 c (a + d) k + uB k + uB = P2(uB) P1 P1 a v c − c c − k=1 c 1 (a + d) − k + uB k + uB = P2(uB) P1 P1 a vB . (118) c − c c − k=0 140 It is surprising that even though, prima facie, it appears from (118) that G(B) is a rational number with a high denominator, say 12c2 f 2, the factor c2 in the denominator drops out and eventually, G(B) is a rational number with at most 12 f in the denominator. The determination of the asymptotic development of log ϑ11(vB uBz, z) as z tends to a/c is done in a rather more complicated manner by Hecke− in his work referred to earlier. We now take up the calculation of G(B) in the case f = (1). We set z∗ = a/c + ir/c and z = (d/c) + i/cr with r > 0 and as before, G(B)v(β ) is − 1

1 2 z∗ lim real part of log (z ω)(z ω′)η (z) . r 0 2πi − − z → We shall show ﬁrst that

1 z∗ 1 lim real part of log (z ω)(z ω′) = . r 0 2πi − − z −4 → For this purpose, we remark in the ﬁrst place that (d/c) < ω < ω < a/c, − ′ since ω(a cω) = 1 implies a/c > ω and ǫ(cω′ + d) = 1 implies d/c < ω′. Now as z =− (d/c) + i/cr and r tends to zero, arg (z ω)(z ω ) tends− to π and − − − ′ Applications to Algebraic Number Theory 125

similarly arg(z∗ ω)(z∗ ω′) tends to zero as z∗ = a/c + ir and r tends to zero. Thus our assertion− above− is proved. Deﬁning the branch of log η2(z) by

πiz ∞ log η2(z) = + 2 log(1 e2πimz), 6 − m=1 where on the right hand side we have taken the principal branches, we see that 141

πiz ∞ ∞ 1 log η2(z) = 2 e2πimnz 6 − n m=1 n=1 and in view of absolute convergence of the series, we have once again,

πiz ∞ 1 Qn log η2(z) = 2 , 6 − n 1 Qn n=1 − where Q = e2πiz. Now

n z∗ 2 z∗ πi ∞ 1 Q log η (Z) = (z∗ z) 2 z 6 − −  n 1 Qn  n=1 z  −    By our remarks above, 1 1 (a + d) G(B)v(β ) = + σ , 1 −4 12 c − 3 where z∗ ∞ 1 Qn σ3 = 2 lim real part of . r 0   2πin 1 Qn   →  n=1 − z      It is easily seen that    

n ∞ 1 Q1 2πiz lim = 0, for Q1 = e . r 0 2πin 1 Qn → n=1 − 1 Thus

n 1 ∞ 1 Q 2 2πiz∗ σ3 = lim real part of with Q2 = e . 2 r 0  2πin 1 Qn  → n=1 2  −    Applications to Algebraic Number Theory 126

To determine σ3, we ﬁrst use the identity,

1 c qc 2k+1 = − , q q 1 qc q c − − k=1 − −

n n n/2 to make the denominators of the terms Q2/(1 Q2) real. Setting q = Q−2 in 142 this identity, we see that −

n r n/2(c 2k) ∞ 1 Q ∞ 1 Q− − 2 = 2 . n nc/2 nc/2 2πin 1 Q 2πin Q− Q n=1 − 2 n=1 k=1 2 − 2 nc/2 an n πr k 2k/c 2πi(a/c)k Now Q−2 = ( 1) P with P = e and Q2 = VkP− where Vk = e . Hence − n c n n(1 2k/c) ∞ 1 Q ∞ 1 V P − 2 = k . 2πin 1 Qn 2πin Pn P n n=1 − 2 k=1 n=1 − − As a consequence,

c n n t n ∞ 1 (V V− )P k = k − k σ3 lim n n , (119) r 0  2πin P P−  → k=1 n=1 −     n n t n 1 (V V− )P k = ∞ k − k where tk 1 2k/c. In (119), the sum n n corresponding − n=1 2πin P P− n n − to k = c is zero since Vc Vc− = 0. Hence eﬀectively k runs from 1 to c 1; but then c k also does the− same when k does so. On the other hand, if−k is replaced by−c k, then V goes to V 1 and t to t so that − k k− k − k c n n t n c n n t n ∞ 1 (V V− )P k ∞ 1 (V V− )P− k k − k = k − k 2πin Pn P n − 2πin Pn P n k=1 n=1 − − k=1 n=1 − − Taking the arithmetic mean, we see that

c n n tkn tkn 1 ∞ 1 (Vk Vk− )(P P− ) σ3 = lim − − 2 r 0 2πin Pn P n → k=1 n=1 − −

We are now in the same situation as in (116) but with uB and vB replaced by 0. By the same analysis as in the former case, we can show that

c c 1 k k − k k σ3 = P1 P1 a = P1 P1 a . c c c c k=1 k=0 Applications to Algebraic Number Theory 127

Thus, for f = (1),

c 1 1 (a + d) − k k G(B)v(β1) = + P2(0) P1 P1 a . (120) −4 c − c c k=0

We now deﬁne 143 1 , if f = (1) v(f) = 4 0, otherwise.   Consolidating (118) and (120) into a single formula, we see that the value of G(B) is given explicitly in terms of elementary arithmetical functions by

Theorem 13. Corresponding to a proper ray class character χ modulo f in Q( √D)(D > 0), with associated v(λ) deﬁned by (104) and a ray class B modulo f, we have the formula

c 1 a + d − k + uB k + uB G(B) = v(β1) P2(uB) P1 P1 a vB v(f) ,  c − c c − −   k=0   (121)   where uB = vB = 0 for f = (1).

Note. Here [β1, β2] is an integral basis of an integral ideal bB in B and uB = S (β1γ), vB = S (β2γ). We recall that G(B) depends only on B and not on the special choice of bB or of its integral basis. The rational integers a, b, c, d are determined by ǫβ2 = aβ2 + bβ1, ǫβ1 = cβ2 + dβ1. The Riemann-Dedekind method used above for the explicit determination of G(B) does not make any speciﬁc use of the fact that z z is a hyperbolic substitution. → ∗ Coming back to our formula for L(1, χ) for a proper ray class character χ modulo f in Q( √D) for which the associated v(λ) is deﬁned by (104), we have as a consequence of Theorems 12 and 13,

π2 L(1, χ) = Λ, (122) T √D where Λ = χ(B)G(B) with B running over all ray classes modulo f. G(B) is B given by (121), and T = χ(λ)e2πiS (λγ) λ mod f

with γ K such that (γ √D) has exact denominator f. 144 ∈ Applications to Algebraic Number Theory 128

We shall now use the elementary formula for L(1, χ) derived above, in order to determine the class number of special abelian extensions of the real quadratic ﬁeld Q( √D). Let F be a relative abelian extension of degree n over K = Q( √D) and let f be the ‘Fuhrer’¨ (conductor) of F relative to K. Let ζ(s, K) and ζ(s, F) be the Dedekind zeta functions of K and F respectively. From class ﬁeld theory, we know that the galois group of F over K is isomorphic to a subgroup of the character group of the group of ray classes modulo f in K. Moreover, the law of splitting in F of prime ideals of K is equivalent to the following decomposition of ζ(s, F), namely ζ(s, F) = ζ(s, K) L(s, χ) (123) χ,1 where χ runs over a complete set of n 1 non-principal ray class characters − modulo fχ with conductor fχ dividing f and L(s, χ) is the L-series in K associated with χ. Multiplying both sides of (123) by s 1 and letting s tend to 1, we have in analogy with Theorem 10, −

2r1 (2π)r2 R H 4rh = L(1, χ). (124) W √ ∆ w √D | | χ,1

In (124), r1 and 2r2 are respectively the number of real and complex conjugates of F over Q, H is the class number of F, W is the number of roots of unity in F, R is the regulator of F, ∆ is the discriminant of F over Q, h is the class number of K, r is the regulator of K and w(= 2) the number of roots of unity in K. We have been able to obtain a formula for L(1, χ) involving elementary arithmetical functions only in the case when the character of signature v(λ) associated with χ is defined by (104). Thus, if we are to obtain for H, a formula involving purely elementary arithmetical functions, then we ought to consider only such abelian extensions F over K, for which, on the right hand side of (124), there occur in the product only characters χ whose associated v(λ) is given by (104). Moreover, suppose that χ1 and χ2 are two such distinct char- 1 acters occuring in that product, then χ1χ2− also occurs in the product and its associated v(λ) is defined by v(λ) = 1 for all λ , 0 in K. But this, again, is 145 a situation which we should avoid, in view of our aim. Thus, in order that we could obtain a formula of elementary type for the class number H of F, we are obliged to consider only those abelian extensions F over K, for which we have the decomposition ζ(s, F) = ζ(s, K)L(s, χ) where χ is a ray class character modulo fχ with fχ as conductor and its associated character of signature is given by (104). This, in the first place, means that F is a quadratic extension of K, i.e. F = Q( √D, √θ) where θ is a non-square number in K. Let ϑ be the Applications to Algebraic Number Theory 129 relative discriminant of F over K. We could assume that (θ) = ϑi2 for an ideal i in K, coprime to ϑ. From class field theory, we know that a prime ideal p in K not dividing f, the conductor of F relative to K, either splits into a product of distinct prime factors or stays prime in F and moreover

1, if p splits in F χ(p) = (125)  1, if p stays prime in F. −  We also know that fχ dividesf. On the other hand, let p be a prime ideal in K not dividing ϑ. We can then find c K such that θ = θc2 is prime to p. Let us define then ∈ ∗ θ ψ(p) = ∗ , p θ where, for p (2), ∗ is the quadratic residue symbol in K and if pa is the | p θ highest power of p dividing (2), then ∗ = +1 or 1, according as the con- p − 2 2a+1 gruence θ∗ ξ ( mod p ) is solvable in K or not. We see that ψ(p) is unambiguously≡ defined for all prime ideals p not dividing ϑ and we extend ψ multiplicatively to all ideals a in the ray classes modulo ϑ in K. From the theory of relative quadratic extensions of algebraic number fields, it follows that for prime ideals p in K not dividing ϑ, χ(p) = ψ(p) and hence χ(a) and ψ(a) coincide on the ideals in the ray classes modulo ϑ. Now, by the law of quadratic reciprocity in K, it can be shown that for all numbers α in K for which α 1( mod ϑ) and α > 0, ψ((α)) = 1. In ≡ ∗ other words, ψ, and hence χ, is a ray class character modulo ϑ. Incidentally 146 we note that fχ divides ϑ, since χ is a ray class character modulo fχ, with fχ as conductor. Again, using the law of quadratic reciprocity, we can show that for integral λ , 0 and coprime to ϑ, χ((λ)) = ψ((λ)) = ψ(λ)v(λ) where ψ(λ)isa prime residue class character modulo ϑ and v(λ) is given by

N(λ) But, for our purposes, we require χ((λ)) to be of the form ψ(λ) . Thus, N(λ) finally, in order that we could calculate H in terms of elementary| arithmetical| functions, we conclude that F should be of the form K( √θ) where θ K and θ > 0. In other words, F is a biquadratic field over Q, which is realized∈ as an imaginary− quadratic extension of the real quadratic field Q( √D). We have in this case, the following formula for the class number H of F, viz.

4π2 R 4rh H = L(1, χ) W √ ∆ w √D | | 4rh π2 = Λ. (126) w √D T √D In order to determine the value of T explicitly, we use the functional equations of ζ(s, F), L(s, χ) and ζD(s) = ζ(s, Q( √D)), viz.

2s 2 s/2 2(1 s) 2 (1 s)/2 (2π)− Γ (s)( ∆ ) ζ(s, F) = (2π)− − Γ (1 s) ( ∆ ) − ζ(1 s, F), | | − × | | − s 2 s/2 (1 s) 2 1 s (1 s)/2 π− Γ (s/2)D ζD(s) = π− − Γ − D − ζD(1 s) (127) 2 −

s 2 s/2 π− Γ ((s + 1)/2)(DN(fχ)) L(s, χ) =

(1 s) 2 1 s (1 s)/2 = π− − Γ − (DN(fχ)) − L(1 s, χ) 2 − ×

χ(q)v(γ √D) N(fχ) , T where γ K such that (γ √D) = q/fχ has exact denominator fχ. Further 147 ∈ 1 1 s ζ(s, F) = ζD(s)L(s, χ) and Γ(s/2)Γ((s + 1)/2) = π 2 2 − Γ(s). These, together with (127), give us

1 2 s 2 D N(fχ) − χ(q)v(γ √D) N(fχ) = . (128) ∆ T | | Since the right hand side is independent of s, we see by setting s = 1/2, that T = N(fχ)χ(q)v(γ √d). Since the right-hand side of (128) is 1, we see again 2 2 by setting s = 3/2 in (128), that D N(fχ) = ∆ . But we know that D N(ϑ) = ∆ . This means that N(ϑ) = N(f ) and since f | divides| ϑ, we obtain incidentally | | χ χ that fχ = ϑ. Thus we have

T = N(ϑ)χ(q)v(γ √D). Applications to Algebraic Number Theory 131

But q q qi2 (κ) γ √ = = = = ( D) 2 fχ ϑ ϑi (θ) where κ = θγ √D. Moreover, since χ is a real character, χ(i2) = 1. Hence χ(q) = χ(qi2) = χ(κ)v(κ) = χ(κ)v(θγ √D) = v(γ √D)χ(κ). Thus T = √N(ϑ)χ(κ). As a result, from (126), we obtain H R w = χ(κ)Λ h r W = χ(θγ √D) χ(B)G(B). (129) B We may summarise the above in Proposition 17. Let F = Q( √D, √θ) be an imaginary quadratic extension of K = Q( √D), D > 0, with an integral ideal f in K for its “conductor”, θ being a totally negative number in K. Then with our earlier notation, the class number H of F is given by r W H = h χ(θγ √D) χ(B)G(B), R w B where B runs over all the ray classes modulo f inK,G(B) is given by (121) and 148 χ is the non-principal ray class character modulo f, associated with F. Remark. We know that G(B) are rational numbers with at most 12 f (where f is the smallest positive rational integer divisible by fχ = f = ϑ) in the denomi- R w H nator, it follows that 12 f is a rational integer. r W h Let η and be respectively the fundamental units in F and Q( √D). Then R = 2 log η and r = log . Now either = η or = η2. In the former case R = 2r; in| the| latter case,| |F may be generated over Q±( √D) by adjoining the square root of (whichever is totally negative) to Q( √D) and in this case, we have R = r.∓ Thus, in any case, since w divides W, we see that 24 f (H/h) is a rational integer. If h is coprime to 24 f , we obtain the interesting result that h divides H. Let us now assume that F is an unramiﬁed imaginary quadratic extension of Q( √D). Then ϑ = (1) and hence the ray class group modulo ϑ is just the narrow class group of Q( √D) and χ is a genus character. It is known that all totally complex unramiﬁed quadratic extensions of Q( √D) are of the form Q( √D1, √D2) where D1 and D2 are coprime negative discriminants (of Applications to Algebraic Number Theory 132

quadratic ﬁelds over Q) which satisfy D1D2 = D. Hence F is obtained from Q( √D) by adjoining either √D1 or √D2, for such a decomposition of D as product of two coprime negative discriminants. On the other hand, since F = Q( √D1, √D2) is an abelian extension of Q with galois group of order 4, we have the decomposition

ζ(s, F) = ζ(s)LDt (s)LD2 (s)LD(s),

where LD1 (s), LD2 (s) and LD(s) are the Dirichlet L-series in Q associated with D D D the Legendre-Jacobi-Kronecker symbols 1 , 2 , and respectively. n n n Moreover ζD(s) = ζ(s)LD(s) and hence 149

L(s, χ) = LD1 (s)LD2 (s), a result due to Kronecker, which we have met already ( 1, Chapter II), χ being a genus character in Q( √D). Therefore §

L(1, χ) = LD1 (1)LD2 (1).

On the other hand, if h1 and h2 are the class numbers of Q( √D1) and Q( √D2) respectively and w1 and w2 the respective number of roots of unity in Q( √D1) and Q( √D2), then

2πh1 2πh2 LD (1) = , LD (1) = . 1 w √ D 2 w √ D 1 | 1| 2 | 2| Hence π2 4π2h h Λ = L(1, χ) = 1 2 √D w w √ D √ D 1 2 | 1| | 2| and as a consequence, we have

Proposition 18. If D1 < 0, D2 < 0, D = D1D2 are discriminants of quadratic ﬁelds over Q and if h1, h2 are the class numbers of and w1, w2 the number of roots of unity in Q( √D1), Q( √D2) respectively, then we have 4h h 1 2 = Λ = χ(B)G(B). (130) w w 1 2 B

(In (130), χ is the genus character in Q( √D) corresponding to the decomposition D = D1D2 and B runs over all the narrow ideal classes in Q( √D)). Applications to Algebraic Number Theory 133

Remark. Formula (130) is interesting in that it gives another arithmetical sig- niﬁcance for Λ. Besides, it is the analogue of Kronecker’s solution of Pell’s equation which we referred to in 3, Chapter II earlier. It has not appeared in literature so far. § From (129) and (130), we have 4hh h W r H = 1 2 . (131) w1w2w R

If we exclude the special cases when 4 D1, D2 < 0, then w1 = w2 = 2 = w. 150 1 − ≤ Moreover, r/R is 1 or 2 , as we have seen earlier. Further if we exclude F from being the cyclotomic ﬁelds of the 8th or 10th or 12th roots of unity or the ﬁelds Q( √D, √ 1) or Q( √D, √ 3), then W = 2. Thus barring these special cases, we have for− the class number− H of F the formula r H = h Λ R and from (131) r H = hh h . (132) R 1 2 In general, R/r = 2 and hence we have, except for some special cases, H 2 = Λ = h h . h 1 2

For D1 = 4, formula (131) was obtained by Dirichlet and in the general case, this was− discovered by Hilbert. A generalization of (132) has been considered by Herglotz. One breaks up D as the product of r( 2) mutually co- ≥ prime discriminants. The ﬁeld K = Q( √D1,..., √Dr) is an unramiﬁed abelian extension of Q( √D) and for the Dedekind zeta function of F, we have the decomposition as a product of L-series for the corresponding genus-characters in Q( √D). One then proceeds as above to obtain the required generalization of (132), involving the class numbers h1,... hr of Q( √D1),... Q( √Dr) respectively.

6 Some Examples

This section is devoted to giving a few interesting examples pertaining to the determination of the class number of totally complex biquadratic extensions of Q with particular reference to Proposition 17 and 18. Examples 1-6 deal with the case when F is a totally complex number field 151 Applications to Algebraic Number Theory 134 which is an unramified imaginary quadratic extension of Q( √D), D > 0(f = (1)). In fact, then F = Q( √D1, √D2) where D = D1D2 is a decomposition of D as product of coprime negative discriminants D1, D2. Formula (129) gives the class number H of F, on computing R, r, h, w, W and G(B). Incidentally we also check (130), by finding h1, h2, w1, w2. On the other hand we know that for any algebraic number field, the class number can be found directly, as follows. Since, in every class of ideals, there exists an integral ideal of norm not exceeding √ ∆ (∆ being the absolute discriminant), it suffices to test for mutual equivalence,| | the integral ideals of norm not exceeding √ ∆ and obtain a maximal set of inequivalent ideals from among these. This will give| | us the class number. Our notation here will be the same as in the last section. The following remark is to effect a simplification of the computation, in the case of Examples 1-6. The ray classes modulo f(= (1)) in Q( √D) are just the narrow ideal classes. Looking at the definition of G(B), we see that for two narrow classes B1 and B2 lying in the same wide class, G(B1) = G(B2) and further χ(B ) = χ(B ) so that χ(B )G(B ) = χ(B )G(B ). Thus − 1 − 2 1 1 2 2

Λ = 2 χ(B)G(B) = 2Λ∗(say) A where A runs over the wide classes in Q( √D) and B is a narrow class contained in A. From (129), we have r W H = 2h Λ∗. R w Further, formula (130) becomes

2h1h2 Λ∗ = . w1w2

As before, ǫ(> 1) is the generator of the group Γf∗ in Q( √D). Example 1. D = 12, D = 3, D = 4 1 − 2 − h = 1, since (2) = (1 + √3)(1 √3), (3) = ( √3)2. −

Similarly h1 = 1, h2 = 1. Further w1 = 6, w2 = 4, w = 2, ǫ = 2 + √3 > 0, 152 2 1 + √3 W = 12. Since ǫ = i , r/R = 1.  1 + i      Applications to Algebraic Number Theory 135

Since h = 1, Λ∗ = G(B1), B1 being the principal narrow class in Q( √12). √ √ We take bB1 = (1); bB1 = [1, 3], β1 = 1, β2 = 3, v(β1) = 1.

√3 2 3 √3 (2 + √3) = a = 2, b = 3, c = 1, d = 2. 1 1 2 1 1 2 + 2 1 1 G(B ) = = . 1 12 1 − 4 12 Thus 12 1 H = 2.1.1 = 1. 2 12 Further 2h1h2 2.1.1 1 = = = Λ∗. w1w2 6.4 12 Example 2. D = 24, D = 3, D = 8, h = h = h = 1, w = 2, w = 6, 1 − 2 − 1 2 1 w2 = 2, W = 6, ǫ = 5 + 2 √6 > 0.

2 Since ǫ = ( √ 3 + √ 2) , r/R = 1. Also Λ∗ = G(B1), B1 being the − − −√ √ √ principal narrow class in Q( 24); bB1 = (1) = [1, 6], β1 = 1, β2 = 6, v(β1) = 1.

√6 5 12 √6 ǫ = , a = 5, b = 12, c = 2, d = 5, 1 2 5 1 1 5 + 5 1 1 Λ∗ = G(B ) = = , 1 12 2 − 4 6 6 1 H = 2.1 1, 2 6 − 2h1h2 2.1.1 = = Λ∗. w1w2 6.2

Example 3. D = 140, D = 4, D = 35, h = 1, w = 4, h = 2, w = 2, 153 1 − 2 − 1 1 2 2 h = 2, w = 2, ǫ = 6 + √35 > 0, W = 4, r/R = 1/2. We take p1 = (1), p2 = (2, 1 + √35) as representatives of the two wide classes in Q( √35) and 2 denote the ray classes containing them by B1, B2 respectively. Since p2 = (2), 35 N(p2) = 2 and χ(B2) = −2 = 1. − √ For B1, bB1 = p1, β1 = 2, β2 = 35, v(β1) = 1, a = 6, b = 35, c = 1, d = 6, G(B1) = 1/12 (6 + 6)/1 1/4 = 3/4. − √ √ For B2, bB2 = p2, β1 = 2, β2 = 1 + 35, v(β1) = 1, ω = (1 + 35)/2, ǫ = 2ω + 5,ǫω = 7ω + 17, a = 7, b = 17, c = 2, d = 5, Applications to Algebraic Number Theory 136

1 7 + 5 1 1 G(B ) = = , 2 12 2 − 4 4

3 1 1 Λ∗ = χ(B)G(B) = = , 4 − 4 2 A 1 4 1 H = 2.2 = 2, 2 2 2 2h1h2 2.1.2 = = Λ∗. w1w2 4.2

Example 4. D = 140, D1 = 7, D2 = 20, h1 = 1, w1 = 2, h2 = 2, w2 = 2, − 1 − √ h = 2, w = 2, W = 2, r/R = 2 , ǫ = 6 + 35 > 0. Let p1, p2, B1, B2 have 7 the same signiﬁcance as in Example 3. Then χ(p2) = − = 1, G(B1) = 3/4, 2 G(B2) = 1/4, Λ∗ = 3/4 + 1/4 = 1, 1 H = 2.2 1.1 = 2, 2 2h1h2 2.1.2 = = Λ∗. w1w2 2.2

Example 5. D = 21, D1 = 3, D2 = 7, h1 = 1, w1 = 6, h2 = 1, w2 = 2, h = 1, w = 2, W = 6, ǫ = (5 + √21)− /2 > 0;−r/R = 1, since ǫ = (( √ 3 + √ 7)/2)2. − − − √ For the principal narrow class B1 in Q( 21), we take bB1 = (1) with integral basis [1,ǫ]; β1 = 1, β2 = ǫ, ω = ǫ, ǫω = 5ω 1, a = 5, b = 1, c = 1, d = 0. − − 1 5 1 1 Λ∗ = G(B ) = = , 1 12 1 − 4 6 6 1 H = 2.1.1 = 1, 2 6 2h1h2 2.1.1 = = Λ∗. w1w2 6.2 154 We might show that H = 1, directly. The discriminant of F is (21)2 and we have therefore to examine the splitting of (2), (3), (5), (7), (11), (13), (17), (19) in F. The ideals (2), (5), (11) and (17) stay prime in Q( √3) and (3), (13), (19) stay prime in Q( √ 7). Further (7) = ( √7)2 in Q( √ 7). The further splitting in F is seen to be as− follows: − 1 + √ 7 1 √ 7 (2) = − − − ,  2   2          Applications to Algebraic Number Theory 137

(3) = ( √ 3)2, − (5) = (2 √ 3 + √ 7)(2 √ 3 √ 7), − − − − − 2 1 2 √ 3 + √ 7 (7) = (1 + ρ) (1 ρ− )− with ρ = − − , − 2 (11) = (2 + √ 7)(2 √ 7), − − − (13) = (1 + 2 √ 3)(1 2 √ 3), − − − 5 √ 3 + √ 7 5 √ 3 √ 7 (17) = − − − − − ,  2   2      (19) = (4 + √ 3)(4 √ 3).  − − − Thus all prime ideals of norm 21 are principal in F and hence H = 1. ≤ Example 6. D = 2021, D = 43, D = 47, h = 1, w = 2. 1 − 2 − 1 1 47 To ﬁnd h2, we observe that − = 1 and so (5) stays prime in Q( √ 47), 5 − − while (2) = p2p2′ , (3) = p3p3′ where

1 + √ 47 1 √47 p = 2, − , p = 3, ± . 2  2  3  2          The integral ideals of norm √ 47 in Q( √47) are p , p , p , p , p2, p 2, p p , ≤ | | 2 2′ 3 3′ 2 ′2 2 3 p2′ p3, p2p3′ , p2′ p3′ . Now

1 + √ 47 2 2 N − = 12 = p p′ p p′ .  2  2 2 3 3     Let p3 be so chosen that 155

2 1 √ 47 p p′ = ± − . 2 3  2      Then p2 p (equivalent in the wide sense). Let now 2 ∼ 3 x + y √ 47 pk = − , 2  2      k being the order of the wide class containing p2 and x, y being rational integers. 2 2 k+2 , k k Then x + 47y = 2 and certainly y 0, for then we would have p2 = p′2 Applications to Algebraic Number Theory 138

i.e. p2 = p2′ , which is not true. But now the smallest k > 0 for which the diophantine equation x2 + 47y2 = 2k+2 is solvable with y , 0 is k = 5 (in 9 + √ 47 fact, 92 + 47 = 128). Hence p5 = − and p4 / (1). Since p2 p , 2  2  2 2 ∼ 3   p 2 p , p p p3 p2 p , p p p , p p p , p p p , we see that ′2 ∼ 3′ 2 3 ∼ 2 ∼ 2 ∼ 3′ 2′ 3 ∼ 2 2 3′ ∼ 2′ 2′ 3′ ∼ 3 (1), p2, p2′ , p3, p3′ form a complete set of mutually inequivalent integral ideals of norm √ 47 and thus h2 = 5. To ﬁnd≤ h,| we| have to consider integral ideals of norm √2021 (i.e. 44). Since ≤ ≤ 2021 = 1 for p = 2, 3, 7, 11, 13, 23, 29, 31, 37, 41, p − these rational primes stay prime also in Q( √2021). We need to consider therefore only the decomposition of (5), (17), (19) and (43) in Q( √2021). Now

2021 43 + √2021 43 √2021 = 0 and (43) = − . 43  2   2          Further     2021 2021 2021 = = = 1 5 17 19 and so (5) = p p , (17) = p p , (19) = p p (say). Actually 5 5′ 17 17′ 19 19′ 1 + √2021 p = 5, . 5  2      Now, 156 39 √2021 39 √2021 p2 = − , 25 and p3 = − . 5  2 −  5  2      Hence p3 (1) but p2 / (1), for this would mean p (1) which is not true, 5 ∼ 5 5 ∼ since 5 is not the norm of any integer in Q( √2021). Now ± (44 + √2021)(44 √2021) = (5)(17) − and (46 + √2021)(46 √2021) = (5)(19) − and by properly choosing p17, p19, we can suppose that pp17′ (1) and p p (1). But 1, p , p2 are inequivalent and p p p , p− ∼p p2. 5 19′ ∼ 5 5 17 ∼ 5 ∼ 19 17′ ∼ 19′ ∼ 5 Applications to Algebraic Number Theory 139

1 √ Hence h = 3. Further w = 2, ǫ = 2 (45 + 2021); w2 = W = 2. Moreover, r/R = 1 since 2 √ 43 + √ 47 ǫ = − − . −  2     √ 2 Let B1, B2, B3 be the narrow classes in Q( 2021) containing (1), p5, p5. Then

1 + √2021 β2 1 + √2021 bB1 = (1), β1 = 1, β2 = ,ω1 = = , 2 β1 2

v(β1) = 1,

ǫω1 = 23ω1 + 505 a = 23, b = 505, c = 1, d = 22 ǫ = ω1 + 22    1 + √2021 b = p , β = 5, β = , v(β ) = 1, B2 5 1 2 2 1

β2 1 + √2021 ω2 = = , β1 10

ǫω2 = 23ω2 + 101 a = 23, b = 101, c = 5, d = 22 ǫ = 5ω2 + 22   157  1 b = p2, β = (39 √2021), β = 25, v(β ) = 1, B3 5 1 2 − 2 − 1 − β2 50 19 + ω1 ω3 = = − = , β1 39 √2021 5 − ǫω3 = 42ω3 + 25 a = 42, b = 25, c = 5, d = 3 ǫ = 5ω3 + 3     43 2 χ(B1) = 1, χ(B2) = χ(p5) = − = 1, χ(B3) = χ(p ) = 1. 5 − 5 Thus 1 23 + 22 1 Λ∗ = 1 12 1 − 4 1 23 + 22 4 k 23k 1 1 P1 P1 + − 12 5 − 5 5 − 4  k=1    Applications to Algebraic Number Theory 140

1 42 + 3 4 k 42k 1 + P1 P1 . 12 5 − 5 5 − 4  k=1    P 23k P 42k  Since 1 5 = 1 5 , we see easily that Λ∗ = 5/2, − 2 5 H = 2.3.1 = 15, 2 2 2h1h2 2.1.5 = = Λ∗. w1w2 2.2 The following two examples deal with the case when F is ramiﬁed (with relative discriminant f , (1)) over Q( √D), (cf. Proposition 17). Now the numbers γ and uB, vB corresponding to the ray classes B in Q( √D) have to be reckoned with.

Example 7. D = 5, F = Q( √5, √θ) where θ = ( √5 5)/2 and θ > 0, h = 1, w = 2, W = 10. The fundamental units in F and Q−( √5) are the− same, viz. ω = (1 + √5)/2 and so r/R = 1/2. It is easy to verify that θ = √5ω′, 158 θ = √5ω = θ ω2, N(ω) = 1. ′ − − 1 1 If we set ρ = ( √θ ω)/2, then ρ− = ( √θ+ω)/2 = ρ ω and so ρ+ρ− = 2 −2 2 − 2 − 2− 1 ω. Therefore ρ + ρ− = ω 2 = ω 1, so that ρ + ρ− + ρ + ρ− + 1 = 0. −Actually, since Re ρ < 0 and− Im ρ > 0,− we have ρ = e4πi/5. As a basis of the integers in F over Q( √5), we have [1, ρ] and the relative discriminant f of F 4 2 over Q( √5) is (θ) = (θ′) = ( √5). Now ϕ(f) = 4 and ω , ω, ω, ω serve as prime residue class representatives modulo f in Q( √5).− In fact, ω4 1, ω 2, ω 3, ω2 4( mod f). For ǫ, we take ω4 = (7 + 3 √5)/2 and≡ then ǫ− > 0≡ as also≡ǫ 1(≡ mod f). We note that v(f) = 0. We take γ ≡= 1/5 so that (γ)f = (1)/( √5) has exact denominator f and q = (1). Further χ(q)v(γ √5) = 1 1 = 1. × − − In the notation of 5 (Chapter II), the set λk consists just of 1, and l consists of 1, 2ω. Therefore,§ since h = 1, we may{ } take as ray class representa-{ } tives modulo f, the ideals p1 = (1) and p2 = (2) and denote the corresponding ray classes by B1 and B2. For B1, χ(B1) = 1, β1 = 1, β2 = ω, v(β1) = 1, a = 5, b = 3, c = 3, d = 2, uB1 = S (γ) = 2/5, vB1 = S (ωγ) = 1/5. For B , χ(B ) = 1, necessarily; β = 2, β = 2ω, v(β ) = 1, a = 5, b = 3, 2 2 − 1 2 1 c = 3, d = 2, uB2 = S (2γ) = 4/5, vB2 = S (2ωγ) = 2/5;

(2/5)2 (2/5) 1 5 + 2 2 k + 2/5 5(k + 2/5) 1 Λ = − + P1 P1  2 12 3 − 3 3 − 5 −  k=0    Applications to Algebraic Number Theory 141

(4/5)2 (4/5) 1 5 + 2 2 k + 4/5 5(k + 4/5) 2 − + P1 P1 −  2 12 3 − 3 3 − 5  k=0  = 2/5;  − r W 1 10 2 H = h ( Λ) = 1 = 1. R w × − 2 2 5 We shall now show that H = 1, directly. Since the absolute discriminant of 159 F is 52 N(f) = 125, we need only to examine the splitting in F of the ideals 5 (2), (3), (5), (7) and (11). Since = 1 for p = 2, 3, 7 the ideals (2), (3), p − θ (7) are prime in Q( √5). Further, = 1 for p = (2), (3), (7) and hence, by p − (125), these stay prime in F too. Again,

(11) = (4 + √5)(4 √5) = (1 √θ)(1 + √θ)(1 ω √θ)(1 + ω √θ) − − − (5) = ( √5)2 = ( √θ)4. Thus, all prime ideals of norm 11 in F are principal and, as a consequence, H = 1. ≤ We might show that H = 1, also by using the following decomposition, due to Kummer, of the Dedekind zeta-function ζ(s, F), viz.

ζ(s, F) = ζ(s)L5(s)P(s)Q(s), where, for σ > 1,

∞ 5 s L5(s) = n− , n n=1

∞ s P(s) = ψ(n)n− , n=1

∞ s Q(s) = ψ(n)n− , n=1 and ψ(n) is deﬁned by

0, n 0( mod 5), ≡ i, n 2( mod 5),  ≡ ψ(n) =  1, n 1( mod 5), − ≡ −  i, n 2( mod 5), − ≡ −  , . 1 n 1( mod 5)  ≡  Applications to Algebraic Number Theory 142

Since ζ(s, F) = ζ(s, Q( √5))L(s, χ) and ζ(s, Q( √5)) = ζ(s)L5(s), we have L(s, χ) = P(s)Q(s). Hence

π2Λ 2π2 P(1)Q(1) = L(1, χ) = = . ( ) − 5 25 ∗

But it is easy to see that 160

∞ 1 i P(1) = + , 5n + 1 5n + 2 n= π −∞ = (cot π/5 + i cot 2π/5), 5 π i(ρ + ρ 1) ρ2 + ρ 2 = − − + − . 5 ρ ρ 1 ρ2 ρ 2 − − − − Similarly π i(ρ + ρ 1) ρ2 + ρ 2 Q(1) = − − − . 5 ρ ρ 1 − ρ2 ρ 2 − − − − Therefore π2 ρ2 + ρ 2 + 2 ρ4 + ρ 4 + 2 P(1)Q(1) = − + − −25 ρ2 + ρ 2 2 ρ4 + ρ 4 2 − − − − π2 ρ2 + ρ 2 + 2 ρ + ρ 1 + 2 = − + − 2 2 1 −25 ρ + ρ− 2 ρ + ρ− 2 2 2 − 2 − π α (α 2) + (α + 2)(α 4) 1 = − − (α = ρ + ρ− = ω) −25 (α2 4)(α 2) − − − π2 10 2π2 = = , −25 × − 5 25 which conﬁrms ( ) above. ∗ Example 8. D = 17, F = Q( √17, √θ) where θ = ( √17 5)/2, θ > 0, h = 1, w = 2, W = 2; ρ = 4 + √17(= 2θ + 9) is the fundamental− unit− in Q( √17) as also in F and so r/R = 1/2.

[1, (1 + √θ)/θ′] is a basis of the integers in F over Q( √17). The relative discriminant f of F over Q( √17)is(θ3); N(f) = 8 and v(f) = 0. Further ϕ(f) = 4 and as representatives of th prime residue classes modulo f in Q( √17), we take 1, 1, 3, 3. It is easy to verify that ρ 3, ρ2 1, ρ 3, ρ2 1( − − ≡ − ≡ − ≡ − ≡ − mod f) and so, in every prime residue class modulo f, there is a unit. Also, it is 161 Applications to Algebraic Number Theory 143

2 3 clear that ǫ = ρ = 33 + 8 √17 is the generator of Γf∗. We take γ = 1/(θ √17); 1 then (γ)( √17) = f− . So q = (1) and moreover, N(γ √17) > 0. In the notation of 5 (Chapter II) again, γk consists of the single element 1 while consists of§ the numbers 1, 3ρ,{ 5ρ,} 7ρ2. Further h = 1 and there { 1} − − are just 4 ray classes modulo f which we denote by B1, B3, B5 and B7 (say) and the corresponding representatives bB1 , bB3 , bB5 , bB7 may be chosen to be (1), (3), (5), (7) respectively. The numbers uB, vB corresponding to the ray classes B1, B3, B5, B7 may be denoted by u1, v1, u3, v3, u5, v5, u7, v7 respectively. If λ 1( mod f), then χ((λ)) = 1 according as N(λ) ≷ 0 and hence ≡ ± ± 2 χ(B1) = 1, χ(B3) = χ(( 3ρ)) = 1, χ(B5) = χ((5ρ)) = 1, χ(B7) = χ(( 7ρ )) = − − −2 − 1. We note incidentally that, for q = 1, 3, 5, 7, χ(Bq) = . q For B1, 23 b = (1), β = 1, β = θ, u = S (γ) = , B1 1 2 1 − 8 1 v = S (θγ) ( mod 1). 1 ≡ 4

For B3, 69 b = (3), β = 3, β = 3θ, u = S (3γ) = , B3 1 2 3 − 8 3 v = S (3θγ) ( mod 1). 3 ≡ 4

For B5, 115 b = (5), β = 5, β = 5θ, u = S (5γ) = , B5 1 2 5 − 8 v = S (5θγ) v ( mod 1). 5 ≡ − 3

For B7, 161 b = (7), β = 7, β = 7θ, u = S (7γ) = , B7 1 2 7 − 8 v = S (7θγ) v ( mod 1). 7 ≡ − 1 For all the 4 classes, a = 7, b = 32, c = 16, d = 73. 162 From (129), we now obtain− −

2H = χ((1))v(γ √17) χ(Bq)G(Bq) q=1,3,5,7 Applications to Algebraic Number Theory 144 i.e. 15 2 33 k + uq k + uq 2H = P2(uq) + P1 P1 7 + vq . q ×  8 16 16  q=1,3,5,7  k=0    It may be veriﬁed that  2 66 f rac338 P2(uq) = ; q 128 q=1,3,5,7 15 k + u1 k + u1 P1 P1 7 + v1 = 16 16 k=0 15 k + u7 k + u7 6256 = P1 P1 7 + v7 = ; 16 16 1282 k=0 15 k + u3 k + u3 P1 P1 7 + v3 = 16 16 k=0 15 k + u5 k + u5 5904 = P1 P1 7 + v5 = . 16 16 − 1282 k=0 Thus 66 6256 + 5904 2H = + 2 = 2 128 1282 i.e. H = 1. We shall now show that H = 1 directly. Since the absolute discriminant of F is 172 8 = 2312(< 492), we have to examine for mutual equivalence all integral ideals× of (absolute) norm < 49 in F. To this end, we ﬁrst remark that the ideals (3), (5), (7), (11), (23), (29), (31), 163 (37) and (41) are prime in Q( √17). Further we have in Q( √17) the following decomposition of other ideals, viz.

(2) = (θ)(θ′), (13) = (1 + 4θ′)(1 + 4θ), (17) = (5 + 2θ)(5 + 2θ′),

(19) = (11 + 2θ)(11 + 2θ′), (43) = (1 + 6θ)(1 + 6θ′),

(47) = (3 2θ)(3 2θ′). − − Now, if p is a prime ideal in Q( √17), then

+1, if p = B B in F, B , B θ 1 2 1 2 χ(p) = =  1, if p is prime in F p − 0, if p = B2 in F.   Applications to Algebraic Number Theory 145

Further, if λ and are integers in Q( √17) such that λ ( mod f), then θ θ ≡ χ((λ)) = = according as N(λ) ≷ 0. (λ) ± () θ For p = (3), (5), (11), (29), (37) in Q( √17), = 1 and so these prime p − ideals stay prime also in F.

2 1 + √θ 1 √θ (2) = (θ)(θ′) = ( √θ) − .  θ′   θ′      Now 1+4θ 3( mod f) and 1+4θ= 1( mod  f) so that χ((1+4θ )) = 1, ′ ≡ ′ − while χ((1 + 4θ)) = +1. So (1 + 4θ′) stays prime in F while (1 + 4θ) splits in F. This gives

(13) = (1 + 4θ′)(1 + θ + √θ)(1 + θ √θ). − Again 5 + 2θ 1( mod f), 5 + 2θ 1( mod f) and so χ((5 + 2θ)) = ≡ ′ ≡ − χ((5 + 2θ′)) = 1. Hence the ideal (17) does not decompose further in F. In view of− the fact that 11 + 2θ 3( mod f), χ((11 + 2θ )) = 1 and ′ ≡ − ′ − so (11 + 2θ′) stays prime in F whereas (11 + 2θ) splits in F. We have in F therefore the decomposition

(19) = (11 + 2θ′)(1 + (5 + θ) √θ)(1 (5 + θ) √θ). − 164 From 1 + 6θ 1( mod f), 1 + 6θ 3( mod f) we know that (1 + 6θ) ′ ≡ − ≡ − stays prime in F while (1 + 6θ′) splits and we obtain the splitting of (43) in F as (43) = (1 + 6θ)(5 + θ + √θ(4 + θ))(5 + θ √θ(4 + θ)). − Both the ideals (3 2θ) and (3 2θ ) split in F and we have − − ′ (47) = (1 + √θ(3 + θ))(1 √θ(3 + θ))(3 + θ + √θ(4 + θ)) − × (3 + θ √θ(4 + θ)). × − The prime ideals (7) and (41) in Q( √17) split in F as

(7) = (θ + 3 + √θ)(θ + 3 √θ), − (41) = (3 + 2θ + 2 √θ(5 + θ))(3 + 2θ 2 √θ(5 + θ)). − Finally, from the decompositions 8 + 3θ √θ (8 + 3θ) + θ(θ √θ)/2 (23)(θ′) = + (θ + √θ) − ,  √θ 2   √θ          Applications to Algebraic Number Theory 146

θ2 θ √θ θ2 θ √θ (62)(1 + 4θ) = 10 7θ + + 10 7θ + , − − 2 2  − − 2 − 2          we can deduce that the ideals (23) and (31) split into two distinct principal prime ideals each in F, if we use the decomposition of (1 + 4θ) and (2) in F. Thus all prime ideals of norm < 49 in F are principal and we have H = 1. Bibliography

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Modular Functions and Algebraic Number Theory

1 Abelian functions and complex multiplications 167 In this section, we shall see how the complex multiplications of abelian functions lead in a natural fashion to the study of the Hilbert modular group. Let z1,..., zn be n independent complex variables and let

z1 . z =  .    zn     Let Cn be the n-dimensional complex euclidean space and G, a domain in Cn. A complex-valued function f (z) deﬁned on G (except, perhaps, for a lower- dimensional subset of G) is meromorphic in G, if there exists a correspondence

a1 . pa(z) a =  .  ( G)   ∈ → qa(z) an     where pa(z) and qa(z) are convergent power-series in z1 a1,... zn an with a − − pa(z) common domain of convergence Ka around a such that for z Ka, f (z) = ∈ qa(z) whenever either qa(z) , 0 or if qa(z) = 0, the pa(z) , 0.

149 Modular Functions and Algebraic Number Theory 150

p1 Let f (z) be a meromorphic function of z in Cn. A complex column p = .  .   pn  0     is a period of f (z), if f (z + p) = f (z) for all z Cn; for example, 0 = . is a ∈  .   0  period of f (z). The periods of f (z) form an additive abelian group; if we regard 168 Cn as a 2n-dimensional real vector space, then the group of periods is, in fact, a closed vector group over the field of real numbers. Either this vector group is discrete or it contains at least one limit point in Cn. In the latter case, f (z) is said to be degenerate; this is equivalent to the fact that f (z) has infinitesimal periods and by a suitable linear transformation of the variables z1,..., zn with complex coefficients, the function f (z) can be brought to depend on strictly less than n complex variables. In the former case, f (z) is said to be non-degenerate and its period-group is a discrete vector-group i.e. a lattice in Cn. We shall consider only the case when the period-lattice has 2n generators linearly independent over the field of real numbers. It is easy to construct an example of a non-degenerate abelian function with 2n independent periods. For example, let P(w) be the Weierstrass’ elliptic function with independent periods ω1, ω2. Then the function f (z) = P(z1) ... P(zn) furnishes the required example, since it has exactly 2n independent periods

ω1 ω2 0 0 0 0 ω ω      1  2       ,   ,  0  ,  0  ,...,   ,   .              0   0       0   0               0   0   0   0  ω1 ω2                      n    Let, then, for a given f (z) meromorphic in C , p1,..., p2n be 2n periods linearly independent over the field of real numbers such that any period p of 2n f (z) is of the form i=1 mi pi with rational integers mi. The n-rowed matrix P = (p1,..., p2n) is called a period-matrix of f (z). It is uniquely determined upto multiplication on the right by a 2n-rowed unimodular matrix. We shall n denote the period-lattice generated by p1,..., p2n in C by L. A meromorphic function f (z) defined in Cn and admitting as periods all the elements of L is called an abelian function for the lattice L. The abelian functions for L constitute a field FL. In FL there exists at least one function f (z) having L exactly as its period-lattice. Moreover, in FL, there exist n functions f1,..., fn which are analytically independent and hence algebraically independent over the field of complex numbers. Also, every abelian function satisfies a 169 polynomial equation of bounded degree with rational functions of f1,..., fn as coefficients. As a consequence, it can be shown that FL is an algebraic function Modular Functions and Algebraic Number Theory 151

field of n variables and one can find n + 1 functions, f0, f1,..., fn in FL such that every function in FL is a rational function of f0, f1,..., fn with complex coefficients. Let P be a complex matrix of n rows and 2n columns such that the 2n columns are linearly independent over the field of real numbers. The problem arises as to when P is a period-matrix of a non-degenerate abelian function or, in other words, a Riemann matrix. It is well known that in order that P might be a Riemann matrix, it is necessary and sufficient that there exists a rational 2n-rowed alternate (skew-symmetric) matrix A such that

(i) PAP′ = 0, (133) 1 (ii) H = i− PAP′ > 0 (i.e. positive hermitian).

These are known as the period relations. If E is the n-rowed identity matrix 0 E and A is the 2n-rowed alternate matrix E 0 , then conditions (i) and (ii) were given by Riemann as precisely the conditions − to be satisﬁed by the periods of a normalised complete system of abelian integrals of the ﬁrst kind on a Riemann surface of genus n. P Setting B = , the two conditions above may be written in the form P 0 iH BAB′ = − , H > 0. (134) iH 0

For n = 1, the conditions (133) reduce to the sole condition that if P = (ω1ω2), 1 then ω1ω2− should not be real. To establish the necessity and sufficiency of these conditions for n > 1, one has to make considerable use of theta-functions. The problem as to when a given P is a Riemann matrix was apparently considered independently also by Weierstrass and he wished to prove to this end that every abelian function can be written as the quotient of two theta- functions. He could not, however, complete the proof of this last statement–it 170 was Appell who gave a complete proof of the same for n = 2, and Poincare,´ for general n. A period matrix P which satisfies the period-relations with respect to an alternate matrix A is said to be polarized with respect to A. The alternate matrix A is not unique; but, in general, A is uniquely determined upto a positive rational scalar factor. Moreover, if P is polarized with respect to A, then from (134), since H > 0, we conclude that A , 0, B , 0. | | | | n Let P be a Riemann matrix and L, the associated lattice in C . Let L1 be a sublattice of L, again of rank 2n over the field of real numbers and let P1 be a period matrix of L1. Then P1 = PR1 for a nonsingular rational integral Modular Functions and Algebraic Number Theory 152

matrix R1. The associated function field FL1 is an algebraic function field of n variables, containing FL. Since FL itself is an algebraic function field of n

variables, we see that FL1 is an algebraic extension of FL.

Conversely, let FL1 be a ﬁeld of abelian functions, P1 an associated Rie- n mann matrix and L1 the corresponding period-lattice in C . Further, let F be a

subﬁeld of FL1 such that FL1 is algebraic over FL. Then F is again a ﬁeld of abelian functions containing at least one non-degenerate abelian function and has a period-lattice L containing L1. This means that if P is a period-matrix associated with F, then P1 = PR1 for a nonsingular rational integral matrix R1. We now prove

Proposition 19. Let P1 and P2 be two Riemann matrices and L1 and L2 the associated period lattices in Cn. A necessary and suﬃcient condition that the

elements of FL1 depend algebraically on those of FL2 and vice versa, is that P1 = PM for a nonsingular rational matrix M.

Proof. Let f0,..., fn be generators of FL1 over the ﬁeld of complex numbers

and let elements of FL1 depend algebraically on those of FL2 . This means that each fi satisﬁes an irreducible polynomial equation

ni (i) ni 1 (i) fi + an 1 fi − + + a0 = 0, i− (i) where am , m = 0, 1,..., ni 1 belong to FL2 . Moreover, it is easy to see (i) − that am for m = 0, 1,..., ni 1 and i = 0, 1,..., n lie in FL1 . Then the field (i) − FL generated by a over the field of complex numbers is a field of abelian 171 { m } functions contained in both FL1 and FL2 and hence admitting for periods, all

the elements of L1 and L2. Since FL1 is algebraic over FL, FL is an algebraic

function ﬁeld of n variables and hence FL2 is also algebraic over FL. Let P be n a period matrix of FL and L, the associated lattice in C . Then L contains both L1 and L2. Hence P1 = PR1 and P2 = PR2 for nonsingular rational integral matrices R1 and R2. Thus P1 = P2 M for a nonsingular rational matrix M.

Conversely, if P1 = P2 M for a nonsingular rational matrix M, then there exists a Riemann matrix P such that P1 = PR1 and P2 = PR2 for nonsingular n rational integral matrices R1 and R2 respectively. If L is the period-lattice in C associated with P, the ﬁeld FL of abelian functions for L, is contained in both

FL1 and FL2 . Moreover FL1 and FL2 are algebraic over FL. Thus the condition is also suﬃcient. n Two lattices L1 and L2 in C , of rank 2n over the ﬁeld of real numbers and having P1 and P2 for period-matrices respectively are said to be commensurable if there is another lattice L of rank 2n containing both L1 and L2 or Modular Functions and Algebraic Number Theory 153

equivalently: P1 = P2 M for a nonsingular rational matrix M. Thus in Propo- sition 19, the necessary and sufficient condition may also be stated that the lattices L1 and L2 associated with P1 and P2 are commensurable. Let f (z) be a non-degenerate abelian function with period matrix P and period lattice L and let m be a scalar. The function f (mz) has period matrix 1 1 m− P. If m is rational, then the period lattice associated with m− P is clearly commensurable with L and hence f (mz) depends algebraically on the field FL, in view of Proposition 19. This is the analogue of the well-known multiplication theorem for elliptic functions. We now ask for all scalars m such that if f (z) FL, f (mz) is algebraic over FL. Let us first consider the case n = 1. ∈ Let p1, p2 be two independent periods of an elliptic function f (z) such that every other period p of f (z) is of the form rp1 + sp2 with rational integral r and s. In order that f (mz) be algebraic over the field of elliptic functions with periods p1 and p2, it is necessary and sufficient that

a b m(p1 p2) = (p1 p2) c d with rational a, b, c, d. This means that m is the root of a quadratic equation 172 with rational coefficients. More precisely, m should be an imaginary quadratic 1 irrationality lying in the field Q(p2 p1− ) where Q is the field of rational numbers. If we require that f (mz) should again be an elliptic funciton with p1 and p2 as periods, then a, b, c, d should be rational integers. This was the principle of complex multiplication for elliptic functions formulated by Abel in his work, “Recharches sur les fonctions elliptiques”. The idea of complex multiplication in its simplest form, however, appears to be contained implicitly in the work of Fagnano, concerning the doubling of an arc of the lemniscate of Bernoulli. If f (z) is a non-degenerate abelian function with period matrix P and period 1 lattice L and Q is an n-rowed complex non-singular matrix, then f (Q− z)isa non-degenerate abelian function with period matrix QP. We formerly wished to find all complex numbers m such that for every f (z) FL, f (mz) is algebraic ∈ over FL. We now ask, more generally, for all n-rowed complex non-singular 1 matrices Q such that for every f (z) FL, f (Q− z) is algebraic over FL. By Proposition 19, a necessary and sufficient∈ condition to be satisfied by Q is that

QP = PM (135) for some rational non-singular 2n-rowed matrix M. The matrix M is called a multiplier of P and Q, a complex multiplication of P. Trivial examples of multipliers are λE, where λ Q and E = E(2n) is the 2n-rowed identity matrix. ∈ Modular Functions and Algebraic Number Theory 154

From QP = PM, it follows that QP = PM, since M is rational. Hence (135) is equivalent to the condition

Q 0 B = BM. 0′ Q Since B is non-singular, we observe that for given M, Q is uniquely determined by 1 Q 0 BMB− = (136) 0′ Q and vice versa. The problem of determining all the complex multiplications Q 173 of a given Riemann matrix P reduces to that of ﬁnding all 2n-rowed rational 1 non-singular matrices M such that BMB− breaks up as in (136) for some n- rowed complex non-singular Q. We now drop the condition that M be non-singular and call M, a multiplier 1 still, if the matrix BMB− breaks up as in (136) or equivalently PM = QP for some Q. We may denote by R, the set of all multipliers M of P. We see ﬁrst that R is a ring containing identity. For, if M , M R, then for some Q and 1 2 ∈ 1 Q2, we have Q1P = PM1 and Q2P = PM2. Hence

P(M1 + M2) = (Q1 + Q2)P,

PM1 M2 = Q1PM2 = Q1Q2P and thus M1 + M2, M1 M2 R. Further the 2n-rowed identity matrix E lies in R and serves as the identity∈ in R. Since for M R and λ Q, λM also lies in R, we see that R is an algebra over Q and in fact,∈ of ﬁnite∈ rank over Q. 1 We now remark that if M R, M∗ = AM′A− also lies in R. This is simple ∈ Q 0 to verify. Let us set G = BAB′, T = ; then 0′ Q 1 1 HQ′H− 0 GT ′G− = .  0 HQ H 1  ′ ′ −     1 1  1 1 Further, since B′T ′ = M′B′ and B′ = A− B− G, we have A− B− GT ′ = 1 1 1 1 M′A− B− G i.e. (GT ′G− )B = B(AM′A− ). This proves that M∗ R. The mapping M M has the following properties, viz. if M , M R,∈ then → ∗ 1 2 ∈ 1 (M1 + M2)∗ = A(M1′ + M2′ )A− = M1∗ + M2∗, 1 1 1 (M1 M2)∗ = AM2′ M1′ A− = AM2′ A− AM1′ A− = M2∗ M1∗, 1 (M∗)∗ = (AM′A− )∗ = M. Modular Functions and Algebraic Number Theory 155

Thus the mapping M M∗ is an involution of R and is usually called the Rosati involution. Further,→ if the matrix Q corresponds to the multiplier M 1 ∈ R, then to M∗ corresponds the complex multiplication Q∗ = HQ′H− . Thus R is an involutorial algebra of finite rank over Q. Such algebras have 174 been extensively studied by A. A. Albert who has also determined the involutorial algebras which can be realized as the rings of multipliers of Riemann matrices. We shall not go into the general theory of such algebras. We are interested only in involutorial algebras which can be realized as the ring of multipliers of a Riemann matrix P and which, in addition, are commutative rings free from divisors of zero. Such a ring of multipliers R is necessarily a field and in fact an algebraic number field k of degree m say, over Q. So R gives a faithful representation of k into M2n(Q), the full matrix algebra of order 2n over Q. Before we proceed further, we need to study this representation of k by R more closely. First, we have the so-called regular representation of k with respect to a (i) basis ω1,...,ωm of k over Q. Namely, let for γ k, γ , i = 1, 2,..., m denote the conjugates of γ over Q. Then we have for l =∈ 1, 2,..., m

m γωl = ωqcql, (137) q=1

where cql Q. If we denote by C the matrix (cql), 1 q, l m with q and l respectively∈ as the row index and column index, then≤ the mapp≤ ing γ C M (Q) is an isomorphism and gives the regular representation of k with→ ∈ m respect to the basis ω1,...,ωm of k over Q. If we change the basis, we get an equivalent representation. Hereafter, we shall always refer to the regular representation of k with respect to a ﬁxed basis ω1,...,ωm of k over Q. Now, from (??), we have

m ( j) ( j) ( j) γ ωl = ωq cql, j = 1, 2,..., m. (137)∗ q=1

Let [γ] denote the m-rowed diagonal matrix [γ(1),...,γ(m)], with γ(1),...,γ(m) as diagonal elements. Further, let us denote by Ω, the m-rowed square matrix ( j) (ωi ) where i and j are respectively the column index and row index. Now (137)∗ reads as [γ]Ω = ΩC.

Or, since Ω is non-singular, 175

1 [γ] = ΩCΩ− . (138) Modular Functions and Algebraic Number Theory 156

Conversely, if G = [γ1,...,γm] is a diagonal matrix with diagonal elements 1 (i) γ1,...,γm such that Ω− GΩ is a rational matrix C, then necessarily γi = γ , i = 1, 2,..., m, for some γ k. In fact, ∈ 1 1 G = ΩCΩ− = ΩC(Ω′Ω)− Ω′.

Now, the matrix Ω′Ω = (S (ωkωl)) is rational and hence, also, the matrix 1 C(Ω′Ω)− = (rkl) is rational. thus m (k) (k) γk = ωp rpqωq p,q=1 with r Q, for k = 1, 2,..., m. If we set pq ∈ m γ = ωprpqωq p,q=1

(i) then we see that γi = γ for i = 1, 2,..., m. We shall make use of this remark later. In the sequel, if C1,..., Cr are square matrices then [C1,..., Cr] shall stand for the direct sum of C1,..., Cr. We shall now prove a theorem concerning an arbitrary faithful representation of k into M2n(Q). Theorem 14. Let k be an arbitrary algebraic number ﬁeld of degree m over Q and let ψ : γ G M (Q) be faithful representation of k, of order 2n. Then → ∈ 2n (i) 2n = qm, for a rational integer q 1 and ≥ (ii) there exists a 2n-rowed non-singular rational matrix T independent of γ such that 1 T − GT = [C,..., C], C being the image of γ under the regular representation of k. m i Proof. Let γ k generate k over Q and let f (x) = ai x (am = 1) be the 176 ∈ i=0 irreducible polynomial of γ over Q. If ψ(γ) = G, we can ﬁnd a complex non- 1 singular matrix W such that W− GW is in the normal form, namely γ 1 0 0 i 0 γi 1 0 1   W− GW = [G1,..., Gh], Gi =  0     1     0 0 γ   i   Modular Functions and Algebraic Number Theory 157

(2n) m i and γ1,...,γh are eigenvalues of G. Using the fact that a0E + i=1 aiG = 0, (2n) m 1 i we see that γi are conjugates of γ over Q. Again, from a0E + i=1 aiW− G W = 1 0, we conclude that W− GW is necessarily in the diagonal form. For, otherwise, even if one Gi is of the form with the elements just above the main m j 1 diagonal equal to 1, then we obtain ja jγi − = 0 contradicting the fact that j=1 1 γi has an irreducible polynomial of degree m. Thus W− GW = [γ1,...,γ2n] where γ1,...,γ2n are conjugates of γ over Q. Moreover, since the characteris- tic polynomial xE G of G has rational coeﬃcients and has for its zeros only those of f (x), it| follows− | that xE G = ( f (x))q for some rational integer q. | − | Thus 2n = mq, which proves (i). Moreover γ1, γ2,...,γ2n are just the numbers γ(1),...,γ(m) repeated q times. We may therefore suppose that

1 W− GW = [[γ],..., [γ]], where [γ] = [γ(1),...,γ(m)]. But from (138), we have

1 1 W− GW = V[C,..., C]V− , (139) where V = [Ω,..., Ω]. m 1 Now, since 1, γ,...,γ − form a basis of k over Q, we have, in order to prove (ii), only to ﬁnd a rational matrix T such that

1 T − GT = [C,..., C].

From (139), we see that there exists a non-singular complex matrix U = (ui j), 177 1 i, j 2n such that ≤ ≤ GU = U[C,..., C]. (140) The matrix equation (140) may be regarded as a system of linear equations in ui j with rational coefficients. Since there exists a solution in complex numbers for this system of linear equations, we know from the theory of linear equations that there exists at least one non-trivial solution for this system, in rational numbers. The rational solutions of this system form a non-trivial vector space of finite dimension over Q. Let U1,..., Ur be the matrices corresponding to a set of generators of this vector space. From the theory of linear equations again, we see that all complex solutions U of (140) are necessarily of the form z1U1 + + zrUr with arbitrary complex z1,..., zr. Now, we know that there exists at least one complex solution U of (140) for which U , 0. This means that the polynomial z U + + z U in z ,..., z does not| vanish| identically. | 1 1 r r| 1 r Hence we can find λ1,...,λr Q such that λ1U1 + + λrUr , 0. If we set T = λ U + + λ U , then T∈ , 0 and | | 1 1 r r | | GT = T[C,..., C], Modular Functions and Algebraic Number Theory 158 i.e. 1 1 T − GT = [C,..., C] = V− [[γ],..., [γ]]V. Our theorem is therefore proved.

The theorem above gives us the exact form of the multipliers when the ring of multipliers is an algebraic number field. With this preliminary investigation regarding the ring of multipliers, we may now proceed to define the generalized half planes and the general modular group. Let A be a 2n-rowed non-singular rational alternate matrix. We denote by BA, the set of all period matrices P which are polarized with respect to A. If P B , it is clear that, for n-rowed complex non-singular Q, QP B . ∈ A ∈ A Now, if A1 and A2 are two such alternate matrices, then we can find a ratio- 178 nal non-singular matrix S such that A1 = S A2S ′. It is easy to verify that BA2 consists precisely of the period matrices of the form PS , for P BA1 . Thus there is no loss of generality in confining ourselves to a fixed 2n-rowed∈ rational 0 E alternate matrix A with A , 0; we might take for A the matrix E 0 where E is the n-rowed identity| matrix.| − Denoting by R, the group of n-rowed complex non-singular matrices, we know that if P B , then for every Q R, QP B . Now, we introduce an ∈ A ∈ ∈ A equivalence relation in BA; namely, two matrices P1, P2 BA are equivalent if P = QP , for Q R. The resulting set R B of equivalence∈ classes is 1 2 ∈ \ A denoted by HA and is called a generalized half-plane. It may be verified that HA can be identified with the space of n-rowed complex symmetric matrices Z = X + iY with Y > 0. Let ΓA be the group of 2n-rowed unimodular matrices U such that UAU′ = A. If P BA, then for U ΓA, PU BA. On BA, the group ΓA acts in the homogeneous∈ manner; namely,∈ if P =∈(FG) B and U = A C Γ where ∈ A B D ∈ A A, B, C, D, F, G are n-rowed square matrices, then PU = (FA +GB FC +GD). A C On HA, ΓA acts in the inhomogeneous way; namely, if Z HA and U = B D 1 ∈ ∈ ΓA then U takes Z to (A + ZB)− (C + ZD). The group ΓA is the well-known modular group of degree n. It has been shown by Siegel that ΓA acts as a discontinuous group of analytic homeomorphisms of HA onto itself and that a nice fundamental domain FA can be constructed for ΓA in HA. It is known that FA is a so-called ‘complex space’. Let R be the ring of multipliers of a Riemann matrix P polarized with respect to A. Corresponding to M R, there is a Q uniquely determined by (136). We denote this set of Q by∈D and shall, hereafter, refer to a Q D corresponding to an M R. ∈ ∈ For the given A and R, we denote by BA,R the set of all n-rowed Riemann matrices P polarized with respect to A and having the property that for every Modular Functions and Algebraic Number Theory 159

M R, PM = QP where Q D corresponds to M R. Clearly BA,R is not empty,∈ since it contains the Riemann∈ matrix P we started∈ with. Moreover, let RR denote the group of K R such that QK = KQ for all Q D. It is obvious ∈ ∈ that if P B R and K RR, then KP B R. We now introduce in B R ∈ A, ∈ ∈ A, A, the equivalence relation that if P , P B R, then P and P are equivalent 179 1 2 ∈ A, 1 2 provided that P = KP for K RR. The resulting set of equivalence classes 1 2 ∈ is denoted by HA,R and called the generalized half-plane corresponding to R. It is to be noted that HA,R may also be deﬁned as follows: If B∗A,R denotes the set of all n-rowed Riemann matrices P polarized with respect to A

and admitting for multipliers all elements of R, then we introduce in B∗A,R the equivalence relation that P , P B are equivalent if P = KP for K R. 1 2 ∈ ∗A,R 1 2 ∈ The resulting set of equivalence classes can be easily identified with HA,R. Let now ΓA,R be the subgroup of ΓA, consisting of unimodular U such that UM = MU for all M R. We call ΓA,R the modular group corresponding to R. Unless R consists of∈ just the trivial multipliers λE where λ Q and E is ∈ the 2n-rowed identity matrix, ΓA,R is, in general, much smaller than ΓA. Now, if P B R and U Γ R, it is easy to see that PU B R. The group Γ R ∈ A, ∈ A, ∈ A, A, acts on H R in the following way: namely, if U Γ R, then U takes the class A, ∈ A, containing P B R into the class containing PU. It is known that in this way ∈ A, ΓA,R acts discontinuously on HA,R and one can construct a fundamental domain FA,R for ΓA,R in HA,R. The space HA,R is the most general half-plane and the group ΓA,R is the most general modular group of degree n. Modular groups of such general type were first considered by Siegel. The modular group of degree n over a totally real algebraic number field arose in a natural way in connection with the researches of Siegel on the analytic theory of quadratic forms over algebraic number fields. Later, the modular group of degree n over a simple involutorial algebra was discussed by Siegel fully, in his work, “Die Modulgruppe in einer einfachen involutorischen Algebra”. As indicated earlier, we are interested in finding HA,R and ΓA,R only when R is a field i.e. an algebraic number field k of degree m, say, over Q. We shall see presently that we are led in a natural fashion to the Hilbert modular group, when k is totally real. First, we might make some preliminary simplification. We might suppose that if M R corresponds to γ k, then M is of the form [C,..., C] where C is the image∈ of γ under the regular∈ representation. In fact, by Theorem 14, 1 we can find a rational matrix T such that T − MT = [C,..., C] uniformly for 180 1 1 all M R. Let us now set A1 = T − AT ′− and let R1 be the set of matrices of the form∈ T 1 MT for M R. Then R gives an equivalent representation of k − ∈ 1 and all the matrices in R1 are of the desired form. Moreover, BA1,R1 consists Modular Functions and Algebraic Number Theory 160

precisely of the matrices of the form PT where P B R. Hence we might as ∈ A, well consider BA1,R1 instead of BA,R and the corresponding half-plane HA1,R1 instead of HA,R. Thus, there is no loss of generality in our assumption regarding the multipliers M. Now if γ k corresponds to M = [C,..., C] R and Q D corresponds ∈ ∈ ∈ B = P to the multiplier M, then for P A,R and B P , we have ∈ 1 Q 0 1 B− B = V− [[γ],..., [γ]]V, (141) 0 Q

(1) (m) where [γ] = [γ ,...,γ ]. Hence the eigenvalues γ1,...,γn of Q are just conjugates of γ and on the other hand, all conjugates of γ occur among the eigenvalues of Q and Q. Moreover, from (??), we see readily that we can ﬁnd a non-singular matrix W uniformly for all Q D such that ∈ 1 W− QW = [γ1,...,γn]. (142)

We claim again that there is no loss of generality in assuming the complex multiplications Q D to be already in the diagonal form as in (142). For, ∈ 1 we could have started with the Riemann matrix W− P instead of P right at the beginning and then the corresponding D would have the desired property. We could then have taken the corresponding half-plane HA,R. We shall now study the eﬀect of the involution in R on the multiplications Q D. The involution in R corresponds to an automorphism γ γ∗ in k =∈Q(γ). This automorphism in k may be extended to all the conjugates→ k(i) (i) (i) of k by prescription (γ )∗ = (γ∗) . Thus, if M, M∗ are the multipliers in R corresponding to γ and γ∗ respectively, in k and Q, Q∗ are the corresponding complex multiplications in D then Q = [γ1,...,γn] and Q∗ = [γ1∗,...,γn∗]. On the other hand we know from p. 173 that

1 Q∗ = HQ′H− , i.e. 181 Q∗H = HQ′, (143)

H being positive hermitian. If h1,..., hn are the diagonal elements of H, then hi > 0 and moreover, from (??), we have

γk∗hk = hkγk, k = 1, 2,..., n, i.e.

γk∗ = γk, k = 1, 2,..., n. Modular Functions and Algebraic Number Theory 161

Thus the automorphism in the conjugate ﬁelds k(i) is given by

(i) (i) (i) γ (γ )∗ = γ . (144) → We now contend that k is either totally real or totally complex. For, if k has at least one real conjugate say k(1), then from (144), we see that the - automorphism is identity on k(1) and hence on all conjugates of k. But then ∗

(i) (i) (i) γ = (γ )∗ = γ . i.e. all conjugates of k are real. Thus k is totally real and in the contrary case, k is totally complex. In the latter case, we know that the -automorphism is not identity on k and has a fixed field k k such that k has∗ degree 2 over k . But 1 ∈ 1 k1 is necessarily totally real since the -automorphism is identity on k1 and all its conjugates. Thus, in the second case,∗ k is an imaginary quadratic extension of a totally real field k1 of degree m/2. In the case when k is totally real, q = 2n/m is necessarily even. For, if M R corresponds to γ k and if Q D corresponds to M R, then the eigenval-∈ ∈ ∈ ∈ ues of Q are γ1,...,γn and the eigenvalues of M are just γ1,...,γn, γ1,..., γn i.e. γ1,...,γn, γ1,...,γn. But we know that the set γ1,...,γn, γ1,...,γn has to run over the full set of conjugates γ(1),...,γ(m) exactly q times. So necessarily that set γ1,...,γn itself has to run over the complete set of conjugates γ(1),...,γ(m) a certain number of times, i.e. in other words, q is necessarily even. Thus when k is totally real, n = (q/2)m and when k is totally complex, 182 n = (m/2)q. We are not interested in the latter case since, as it may be verified, the space HA,R in this case, is of complex dimension zero and we can have no useful function-theory in this space. We shall therefore consider only the case when the ring of multipliers R of the given n-rowed Riemann matrix P, gives a faithful representation of a totally real algebraic number field k of degree m = n/(q/2) and moreover, we may suppose that m is as large as possible i.e. q = 2 and m = n. The other cases may be dealt with in a similar fashion. We now prove the following

Theorem 15. Let P be an n-rowed Riemann matrix polarized with respect to a rational alternate matrix A and let the ring of multipliers R of P give a faithful representation of a totally real algebraic number ﬁeld k of degree n over Q. Then the generalized half-plane HA,R is the product of n complex half-planes and the modular group ΓA,R is of the type of the Hilbert modular group. Modular Functions and Algebraic Number Theory 162

Proof. (We shall retain, in the course of this proof, the simpliﬁcation carried out already regarding R and D and shall follow our earlier notation throughout). We know that if k = Q(λ), then, for P B R, ∈ A, 1 1 [λ]PV− = PV− [[λ][λ]], (145) where [λ] = [λ(1),...,λ(n)] and V = [Ω, Ω]. 1 Let us write PV− as ((ri j)(si j)) where (ri j) and (si j) are n-rowed square matrices. It follows from (145) then, that

( j) (i) ri jλ = λ ri j, ( j) (i) si jλ = λ si j.

(i) ( j) 1 Since, for i , j, λ , λ we see that PV− is necessarily of the form ([ξ][η]) where [ξ] = [ξ1,...,ξn] and [η] = [η1,...,ηn].

Moreover, since M∗ = M, we have MA = AM′ for all M R. In particular, we see that ∈ [[λ], [λ]]VAV′ = VAV′[[λ], [λ]]. K L Splitting up VAV′ and M N with n-rowed square matrices K, L, M, N, we see 183 again, by the same argument as above, that [] [ν] VAV′ = [κ] [ρ] where [], [ν], [κ] and [ρ] and n-rowed diagonal matrices. Since A is alternate, necessarily we have 0 [ν] VAV′ = , [ν] 0 − 1 1 where [ν] = [ν1,...,νn]. But now since A is rational, Ω− [ν]Ω′− is rational. 1 As a consequence, Ω− [ν]Ω is again rational and hence by an earlier argument (see p. 174), we have [ν] = [ν(1),...,ν(n)] for a ν k. Thus ∈ 1 1 0 Ω− [ν]Ω′− A = 1 1 . Ω− [ν]Ω′− 0 − 1 1 It is to be noted that ν , 0 and further Ω− [ν]Ω′− is not diagonal, in general. Let now P BA,R be the representative of a point in HA,R. Then from the 1 ∈ fact that i− PAP′ > 0, we obtain

1 1 1 i− PV− VAV′(PV− )′ > 0, Modular Functions and Algebraic Number Theory 163 i.e. 1 0 [ν] i− ([ξ][η]) ([ξ][η]) > 0, [ν] 0 − i.e. (k) 1 0 < h = ν i− (ξ η η ξ ), for k = 1, 2,..., n. k k k − k k 1 If we set τ j = η jξ−j , j = 1, 2,..., n and [τ] = [τ1,...,τn], then

1 1 (n) [ξ]− PV− = (E [τ]) (146) and the conditions above are just

( j) ν Im(τ j) > 0, j = 1, 2,..., n.

Now [ξ] commutes with all the multiplications Q Q since they are in the 184 1 ∈ diagonal form. Hence we could take [ξ]− P instead of P as a representative of the corresponding point in HA,R. Thus, in view of (146), since V is independent of P, the space HA,R may be identiﬁed as the set of n-tuples (z1,..., zn) with (i) complex zi satisfying ν Im(zi) > 0 i.e. HA,R is a product of n half-planes. If ν > 0, then HA,R is just the product of n upper half-planes. For the general theory, we should take HA,R as the product of n (not necessarily upper) half- planes. The ﬁrst statement in our theorem has thus been proved. We shall now determine in explicit terms, the group ΓA,R which consists of 2n-rowed unimodular matrices U such that UAU′ = A and UM = MU for all M R. ∈Since UM = MU for all M R, we have, in particular, ∈ 1 1 UV− [[λ][λ]]V = V− [[λ][λ]]VU, i.e. 1 1 VUV− [[λ][λ]] = [[λ][λ]]VUV− . By the same argument as above, we have necessarily

1 [δ] [β] VUV− = , [γ] [α] where [α], [β], [γ], [δ] are diagonal matrices. Again, since the elements of 1 Ω− [δ]Ω are rational integers, it follows by an earlier remark (see p. 174) that [δ] = [δ(1),...,δ(n)] for a δ k. Similarly, [β] = [β(1),...,β(n)], [γ] = [γ(1),...,γ(n)] and [α] = [α(1),...,α∈ (n)] for α, β, γ k. Let us denote by D the order in k, consisting of numbers ρ k for which the∈ matrix Ω 1[ρ(1),...,ρ(n)]Ω ∈ − Modular Functions and Algebraic Number Theory 164 is a rationla integral matrix. Then α, β, γ, δ D. The order D coincides with ∈ the ring o of all algebraic integers in k, when ω1,...,ωn is a basis of o. Now, from UAU′ = A, we have

[δ] [β] 0 [ν] [δ] [γ] 0 [ν] = [γ] [α] [ν] 0 [β] [α] [ν] 0 − − i.e. 185 ν(k)(α(k)δ(k) β(k)γ(k)) = ν(k), k = 1, 2,..., n, − i.e. αδ βγ = 1. − Thus ΓA,R consists of 2n-rowed matrices of the form

1 [δ] [β] V− V [γ] [α] where α, β, γ, δ are elements of D satisfying αδ βγ = 1. − We know how Γ R acts on B R namely, if U Γ R, then U takes P B R A, A, ∈ A, ∈ A, to PU. Now we may describe the action of ΓA,R on HA,R as follows. If the point 1 (τ ,...,τ ) H R corresponds to the matrix PV with P B R, then the 1 n ∈ A, − ∈ A, element U of ΓA,R maps (τ1,...,τn) on the point in HA,R which corresponds to 1 PUV− . But now

1 1 1 PUV− = PV− VUV− [δ] [β] = [ξ](E[τ]) [γ] [α] = [ξ]([δ] + [γ][τ][β] + [α][τ]).

1 Hence to PUV− corresponds the point

α(1)τ + β(1) α(n)τ + β(n) 1 ,..., n (1) (1) (n) (n) γ τ1 + δ γ τn + δ in HA,R. Thus ΓA,R is represented as the group of fractional linear transformations α(1)τ + β(1) α(n)τ + β(n) τ ,...,τ 1 ,..., n ( 1 n) (1) (1) (n) (n) → γ τ1 + δ γ τn + δ

(of H R onto itself) with α, β, δ D such that αδ βγ = 1. If ω ,...,ω is A, ∈ − 1 n a basis of o, ΓA,R is the well-known Hilbert modular group. In other cases, ΓA,R is seen to be commensurable with the Hilbert modular group. Modular Functions and Algebraic Number Theory 165

Theorem 15 is therefore completely proved. 186 We have considered above only the case m = n. The case when n/m = p > 1 can be dealt with in a similar fashion. Here HA,R is a product of n generalized half-planes of degree p and ΓA,R is the group of mappings

Z = (Z ,..., Z ) H R (Z∗,..., Z∗) H R 1 n ∈ A, → 1 n ∈ A, (i) (i) (i) (i) 1 A B where Zi∗ = (A Zi + B )(C Zi + D )− and C D is a modular matrix of degree p, with elements which are algebraic integers in k. The field of modular functions belonging to ΓA,R in this case has been systematically investigated by I.I. Piatetskii-Shapiro in recent years. In the succeeding sections, we shall define the Hilbert modular function and one of the main results which we wish to prove is that the Hilbert modular functions form an algebraic function field of n variables. Our proof will run essentially on the same lines as that of K.B. Gundlach. But first we need to construct a workable fundamental domain in the space HA,R for the Hilbert modular group. This shall be done in the next section.

2 Fundamental domain for the Hilbert modular group

Let K be a totally real algebraic number field of degree n over Q, the field of rational numbers and let K(1)(= K), K(2),..., K(n) be the conjugates of K. For our purposes in future, we first extend K to K by adding an element which satisfies the following conditions, namely ∞ a + = , for a complex number a ∞ ∞ a = , for complex a , 0 ∞ ∞ = ∞ ∞ ∞ 1 = 0 ∞ 1 = 0. ∞ (The expressions , 0 and / are undefined). We define the conjugates (i) of to∞ be ± ∞, again. ∞ ∞ ∞ ∞ ∞ ∞ (i) Corresponding to each K , i = 1, 2,..., n, we associate a variable zi in 187 the extended complex plane subject to the restriction that if one z is , all the i ∞ other zi are also . We shall denote the n-tuple (z1,..., zn) by z and for λ K, we shall denote∞ the n-tuple (λ(1),...,λ(n)) again by λ, when there is no risk∈ of Modular Functions and Algebraic Number Theory 166

confusion. For given z = (z1,..., zn) the norm N(z) shall stand for z1,..., zn and the trace S (z) for z1 + + zn. If λ K, N(λ) and S (λ) coincide with the usual norm and trace in K respectively. ∈ α β Let S denote the group of 2-rowed square matrices S = γ δ with α, β, γ, δ K and αδ βγ = 1. Let E denote the normal subgroup of S consist- ∈ 1 0 − 1 0 ing of 0 1 and −0 1 . Then, for the factor group S/E, we have a faithful representation as the− group of mappings

z = (z ,..., z ) z = (z∗,..., z∗) 1 n → M 1 n with α( j)z + β( j) = j z∗j ( j) ( j) γ z j + δ corresponding to each M = α β S. We shall denote z as (αz + β)/(γz + δ) γ δ ∈ M symbolically. Let, further, z j = xj + iy j, z∗j = x∗j + iy∗j. Writing z = x + iy where x = (x1,..., xn) and y = (y1,..., yn) we shall denote (y1∗,..., yn∗) by yM and (x ,..., x ) by x so that z = x + iy . It is easy to see that for M , M S, 1∗ n∗ M M M M 1 2 ∈

zM1 M2 = (zM2 )M1. (1) (n) α β α α As usual, for M = S, M = ,..., . γ δ ∈ ∞ γ(1) γ(n) α β Let M be the subgroup of S consisting of γ δ with α, β, δ o (the ring of integers in K). The factor group M/E is precisely the (inhomogeneous)∈ Hilbert modular group, which we shall denote hereafeter by Γ. One can consider more generally than M, the group M0 of 2-rowed square matrices α β where α, β, γ, δ o and αδ βγ = ǫ with ǫ being a totally γ δ ∈ − positive unit in K. Let E0 be the subgroup of M0 consisting of matrices of the 188 ǫ 0 form 0 ǫ where ǫ is a unit in K. Further, let ρ1(= 1), ρ2,...,ρν be a complete set of representatives of the group of units ǫ > 0 in K modulo the subgroup of squares of units in K. Then it is clear that M0/E0 is isomorphic to the group of substitutions z ρz where z z is a Hilbert modular substitution and → M → M ρ = ρi for some i. Thus the study of M0/E0 can be reduced to that of Γ. The group Γ is, in general, “smaller than” M0/E0 and is called, therefore the narrow Hilbert modular group, usually. Two elements λ, in K are equivalent (in symbols, λ ), if for some ∼ M = α β M, γ δ ∈ αλ + β = λ = . M γλ + δ This is a genuine equivalence relation. We shall presently show that K falls into h equivalence classes, where h is the class number of K. Modular Functions and Algebraic Number Theory 167

Proposition 20. There exist λ ,...,λ K such that for any λ K, we have 1 h ∈ ∈ λ λi for some i uniquely determined by λ. ∼ Proof. Any λ K is of the form ρ/σ where ρ, σ o. (If λ = , we may take ρ = 1 and∈σ = 0). With λ, we may associate the∈ integral ideal∞a = (ρ, σ). We ﬁrst see that if a1 = (θ)a is any integral ideal in the class of a, then a1 is of the form (ρ1,σ1) where ρ1 and σ1 are in o such that λ = ρ1/σ1. This is quite obvious, since a1 = (ρθ, σθ) and taking ρ1 = ρθ, σ1 = σθ, our assertion is proved. Conversely, any integral ideal a1 associated with λ K in this way is necessarily in the same ideal-class as a. In fact, let λ be written∈ in the form ρ /σ with ρ , σ o and let a = (ρ ,σ ). Then we claim a is in the same 1 1 1 1 ∈ 1 1 1 1 class as a. For, since λ = ρ/σ = ρ1/σ1, we have ρσ1 = ρ1σ and hence (ρ) (σ ) (ρ ) (σ) 1 = 1 a a1 a1 a

Now (ρ)/a and (σ)/a as also (ρ1)/a1 and (σ1)/a1 are mutually coprime. Hence we have (ρ)/a = (ρ1)/a1 and similarly (σ)/a = (σ1)/a1. This means that for aθ K, ρ = ρθ and hence σ = σθ. Thus a = (θ)a, for θ K. ∈ 1 1 1 ∈ We choose in the h ideal classes, fixed integral ideals a1,..., ah such that 189 ai is of minimum norm among all the integral ideals of its class. (Perhaps, the ideal ai is not uniquely fixed in its class by this condition but there are at most finitely many possibilities for ai and we choose from these, a fixed ai). It follows from the above that to a given λ K, we can make correspond an ∈ ideal ai such that λ = ρ/σ and ai = (ρ, σ) for suitable ρ, σ o. Now, if αλ + β ∈ = λ, then to again corresponds the ideal (αρ + βσ, γρ + δσ) γλ + δ ∼ α β which is just ai since γ δ is unimodular. Thus all elements of an equivalence class in K correspond to the same ideal. We shall now show that if the same ideal ai corresponds to λ, λ∗ H, then ∈ necessarily λ λ∗. Let, in fact, λ = ρ/σ, λ∗ = ρ∗/σ∗ and ai = (ρ, σ) = (ρ∗,σ∗). ∼ 1 It is well-known that there exist numbers ξ, η, ξ∗, η∗ in ai− such that ρη σξ = 1 ρ ξ − and ρ∗η∗ σ∗ξ∗ = 1. Let A = σ η and − ρ∗ ξ∗ A∗ = ; σ∗ η∗ then A, A S and moreover ∗ ∈

1 ρ∗ ξ∗ η ξ α β A∗A− = − = σ∗ η∗ σ ρ γ δ − Modular Functions and Algebraic Number Theory 168 where

α = ρ∗η ξ∗σ, β = ρ∗ξ + ξ∗ρ, γ = σ∗η η∗σ, δ = σ∗ξ + η∗ρ − − − − clearly are integers in K satisfying αδ βγ = 1. Hence A A 1 M and thus − ∗ − ∈ ρ∗ αρ + βσ αλ + β λ∗ = = = λ. σ∗ γρ + δσ γλ + δ ∼ Thus, we see that to different equivalence classes in H correspond different ideals ai. It is almost trivial to verify that to each ideal ai, there corresponds an equivalence class in K. As a consequence, there are exactly h equivalence classes in K and our proposition is proved. We now make a convention to be followed in the sequel. We shall assume 190 that λ = 1/0 = , without loss of generality. Moreover, with λ = ρ /σ , 1 ∞ i i i we associated a fixed matrix, A = ρi ξi S; let us remark that it is always i σi ηi ∈ 1 possible to find, though not uniquely, numbers ξi, ηi in ai− such that ρiηi ξiσi = 1 0 − 1. Further, we shall suppose that A1 = 0 1 . Let, for λ K, Γλ denote the group of Hilbert modular substitutions z (αz + β)/(γz +∈δ) such that (αλ + β)/(γλ + δ) = λ. For our use later, we need→ to determine Γλ explicitly. It clearly suffices to find Γλi (for i = 1, 2,..., h) since λ = (λ ) for some M Γ and Γ = MΓ M 1. Let then for i M ∈ λ λi − α β αλi + β M = M, = λi. γ δ ∈ γλi + δ

We have (αρi + βσi, γρi + δσi) = ai = (ρi,σi). And since (αρi + βσi)σi = (γρi + δσi)ρi, we have (αρ + βσ ) (σ ) (γρ + δσ ) (ρ ) i i i = i i i . ai ai ai ai As before, (αρ + βσ ) (ρ ) (γρ + δσ ) (σ ) i i = i and i i = i . ai ai ai ai Hence, for a unit ǫ in K, we have

αρi + βσi = ǫρi, γρi + δσi = ǫσi. This means that ρi ξi∗ ǫ 0 MAi = 1 , σi ηi∗0 ǫ− Modular Functions and Algebraic Number Theory 169

1 where ξi∗ = (αξi+βηi)ǫ and ηi∗ = (γξi+δηi)ǫ lie in ai− . Further since ρiηi σiξi = 1 = ρ η σ ξ , we have − i i∗ − i i∗

ρ (η∗ η ) = σ (ξ∗ ξ ), i i − i i i − i i.e. (ρ ) (η∗ ηi) (σ ) (ξ∗ ξi) i i − = i i − . a 1 a 1 i ai− i ai−

(ξi∗ ξi) ρ /a σ /a ρ /a − 191 Again, since ( i) i is coprime to ( i) i, we see that ( i) i divides 1 , ai− i.e. (ξ ξ ) = a 2i(ρ ) for an integral ideal i. In other words, i∗ − i i− i 2 ξ∗ = ξ + ρ ζ, ζ a− . i i i ∈ i As a result, we obtain also

ηi∗ = ηi + σiζ. We now observe that 1 ǫ ζǫ− MAi = Ai 1 . 0 ǫ−

Thus Γλi consists precisely of the modular substitutions z zM, where ǫ ζ 1 2 → M = Ai 1 Ai− M with ζ ai− and ǫ any unit in K. 0 ǫ− ∈ ∈ Let Hn denote the product of n upper half-planes, namely, the set of z = (z1,..., zn) with z j = x j + iy j, y j > 0. The Hilbert modular group Γ has a representation as a group of analytic homeomorphisms z (αz + β)/(γz + δ) → of H onto itself. In the following, we shall freely identify M = α β M with n γ δ ∈ the modular substitution z zM in Γ and speak of M belonging toΓ, without risk of confusion. → The points λ = (λ(1),...,λ(n)) for λ K lie on the boundary of H and ∈ n are called (parabolic) cusps of Hn. By Proposition 20, there exist h cusps, λ = ( ,..., ), λ = (λ(1),...,λ(n)),...,λ = (λ(1),...,λ(n)) which are not 1 ∞ ∞ 2 2 2 h h h equivalent with respect to Γ and such that any other (parabolic) cusp of Hn is equivalent to exactly one of them; λ1,...,λh are called the base cusps. n If V C , then for given M Γ, VM shall denote the set of all zM for z V. For any z⊂ H , Γ shall stand for∈ the isotropy group of z in Γ, namely the group∈ ∈ n z of M Γ, for which zM = z. We shall conclude from the following proposition that Γ∈is ﬁnite, for all z H . z ∈ n Proposition 21. For any two compact sets B, B in H , the number of M Γ ′ n ∈ for which BM intersects B′ is ﬁnite. Modular Functions and Algebraic Number Theory 170

Proof. Let Λ denote the set of M = α β Γ such that B intersects B γ δ ∈ M ′ i.e. given M Λ, there exists w = (w1,..., wn) in B such that wM = (w1′ ,..., wn′ ) 192 B . Let w ∈= u + iv and w = u + iv ; then v = v γ( j)w + δ( j) 2. Since B ∈ ′ j j j ′j ′j ′j ′j j| j |− and B′ are compact, we see that

γ( j)w + δ( j) c, j = 1, 2,..., n | j | ≤ for a constant c depending only on B and B′. But then it follows immediately 0 1 that γ, δ belong to a finite set of integers in K. Let I = 1 0 and let B∗ and − B′∗ denote the images of B and B′ respectively under the modular substitution 1 z zI = z− . The images B∗ and B′∗ are again compact. Further noting → − α β 1 δ γ that if M = γ δ Γ, then IMI′ = M′− = β− α , we can show easily , ∈ , − that BM B′ if and only if B∗M 1 B′∗ . Now applying the same ∩ ∅ ′− ∩ ∅ argument as above to the compact sets B∗ and B′∗, we may conclude that if, for = α β Γ , M γ δ ,(B∗)M′ 1 B′∗ , then α, β belong to a finite set of integers in K. As a result,∈ we obtain− ∩ finally∅ that Λ is finite.

Taking a point z Hn instead of B and B′, we deduce that Γz is ﬁnite. We are now in a∈ position to prove the following

Theorem 16. The Hilbert modular group Γ acts properly discontinuously on H ; in other words, for any z H , there exists a neighbourhood V of z such n ∈ n that only for finitely many M Γ, VM intersects V and when VM V , , then M Γ . ∈ ∩ ∅ ∈ z Proof. Let z H and V a neighbourhood of z such that the closure V of V ∈ n in Hn is compact. Taking V for B and B′ in Proposition 21, we note that only for finitely many M Γ, say M ,..., M , V intersects V. Among these M , ∈ 1 r M i let M1,..., Ms be exactly those which do not belong to Γz. Then we can find a neighbourhood W of z such that WMi W = , i = 1, 2,..., s. Now let U = W V; then U has already the property that∩ for M∅ , M (i = 1, 2,..., r)U U = ∩. i M ∩ ∅ Further UMi U = , for i = 1, 2,..., s. Thus U satisfies the requirements of the theorem.∩ ∅

If z is not a fixed point of any M in r except the identity of in other words, 1 0 if Γz consists only of 0 1 , then there exists a neighbourhood V z such that for M , 1 0 , V V = . 0 1 M ∩ ∅ Let λ= ρ/σ be a cusp of Hn and a = (ρ, σ) be the integral ideal among 193 2 a1,..., ah which is associated with λ. Let α1,...,αn be a minimal basis for a− ρ ξ and further, associated with λ, let us choose a fixed A = σ η S with ξ, η 1 ∈ lying in a− . Moreover, let ǫ1,...,ǫn 1 be n 1 independent generators of the − − Modular Functions and Algebraic Number Theory 171 group of units in K. As a first step towards constructing a fundamental domain for Γ in Hn, we shall introduce ‘local coordinates’ relative to λ, for every point z in Hn and then construct a fundamental domain Gλ, for Γλ in Hn. Let z = (z ,..., z ) be any point of H and z 1 = (z ,..., z ) with z = 1 n n A− 1∗ n∗ ∗j + 1 = ,..., x∗j iy∗j. Denoting yA− (y1∗ yn∗) by y∗, we define the local coordinates of z relative to λ by the 2n quantities

1/ N(y∗), Y1,..., Yn 1, X1,..., Xn, − where Y1,..., Yn 1, X1,..., Xn are uniquely determined by the linear equations −

(k) (k) 1 yk∗ Y1 log ǫ + + Yn 1 log ǫ = log , 1 − n 1 n | | | − | 2 N(y∗)   (147) k = 1, 2,..., n 1  −  (1) (1)  X1α + + Xnα = x∗, l = 1, 2,..., n. 1 n l   The group Γλ consists of all the modular substitutions of the form z zM 1 1 ǫζǫ− 2 → where M = AT A− and T = 1 with ǫ, a unit in K and ζ a− . The 0ζǫ− ∈ 1 1 = transformation z zM is equivalent to the transformation zA− zT A− 2 → → 1 + Γ 1 ǫ zA− ζ. It is easily verified that λ is generated by the ‘dilatations’ zA− 2 → ǫ 1 = , ,..., 1 1 + α = , ,..., i zA− , i 1 2 n 1 and the ‘translations’ zA− zA− j, j 1 2 n. − → 1 1 ǫζǫ− Let now z zM with M = AT A− , T = 1 be a modular substi- → 0ζǫ− k1 kn 1 tution in Γλ and let ǫ = ǫ1 ...ǫn −1 and ζ = m1α1 + + mnαn, where ± − k1,..., kn 1, m1,..., mn are rational integers. It is obvious that the first co- − ordinate 1/ N(y∗) of z is preserved by the modular substitution z zM since 194 2 → 1 = 1 = 1 = ff N(yA− M) N(yT A− ) N(ǫ )N(yA− ) N(y∗). The e ect of the substitution z zM on Y1,..., Yn 1 is given by → − (Y1, Y2,..., Yn 1) (Y1 + k1, Y2 + k2,..., Yn 1 + kn 1), (148) − → − − as can be verified from (147). If ǫ2 = 1, then the effect of the substitution z z on X ,..., X is again just a ‘translation’, → M 1 n (X ,..., X ) (X + m ,..., X + m ). (149) 1 n → 1 1 n n 2 If ǫ , 1, then the effect of the substitution z zM on X1,..., Xn is not merely a ‘translation’ but an affine transformation given→ by

(X ,..., X ) (X∗ + m ,..., X∗ + m ), 1 n → 1 1 n n 1 where (X1∗X2∗ ... Xn∗) = (X1X2 ... Xn)UVU− , U denotes the n-rowed square ( j) (1)2 (n)2 matrix (αi ) and V is the diagonal matrix [ǫ ,...,ǫ ]. Modular Functions and Algebraic Number Theory 172

We define a point z H to be reduced with respect to Γ , if ∈ n λ 1 1 Yi < , i = 1, 2,..., n 1, −2 ≤ 2 −  (150) 1 1  X < , j = 1, 2,..., n.  2 j 2  − ≤   It is, first of all, clear that for any z Hn, there exists an M Γλ such that the ∈ ∈ k1 kn 1 equivalent point zM is reduced with respect to Γλ. In fact, for ǫ = ǫ1 ...ǫn −1 ǫ 0 1 ± − and M1 = A 1 A− , the effect of the substitution z zM is given by 0 ǫ− → 1 (??) and hence, by choosing k1,..., kn 1 properly, we could suppose that the − coordinates Y1,..., Yn 1 of zM satisfy (150). Again, since for ζ = m1α1 + − 1 + m α and M = A 1 ζ A 1, the effect of the substitution z z on n n 2 0 1 − → M2 the coordinates X1,..., Xn of zM1 is given by (149), we could suppose that for suitable m1,..., mn, the coordinates X1,..., Xn of zM2,M1 satisfy (150). It is now clear that zM2 M1 is reduced with respect to Γλ. On the other hand, let z = x + iy, w = u + iv Hn be reduced and equivalent with respect to Γλ and let the local coordinates∈ of z and w relative to λ be respectively

1/ N(yA 1 ), Y1,..., Yn 1, X1,..., Xn, − − 1 1/ N(vA− ), Y1∗,..., Yn∗ 1, X1∗,..., Xn∗. − 2 1 = 1 + 195 Further, let zA− ǫ wA− ζ. Then, in view of the fact that 1 1 Y∗ Y ( mod 1), Y , Y∗ < , i = 1, 2,..., n 1, i ≡ i −2 ≤ i i 2 − 2 we have ﬁrst Yi∗ = Yi, i = 1, 2,..., n 1 and hence ǫ = 1. Again, since we have − 1 1 X∗ X ( mod 1), X , X∗ < , j = 1, 2,..., n, j ≡ j −2 ≤ j j 2

= = , ,..., ζ = 1 = 1 we see that X∗j X j, j 1 2 n and hence 0. Thus zA− wA− , i.e. z = w. Denoting by Gλ the set of z Hn whose local coordinates Y1,..., Yn 1, X1,..., Xn satisfy (150), we conclude that∈ any z H is equivalent with respect to−Γ ,toa ∈ n λ point of Gλ and no two distinct points of Gλ are equivalent with respect to Γλ. Thus Gλ is a fundamental domain for Γλ in hn. For n = 1, the fundamental domain G is just the vertical strip 1/2 x < λ − ≤ ∗ 1 = + 1/2, y∗ > 0 with reference to the coordinate zA− x∗ iy∗. Going back to the coordinate z, the vertical strips x∗ = 1/2, y∗ > 0 are mapped into semi-circles passing through λ and orthogonal to± the real axis and the fundamental domain looks as in the ﬁgure. Modular Functions and Algebraic Number Theory 173

Figure, page 195

If z Gλ and if c1 N(y 1 ) c2, then we claim that z lies in a compact ∈ ≤ A− ≤ set in Hn depending on c1, c2 and on the choice of ǫ1,...,ǫn 1 and α1,...,αn in K. As a matter of fact, from (150), we know that −

yi∗ log c3 for i = 1, 2,..., n 1,  n  N(y∗) ≤ −     and x c for j = 1, 2,..., n. Hence we have | ∗j| ≤ 4

y∗ c i c , for i = 1, 2,..., n 1. 5 n 6 ≤ N(y∗) ≤ − Since c N(y ) c , we have indeed for all i = 1, 2,..., n, 196 1 ≤ ∗ ≤ 2 y∗ c i c . 7 n 8 ≤ N(y∗) ≤ As a result, we obtain for i, j = 1, 2,..., n,

c y∗ c , 9 ≤ i ≤ 10 x∗ c , | j| ≤ 4 where c9, c10 and c4 depend only on c1, c2 and the choice of ǫ1,...,ǫn 1, − 1 H α1,...,αn in K. Thus zA− and therefore z lies in a compact set in n, depending on c1, c2 and K. Now, we introduce the notion of “distance of a point z H from a cusp λ ∈ n of Hn”. We have already in Hn a metric given by

n dx2 + dy2 2 = i i ds 2 i=1 yi which is non-euclidean in the case n = 1 and has an invariance property with respect to Γ. But since the cusps lie on the boundary of Hn, the distance (relative to this metric) of an inner point of Hn from a cusp is inﬁnite. Hence, this metric is not useful for our purposes. ρ ξ For z Hn and a cusp λ = ρ/σ with associated A = σ η S, we deﬁne the distance∈ ∆(z,λ) of z from λ by ∈

1 1 2 1 ∆(z,λ) = (N(y 1 ))− 2 = (N(y− σz + ρ )) 2 A− | − | Modular Functions and Algebraic Number Theory 174

2 1 2 1 = (N(( σx + ρ) y− + σ y)) 2 . − For λ = , ∆(z, ) = 1/ N(y); hence the larger the N(y), the ‘closer’ is z to ∞ ∞ . ∞ Now ∆(z,λ) has an important invariance property with respect to Γ. Namely, for M Γ, we have ∈ ∆(zM,λM) =∆(z,λ). (151)

This is very easy to verify, since, by deﬁnition 197

1 ∆ = 1 1 2 (zM,λM) (N(Im(zM)A− M− ))− 1 = 1 2 =∆ (N(yA− ))− (z,λ). Moreover, ∆(z,λ) does not depend on the special choice of A associated ρi ξ1 1 with λ. For, if A1 = η S is associated with λ = ρ1/σ1, then A− A1 = ∼1 1 ∈ ǫ ζ 2 T = 1 where ǫ is a unit in K. Now N(y 1 ) = N(yA 1 )N(ǫ− ) = N(yA 1 ) 0 ǫ− A1− − − and hence our assertion is proved. Let, for a given cusp λ and r > 0, Uλ,r denote the set of z Hn such that ∆(z,λ) < r. This deﬁnes a ‘neighbourhood’ of λ and all points∈ z H ∈ n which belong to Uλ,r are inner points of the same. The neighbourhoods, Uλ,r for 0 < r < cover the entire upper half-plane Hn. Each neighbourhood Uλ,r is left invariant∞ by a modular substitution in Γ . For, by (151), if M Γ , then λ ∈ λ

∆(zM,λ) =∆(zM,λM) =∆(z,λ) and so, if z U , then again ∆(z ,λ) < r i.e. z U . A fundamental ∈ λ,r M M ∈ λ,r domain for Γλ in aλ,r is obviously given by Gλ Uλ,r. We shall now prove some interesting facts concerning∩ ∆(z,λ) which will be useful in constructing a fundamental domain for Γ in Hn. (i) For z = x + iy H , there exists a cusp λ of H such that for all cusps ∈ n n of Hn, we have ∆(z,λ) ∆(z, ). (152) ≤ Proof. If λ is a cusp, then λ = ρ/σ for ρ, σ o such that (ρ, σ) is one of the h ∈ ideals U1,..., Uh. Then

2 1 2 1 ∆(z,λ) = (N(( σx + ρ) y− + σ y)) 2 . − Modular Functions and Algebraic Number Theory 175

2 1 2 1 Let us consider the expression (N(( σx + ρ) y− + σ y)) 2 as a function of the pair of integers (ρ, σ). It remains unchanged− if ρ, σ are replaced by ρǫ, σǫ for any unit ǫ in K. We shall now show that there exists a pair of integers ρ1, σ1 in K such that

2 1 2 1 2 1 2 1 (N(( σ x + ρ ) y− + σ y)) 2 (N(( σx + ρ) y− + σ y)) 2 (153) − 1 1 1 ≤ − for all pairs of integers (ρ, σ). In order to prove (153), it obviously suﬃces to 198 show that for given c1 > 0, there are only ﬁnitely many non-associated pairs of integers (ρ, σ) such that

2 1 2 1 (N(( σx + ρ) y− + σ y)) 2 c . (154) − ≤ 11 Now, it is known from the theory of algebraic number ﬁelds that if α = (α1,...,αn) is an n-tuple of real numbers with N(α) , 0, then we can ﬁnd a unit ǫ in K such that α ǫ(i) c N(α) 1/n (155) | i | ≤ 12| | for a constant c12 depending only on K. In view of (154), we can suppose after multiplying ρ and σ by a suitable unit ǫ, that already we have

(i) (i) 2 1 (i)2 ( σ xi + ρ ) y− + σ c13, i = 1, 2,..., n, − i yi ≤ (i) (i) for a constant c13 depending only on c11 and c12. This implies that ( σ xi+ρ ) and σ(i) and as a consequence ρ(i) and σ(i) are bounded, for i =−1, 2,..., n. Again, in this case, we know from the theory of algebraic numbers that there are only finitely many possibilities for ρ and σ. Hence (154) is true only for finitely many non-associated pairs of integers (ρ, σ). From these pairs, we 2 1 2 1 choose a pair (ρ ,σ ) such that the value of N(( σ x + ρ ) y + σ y)) 2 is a 1 1 − 1 1 − 1 minimum. This pair (ρ1,σ1) now obviously satisfies (153). Let (ρ ,σ ) = b and let b = a (θ) 1 for some a and a θ K. Then a = 1 1 i − i ∈ i (ρ1θ,σ1θ). Now, in (153), if we replace ρ, σ by ρ1θ, σ1θ respectively, we get N(θ) 1. On the other hand, since ai is of minimum norm among the integral ideals| | ≥ of its class, N(a ) N(b) and therefore N(θ) 1. Thus N(θ) = 1. i ≤ | | ≤ | | Let now ρ = ρ1θ, σ = σ1θ and λ = ρ/σ. Then ai = (ρ, σ) and by definition, 1 1 2 1 2 2 2 1 2 2 ∆(z,λ) = (N(( σx + ρ) y− + σ y)) = (N(( σ1 x + ρ1) y− + σ1y)) . If we use (153), then− we see at once that ∆(z,λ) ∆−(z, ) for all cusps . For given z H , we define ≤ ∈ n ∆(z) = ∆(z,λ). λ Modular Functions and Algebraic Number Theory 176

By (i), there exists a cusp λ such that ∆(z,λ) =∆(z). In general, λ is unique, but 199 there are exceptional cases when the minimum is attained for more than one λ. We shall see presently that there exists a constant d > 0, depending only on K such that if ∆(z) < d, then the cusp λ for which ∆(z,λ) =∆(z) is unique.

(ii) There exists d > 0 depending only on K, such that, if for z = x + iy Hn, ∆(z,λ) < d and ∆(z, ) < d, then necessarily λ = . ∈

Proof. Let λ = ρ/σ and = ρ1/σ1 and let for a real number d > 0,

2 1 2 1 ∆(z,λ) = (N(( σx + ρ) y− + σ y)) 2 < d, − 2 1 2 1 ∆(z, ) = (N( σ x + ρ ) y− + σ y) 2 < d. − 1 1 1 After multiplying ρ and σ by a suitable unit ǫ in K, we might assume in view of (155) that

(i) (i) 2 1 (i)2 2/n (σ x ρ ) y− + σ y < c d , i = 1, 2,..., n. i − i i 12 Hence

1 σ(i) x + ρ(i) y− 2 < √c d1/n, | − i | i 12 1 σ(i) y 2 < √c d1/n. | | i 12 Similarly we have

(i) (i) 1 σ x + ρ y− 2 < √c d1/n, | − 1 i 1 | i 12 1 σ(i) y 2 < √c d1/n. | 1 | i 12 Now

(i) (i) 1 (i) 1 (i) (i) 1 1 ρ(i)σ ρ σ(i) = ( σ(i) x + ρ(i))y− 2 σ y 2 ( σ x + ρ )y− 2 σ(i)y 2 1 − 1 − i i 1 i − − 1 i 1 i i and hence N(ρσ σρ ) < (2c d2/n)n. | 1 − 1 | 12 n/2 If we set d = (2c12)− , then N(ρσ1 σρ1) < 1. Since ρσ1 σρ1 is an integer, 200 it follows that ρσ σρ = 0| i.e. λ =−. | − 1 − 1 Thus for d = (2c ) n/2, the conditions ∆(z,λ) < d, ∆(z, ) < d for a z H 12 − ∈ n imply that λ = . Therefore, the neighbourhoods Uλ,d for the various cusps λ are mutually disjoint. We shall now prove that for z Hn, ∆(z) is uniformly bounded in Hn. To this end, it suﬃces to prove ∈ Modular Functions and Algebraic Number Theory 177

(iii) There exists c > 0 depending only on K such that for any z = x+iy Hn, there exists a cusp λ with the property that ∆(z,λ) < c. ∈

Proof. We shall prove the existence of a constant c > 0 depending only on K and a pair of integers (ρ, σ) not both zero such that

2 1 2 1 (N(( σx + ρ) y− + σ y)) 2 < c. −

Let ω1,...,ωn be a basis of o. Consider now the following system of 2n linear inequalities in the 2n variables a1,..., an, b1,..., bn, viz.

1 1 2 (k) (k) 2 (k) (k) y− (ω a1 + + ω an) xky− (ω b1 + + ω bn) αk k 1 n − k 1 n ≤ for k = 1 , 2,..., n and

1 2 (k) (k) y (ω b1 + + ω bn) βk k 1 n ≤ for k = 1,..., n.

( j) 2 The determinant of this system of linear forms is (ωi ) = ∆ where ∆ is the discriminant of K. By Minkowski’s theorem on linear| | forms,| | this system of linear inequalities has a nontrivial solution in rational integers a1,..., an, 2n b1,..., bn if α1,...,αn β1,...,βn ∆ . Taking αi = β j = √ ∆ i = 1, 2,..., n, j = 1, 2,..., n, in particular, this system≤ | of| inequalities has a nontrivia| | l solution in rational integers, say, a1′ ,..., an′ , b1′ ,..., bn′ . Let us take ρ = a1′ ω1 + + an′ ωn and σ = b ω + + b ω . Then we obtain 1′ 1 n′ n (i) (i) 2 1 (i)2 2n ( σ x + ρ ) y− + σ y ∆ , i = 1, 2,..., n. − i i i ≤ | | 2 1 2 1 n/2 1 Hence (N(( σx + ρ) y− + σ y)) 2 2 ∆ 2 = c (say). 201 − 1≤ | | Now let (ρ, σ) = b; b = ai(θ)− for some ai and for some θ K. Since ai is of minimum norm among the integral ideals of its class, N(θ)∈ 1. Further | | ≤ ai = (ρθ, σθ). Now, for the cusp λ = ρθ/σθ = ρ/σ, we have

2 1 2 1 ∆(z,λ) = N(θ) (N(( σx + ρ) y− + σ y)) 2 c, | | − ≤ which was what we wished to prove. From (iii), we deduce that Hn = Uλ,c. λ (iv) For z H and M Γ, ∆(z ) =∆(z). ∈ n ∈ M Modular Functions and Algebraic Number Theory 178

Proof. In fact,

∆(zM) = inf ∆(zM,λ) λ

= inf ∆(z,λM 1 ) ((by 151)) λ − = inf ∆(z,λ) λ =∆(z).

We now have all the necessary material for the construction of a fundamental domain for Γ in Hn. A point z Hn is semi-reduced (with respect to a cups λ), if ∆(z) =∆(z,λ). If z is semi-reduced∈ with respect to λ, then for all cusps , we have ∆(z, ) ∆(z,λ). ≥ Let λ (= ( ,..., )), λ ,...,λ be the h inequivalent base cusps of H . 1 ∞ ∞ 2 h n We denote by Fλi , the set of all z Hn which are semi-reduced with respect to λ . Clearly F Ui , in view of∈ (iii) above. The set F is invariant under the i λi ⊂ λi,c λi modular substitution z zM, for M Γλi . For, by (iv) above, ∆(zM) = ∆(z) and further. → ∈ ∆(z) =∆(z,λi) =∆(zM, (λi)M) =∆(zM,λi). Thus ∆(z ) =∆(z ,λ ) and hence z F , for M Γ . M M i M ∈ λi ∈ λi Let G denote the closure in H of the set G and q = F G . Then λi n λi i λi ∩ λi q is explicitly deﬁned as the set of z H whose local coordinates X ,..., X , 202 i ∈ n 1 n Y1,..., Yn 1 relative to the cusp λi satisfy the conditions − 1 1 1 1 X , Y (k = 1, 2,..., n; l = 1, 2,..., n 1) −2 ≤ k ≤ 2 −2 ≤ 1 ≤ 2 − and further, for all cusps ,

∆(z, ) ∆(z,λ ). ≥ i Let F = h q . We see that the q as also F are closed in H . ∪i=1 i i n A point z qi is an inner point of qi if in all the conditions above, strict inequality holds,∈ viz. 1 1 1 1 < X < , < Y < (k = 1, 2,..., n, l = 1, 2,..., n 1) −2 k 2 −2 l 2 − ∆(z, ) > ∆(z,λi), for , λi. (150)∗ Modular Functions and Algebraic Number Theory 179

If equality holds even in one of these conditions, then z issaidtobea boundary point of qi. The set of boundary points of qi constitute the boundary of qi, which may be denoted by Bd qi. We denote by Ri, the set of inner points of qi. It is clear that the qi do not overlap and intersect at most on their boundary. A point z F may now be called an inner point of F, if z is an inner point ∈ of some qi; similarly we may deﬁne a boundary point of F and denote the set of boundary points of F by Bd F. We say that a point z H is reduced (with respect to Γ) if, in the ﬁrst place, ∈ n z is semi-reduced with respect to some one of the h cusps λ1,...,λh, say λi and then further z Gλi . Clearly F is just the set of all z Hn reduced with respect to Γ. ∈ ∈ Before we proceed to show that F is a fundamental domain for Γ in Hn, we shall prove the following result concerning the inner points of F, namely, The set of inner points of F is open in Hn. Proof. It is enough to show that each R is open in H . Let, then, z = x +iy j n 0 0 0 ∈ R j; we have to prove that there exists a neighbourhood V of z0 in Hn which is wholly contained in R j.

Now, for each z = x + iy Hn and a cusp = ρ/σ, we have ∆(z, ) = 203 2 1 2 1 ∈ (N(( σx + ρ) y− + σ y)) 2 . Using the fact that for each j = 1, 2,..., n, (−j) ( j) 2 1 ( j)2 ( j) ( σ x j + ρ ) y− + σ y j is a positive-deﬁnite binary quadratic form in σ and− ρ( j), we can easily show that

2 2 1 ∆(z, ) α(N(σ + ρ )) 2 ≥ where α = α(z) depends continuously on z and does not depend on . We can now find a sufficiently small neighbourhood W of z such that for all z W, 0 ∈ 1 2 2 1 ∆(z, ) α (N(σ + ρ )) 2 ≥ 2 0 where α = α(z ). Moreover, we can assume W so chosen that for all z W, 0 0 ∈ ∆(z,λ ) 2∆(z ,λ ). j ≤ 0 j Thus, for all cusps = ρ/σ and z W, ∈ 1 2 2 1 ∆(z, ) ∆(z,λ ) α (N(σ + ρ )) 2 2∆(z ,λ ). − j ≥ 2 0 − 0 j Now, employing an argument used already on p. 198, we can show that there are only finitely many non-associated pairs of integers(ρ, σ) such that

1 2 2 1 α (N(σ + ρ )) 2 2∆(z ,λ ). 2 0 ≤ 0 j Modular Functions and Algebraic Number Theory 180

It is an immediate consequence that except for ﬁnitely many cusps 1,...,r, we have for , λ j, ∆(z, ) > ∆(z,λ j).

Now, since ∆(z0, ) > ∆(z0,λ j) for all cusps , λ j, we can, in view of the continuity in z of ∆(z, i) (for i = 1, 2,..., r) ﬁnd a neighbourhood U of z0 such that ∆(z, k) > ∆(z,λ j) (k = 1, 2,..., r) for z U and , λ . We then have ﬁnally for all z U W 204 ∈ k j ∈ ∩

∆(z, ) > ∆(z,λ j) provided , λ j. Further, we could have chosen U such that for all z U, (150) is satisﬁed, in addition. Thus the neighbourhood V = U W ∈of z ∗ ∩ 0 satisﬁes our requirements and so R j is open. It may now be seen that the closure R of R is just q . In fact, let z q and i i i ∈ i 1 1 , let λi = Ai , z∗ = zA , A = ν = ρ/σ . Then we have ∞ i− i− ∞

∆(z, ) ∆(z∗, ν) 2 2 1 σ , 0, = = (N(( σx∗ + ρ) + (σy∗) )) 2 . ∆(z,λ ) ∆(z , ) − i ∗ ∞ If y∗ is replaces by ty∗ where t is a positive scalar factor, then the expression above is a strictly monotonic increasing function of t, whereas the coordinates X1,..., Xn, Y1,..., Yn 1 remain unchanged. Thus, if z qi and zA 1 = z∗ = x∗ + − ∈ i− (t) , iy∗, then for z = (x∗ + ity∗)Ai , t > 1, the inequalities ∆(z, ) > ∆(z,λi)( λi) are satisfied and moreover for a small change in the coordinates X1,..., Xn, Y1,..., Yn 1, the inequalities (150)∗ are also satisfied. As a consequence, if − z qi, then every neighbourhood of z intersects Ri; in other words, Ri = qi. ∈ (t) From above, we have that if z qi, then the entire curve defined by z = (x + ity ) , t 1 lies in q and hence∈ for z R , in particular, z(t) for large ∗ ∗ Ai ≥ i ∈ i t > 0 belongs to Ri, Using this it may be shown that qi and similarly Ri are connected. The existence of inner points of q j is an immediate consequence of result (ii) on p. 199. For j = 1, it may be verified that z = (it,..., it), t > 1 is an inner point of q1. Now we proceed to prove that F is a fundamental domain for Γ in Hn. We have first to show that

(α) the images F of F for M Γ cover H without gaps. M ∈ n Modular Functions and Algebraic Number Theory 181

Proof. This is obvious from the very method of construction of F. First, for any z Hn, there exists a cusp λ such that ∆(z) = ∆(z,λ). Let λ = (λi)M for some λ∈ and M Γ. We have then i ∈

∆ 1 =∆ =∆ =∆ 1 (zM− ) (z) (z,λ) (zM− ,λi) and hence z 1 Fλ . Now we can ﬁnd N Γλ such that (z 1 )N is reduced 205 M− ∈ i ∈ i M− with respect to Γλ and thus z 1 F. i NM− ∈ Next, we show that

(β) the images F of F for M Γ cover H withtout overlaps. M ∈ n 1 0 Proof. Let z1, z2 F such that z1 = (z2)M for an M , 0 1 in Γ and let z q and z q .∈ Now, since z F , we have ± 1 ∈ i 2 ∈ j 1 ∈ λi ∆(z ,λ ) ∆(z , (λ ) ) =∆(z ,λ ). 1 i ≤ 1 j M 2 j Similarly ∆(z2,λ j) ∆(z2, (λi) 1 ) =∆(z1,λi). ≤ M− Therefore, we obtain

∆(z1,λi) =∆(z2,λ j) =∆(z1, (λ j)M).

Let us first suppose that ∆(z1,λi) =∆(z2,λ j) < d. Then, since ∆(z1,λi) < d as also ∆(z1, (λ j)M) < d, we infer from the result (ii) on p. 199 that λi = (λ j)M. But λi and λ j, for i , j are not equivalent with respect to Γ. Therefore i = j and M Γλi . Again, since both z1 and z2 are in Gλi and further z1 = (z2)M with M Γ ∈, we conclude that necessarily z and z belong to BdF and indeed their ∈ λi 1 2 local coordinates X1,..., Xn, Y1,..., Yn 1 relative to λi satisfy at least one of the − finite number of conditions X1 = 1/2,..., Xn = 1/2, Y1 = 1/2,..., Yn 1 = ± ± ± − 1/2. Further M clearly belongs to a finite set M1,..., Mr of elements in ±h

Γλi . i=1 We have now to deal with the case

d ∆(z ,λ ) c and d ∆(z ,λ ) c ≤ 1 i ≤ ≤ 2 j ≤ where we may suppose that λi , (λ j)M. Let, for i = 1, 2,..., h, Bi denote the set of z H for which d ∆(z,λ ) c and z G . Then, by our remark on p. ∈ n ≤ i ≤ ∈ λi Modular Functions and Algebraic Number Theory 182

h 196 B is compact and so is B = B . Now, both z and z = (z ) belong to i i 2 1 2 M i=1 the compact set B. We may then deduce from Proposition 21 that M belongs to a ﬁnite set of elements Mr+1,..., Ms in Γ, depending only on B and hence 206 only on K. Moreover z1 satisﬁes

∆(z1,λi) =∆(z1, (λ j)M) with (λ j)M , λi. Hence z1 and similarly z2 belongs to BdF. As a result, arbitrarily near z1 and z2, there exist points z such that ∆(z,λi) , ∆(z, (λ j)M) for M = Mr+1,..., Ms. Thus, ﬁnally, no two inner points of F can be equivalent with respect to

Γ. Further F intersects only ﬁnitely many of its neighbours FM1 ,..., FMs and indeed only on its boundary. We have therefore established (ii). From (α) and (β) above, it follows that F is a fundamental domain for Γ in Hn. It consists of h connected ‘pieces’ corresponding to the h inequivalent base cusps λ1,...,λh and is bounded by a ﬁnite number of ‘manifolds’ of the form

∆(z,λi) =∆(z, (λ j)M), i, j = 1, 2,..., h, M = Mr+1,..., Ms (156) and the hypersurfaces deﬁned by 1 1 X(k) = , Y(k) = , i = 1, 2,..., n; j = 1, 2,..., n 1 i ±2 j ±2 −

(k) (k) (k) (k) where X1 ,..., Xn , Y1 ,..., Yn 1 are local coordinates relative to the base − cusp λk. The ‘manifolds’ deﬁned by (156) are seen to be generalizations of the ‘isometric circles’, in the sense of Ford, for a fuchsian group. In fact, if λi = ρi/σi and (λ j)M = ρ/σ, then the condition (156) is just

N( σ z + ρ ) = N( σz + ρ ). | − i i| | − |

If we set n = 1 and λi = or equivalently ρi = 1 and σi = 0, then the condition reads as ∞ σz + ρ = 1 | − | which is the familiar ‘isometric circle’ corresponding to the transformation ηz ξ z − of H onto itself. → σz + ρ 1 The− conditions ∆(z,λ) ∆(z,λ ) by which F was deﬁned, simply mean ≥ i λi for n = 1 and λ = that γz + δ 1 for all pairs of coprime rational integers 207 i ∞ | | ≥ (γ,δ). Thus, just as the points of the well-known fundamental domain in H1 Modular Functions and Algebraic Number Theory 183 for the elliptic modular group lie in the exterior of the the isometric circles γz + δ = 1 corresponding to the same group, F lies in the ‘exterior’ of the | | λi generalized isometric circles ∆(z,λi) =∆(z,λ). We now prove the following important result, which will be used later. Proposition 22. Let F denote the set of z F for which ∆(z,λ ) e > 0, ∗ ∈ i ≥ i i = 1, 2,..., h. Then F∗ is compact in Hn.

The proof is almost trivial in the light of our remark on p. 196. Let Bi denote the set of z G for which e ∆(z,λ ) c. Then B as also B = h B is ∈ λi i ≤ i ≤ i ∪i=1 i compact in Hn. Hence F∗ which is closed and contained in B, is again compact. Suppose instead of H , we consider the space H of (z ,..., z ) Cn with n n∗ 1 n ∈ Im(z1) < 0 and Im(zi) > 0 for i = 2,..., n. Then by means of the mapping (z ,..., z ) (z , z ,..., z ) of h onto H , F goes over into a fundamental 1 n → 1 2 n n n∗ domain for Γ in Hn∗. The general case when we take instead of Hn, the product of r uppper half-planes and n r lower half-planes, can be dealt with, in a similar fashion. − It was Blumenthal who first gave a method of constructing a fundamental domain for Γ in Hn but his proof contained an error since he obtained a fundamental domain with just one cusp and not h cusps. This error was set right by Maass. We remark finally that our method of constructing the fundamental domain F is essentially different from the well-known method of Fricke for constructing a ‘normal polygon’ for a discontinuous group of analytic automorphisms of a bounded domain in the complex plane. Our method uses only the notion of “distance of a point of Hn from a cusp”, whereas we require a metric invariant under the group, for Fricke’s method which is as follows. Let G be a bounded domain in the complex plane and Γ, a discontinuous group of analytic automorphisms of G. Further let G possess a metric D(z1, z2) for z1, z2 G, having the property that D((z ) , (z ) ) = D(z , z ) for M Γ. We choose∈ a 1 M 2 M 1 2 ∈ point z0 which is not a fixed point of any M Γ, other than the identity and a fundamental domain for Γ in G is given by the∈ set of points z G for which ∈ D(z, (z ) ) D(z, z ) forall M Γ. 0 M ≥ 0 ∈ 208 Now we know that Hn carries a Riemannian metric which is invariant under Γ. Suppose we use this metric and try to adapt Fricke’s method of constructing a fundamental domain. For our later purposes, we require a fundamental domain whose nature near the cusps should be well known. Therefore we see in the first place that the adaptation of Fricke’s method to our case is not practical in view of the fact that the distance of a point of Hn from the cusps, relative to Modular Functions and Algebraic Number Theory 184 the Riemannian metric, is infinite. In the second place, it is advantageous to adapt Fricke’s method only when the fundamental domain is compact, whereas we know that the fundamental domain is not compact in our case. Moreover, our method of construction of the fundamental domain uses the deep and in- trinsic properties of algebraic number fields. The following proposition is necessary for our later purposes.

Proposition 23. For any compact set C in Hn, there exists a constant b = b(C) > 0 such that C U , for all cusps . ∩ ,b ∅ Proof. Since C is compact, we can ﬁnd α, β > 0 depending only on C such that for z = x + iy C, we have β N(y) α. Now, for any cusp λ = (ρ/σ)(ρ, σ ∈ ≤ ≤ 2 1 2 1/∈2 o) and z = x + iy C, it is clear that ∆(z,λ) = (N(( σx + ρ) y− + σ y)) satisﬁes ∈ − 1 1 N(σ) N(y) 2 β 2 , for σ , 0 ∆(z,λ) | | 1 ≥ 1 ≥  N(ρ) N(y)− 2 α− 2 , for σ = 0. | | ≥  1 1 If we choose b for which 0 < b < min(α− 2 , β 2 ), then it is obvious that for all cusps λ, U C = . λ,b ∩ ∅ 3 Hilbert modular functions

We shall first develop here some properties of (entire) Hilbert modular forms and, in particular, show that a Hilbert modular form satisfies the regularity condition at the cusps of the fundamental domain F, automatically for n 2. We ≥ shall then prove the existence of n+1 (entire) modular forms f0(z), f1(z),..., fn(z) 209 f (z) f (z) of ‘large’ weight k such that the functions 1 ,..., n are analytically in- f0(z) f0(z) dependent. Next, we shall obtain estimates for the dimensions of the space of entire modular forms of ‘large’ weight. Finally, we shall show that every Hilbert modular function can be expressed as the quotient of two Hilbert modular forms of the same weight and use this result to prove the main theorem that the Hilbert modular functions form an algebraic function field of n variables over the field of complex numbers. Let K be a totally real algebraic number field of degree n over R and let Γ be the corresponding Hilbert modular group. Further let k be a rational integer. A complex-valued function f (z) defined in Hn is called an entire Hilbert modular form of weight k (or of dimension-k) belonging to the group Γ, if

(1) f (z) is regular in Hn,

α β k (2) for every M = γ δ Γ, f (zM)N(γz + δ)− = f (z), (??) ∈ Modular Functions and Algebraic Number Theory 185

k for every cusp λi = ρi/σi of F, f (z)N( σiz + ρi) is regular at the cusp λi • − 2πiS (βzA 1 ) i.e. it has a Fourier expansion of the form c(0) + c(β)e i− where, β in the summation, only terms corresponding to β > 0 can occur. It will be shown later, as a consequence of a slightly more general result, that for n 2, conditions 1) and 2) together imply 3). However, for n = 1, it is clear that 1)≥ and 2) do not imply 3) and one has necessarily to impose condition 3) also in the deﬁnition of an entire modular form. We shall hereafter refer to entire Hilbert modular forms belonging to Γ merely as modular forms, without any risk of misunderstanding. α β Now, if in (??), we take M = −γ −δ instead of M, we have − − − k k f (z )N( γz δ)− = f (z) = f (z )N(γz + δ)− , M − − M i.e. k nk f (z )N(γz + δ)− (( 1) 1) = 0. M − − i.e. 210 f (z)(( 1)nk 1) = 0. − − Therefore, if nk is odd, necessarily f (z) is identically zero. We are not interested in this case and we shall suppose hereafter that

nk 0( mod 2). ≡ ρ ξ Let λ = ρ/σ be a cusp of Hn, a = (ρ, σ) and A = σ η S. Further let ϑ be the different of K. We then have ∈ Theorem 17. Let f (z) be regular in an annular neighbourhood of a cusp λ of Hn defined by z Hn, 0 < r < ∆(z,λ) < R and let for every M Γλ, (??) be satisfied by f (z).∈ Then, for n 2, f (z) is regular in 0 < ∆(z,λ)∈< R and we have ≥ k 2πiS (βz 1 ) f (z)N( σz + ρ) = c(0) + c(β)e A− , − a2ϑ1 β>0 − | where the summation is over all β a2ϑ 1 with β > 0 and the series converges ∈ − absolutely, uniformly over any compact subset of Hn with 0 < ∆(z,λ) < R. α β 1 = 1 = = = = Proof. Let w zA− and g(w) f (wA) f (z). Consider M γ δ AT A− 1 ∈ ǫ ζǫ− 2 Γλ with T = 1 ǫ being a unit in K and ζ a− . Then zM = (wT )A and 0 ζǫ− ∈ k f (zM) = g(wT ). Moreover, if h(w) = N(σw + η)− , then it is easy to verify that

k h(wT ) = N(σwT + η)− Modular Functions and Algebraic Number Theory 186

k k k = N(ǫ)− N(γwA + δ)− N(σw + η)− k k = N(ǫ)− N(γz + δ)− h(w).

k Let us now deﬁne G(w) = g(w)h(w). Then, from f (zM)N(γz + δ)− = f (z), we obtain 2 k G(ǫ w + ζ) = G(wT ) = N(ǫ) G(w) In particular, we have G(w + ζ) = G(w) (158) G(ǫ2w) = N(ǫ)kG(w). 211

2 Let α1,...,αn be a minimal basis of a− . Further, let W = (W1,..., Wn) where Wi are deﬁned by

n (k) wk = αi Wi, k = 1, 2,..., n. i=1 From (158), we then see that the function H(W) = G(w) has period 1 in each variable Wi. Let wk = uk +ivk and Wk = Xk +iYk, for k = 1, 2,..., n. Further let D be the 2 2 domain in the W-space defined by v1 > 0,..., vn > 0, R− < v1,..., vn < r− . (For n = 2, v1 and v2 are bounded between the branches of two hyperbolas as in the figure. The mapping wk Wk is just an affine transformation of the w-space). It is clear from the hypothesis→ that the function H(W) is regular in D. Figure, page 211

Under the mapping W q = (q , q ,..., q ) with q = e2πiW j ( j = 1, 2,..., n) → 1 2 n j the domain D goes into a Reinhardt domain D∗ in the q-space and H∗(q) = H(W) is regular in D . If now q D , we can ﬁnd a Reinhardt domain V ∗ ∗ ∈ ∗ deﬁned by ak < qk < bk, k = 1, 2,..., n such that q∗ V D∗. Since H∗(q) | | ∈ ∈ r r 1 n 212 is regular in V, it has a Laurent expansion cr1,...,rn q1 ... qn which

2πi(r1W1+ +rnWn) H(W) = H∗(q) = cr1,...,rn e .

Since D∗ is a Reinhardt domain, it is possible to continue H∗(q) analytically from q to any point q D through a ﬁnite chain of overlapping Reinhardt ∗ ∈ ∗ Modular Functions and Algebraic Number Theory 187

domains of the type of V and if, in addition, we use the fact that the Laurent expansion is unique, we see that for all W D, ∈

2πi(r1W1+ +rnWn) H(W) = cr1,...,rn e , (159) 0,..., vn > 0, 2 2 R− < v1 ... vn < r− , we have G(w) = H(W) = c(β)e2πiS (βw). (160) β a2ϑ 1 ∈ − If ǫ is a unit in K, then the transformation w ǫ2w corresponds to an aﬃne transformation of the W-space which takes D→into itself. We therefore have from (160) that

2 G(ǫ2w) = c(β)e2πiS (βǫ w). β a2ϑ 1 ∈ − On the other hand, from (158) and (160),

G(ǫ2w) = N(ǫ)k c(β)e2πiS (βw). β a2ϑ 1 ∈ − 2 1 2 2 1 For β a ϑ− , βǫ is also in a ϑ− . On comparing the Fourier coeﬃcients of 213 G(ǫ2w∈), we obtain 2 k c(βǫ ) = N(ǫ)− c(β), i.e. c(βǫ2) = c(β) . | | | | Thus c(βǫ2m) = c(β) for any rational integer m. | | | | Let us consider a unit cube U in D, deﬁned as the set of W = (W1,..., Wn), W j = X j + iY∗j and 0 X j 1. Further let H(W) M on U. From (159), we have for β = r β + ≤ + r ≤β , | | ≤ 1 1 n n Modular Functions and Algebraic Number Theory 188

2πi(r1W1+ +rnWn) c(β) = cr ,...,r = ... H(W)e− dX1 ... dXn 1 n W U ∈ 1 1 2πS (βv∗) 2πi(r1 X1+ +rn Xn) = e ... H(W)e− dX1 ... dXn, 0 0 W U ∈ i.e. c(β) e2πS (βv∗) M, | | ≤ n ( j) where v∗ = (v1∗,..., vn∗) with v∗j = αi Yi∗ > 0. i=1 Taking βǫ2m instead of β, we have

2m c(βǫ2m) e2πS (βǫ v∗) M, | | ≤ i.e. 2m c(β) e2πS (βǫ v∗) M. (161) | | ≤ 2 1 (i) Let now β a ϑ− with β < 0 for some i. We claim that c(β) = 0, necessarily. For,∈ using the existence of n 1(n 2!) independent units in K we can ﬁnd a unit ǫ in K such that − ≥

ǫ(i) > 1, ǫ( j) < 1 for j , i. | | | | 2m Then, as m tends to inﬁnity, S (βǫ v∗) tends to . Hence, from (161), we see 214 that unless β > 0 or β = 0, c(β) = 0. Thus −∞

G(w) = c(0) + c(β)e2πiS (βw), a2ϑ1 β>0 − | i.e. k 2πiS (βz 1 ) f (z)N( σz + ρ) = c(0) + c(β)e A− , (162) − a2ϑ1 β>0 − | where the series on the right hand side converges absolutely, uniformly in every compact subset of the region r < ∆(z,λ) < R. 2 ∆ 1 Let now 0 < (z,λ) r i.e. r− N(yA− ) < . We can then ﬁnd a ≤ ≤2 ∞ 2 1 number t with 0 < t < 1 such that R− < N(tyA− ) < r− . Hence the series c(0) + c(β)e2πiS (βwt) converges absolutely. Now, since a2ϑ 1 β>0 −| c(0) + c(β)e2πiS (βw) c(0) + c(β) e2πiS (tβw) , ≤ | | | || | a2ϑ1 β>0 β − | Modular Functions and Algebraic Number Theory 189 it converges absolutely. It is now clear that the series on the right hand side of (162) converges absolutely, uniformly over every compact subset in Hn for which 0 < ∆(z,λ) r and provides the analytic continuation of f (z)N( σz + ρ)+k in 0 < ∆(z,λ)≤< R. Thus f (z) is regular in the full neighbourhood− 0 < ∆(z,λ) < R, of the cusp λ and moreover, in this neighbourhood,

k 2πiS (βz 1 ) f (z)N( σz + ρ) = c(0) + c(β)e A− . (163) − a2ϑ1 β>0 − | From the theorem above, we can deduce two important corollaries.

(a) Let Vi denote the neighbourhood 0 < ∆(z,λi) < di < c of the cusp h λi(i = 1, 2,..., h) of F and let F∗ = F Vi. Then a function f (z) − i=1 which is regular in F and satisfies (??) for all M Γ is automatically ∗ ∈ regular in the whole of F and hence in the whole of hn and is an entire modular form, for n 2. ≥ (b) For n 2, if a function f (z) satisfies conditions (1) and (2) in the defi- ≥ nition of a modular form, it satisfies condition (3) automatically and is 215 therefore regular at the cusp F.

The proofs of (a) and (b) are very simple, in view of Theorem 17. The second of the two results above was ﬁrst proved by Gotzky in the special case n = 2 and K = Q( √5), the class number h being 1. In the general case, it was proved by Gundlach. An analogue of this result in the case of modular forms of degree n was established by Koecher. In the theorem above, we might, instead of Hn, take, for example, the space n Hn∗ of (z1,..., zn) C with Im(z1) < 0, Im(z2) > 0,..., Im(zn) > 0. Then, under the same conditions∈ as in the theorem above, we have for f (z)N( σz+ρ)k 2πiS (βz 1 ) − an expansion of the form c(0) + c(β)e A− , where β runs over all 2 1 a ϑ− β>0 2 1 (1) | (2) (n) numbers of a ϑ− for which β < 0, β > 0,...,β > 0. For other types of such spaces again we have similar expansions for f (z)N( σz + ρ)k at the cusp ρ/σ. − So far, there was no restriction on k except that k should be integral and kn should be even. In view of the following proposition however, we shall henceforth be interested only in modular forms of weight k > 0.

Proposition 24. A modular form of weight k < 0 vanishes identically and the only modular forms of weight 0 are constants. Modular Functions and Algebraic Number Theory 190

Proof. First, let f (z) be a modular form of weight k < 0. Consider ϕ(z) = N(y)k/2 f (z) for z = x + iy H . We shall prove that ϕ(z) is bounded in the | | ∈ n whole of Hn. Let, in fact, λi = ρi/σi be a cusp of F and Vi, the neighbourhood k/2 k/2 of λi deﬁned by 0 < ∆(z,λi) < d. Now, since N(y) = N(y 1 ) N( σiz + Ai− k | − ρi) , we have | k/2 k ϕ(z) = N(yA 1 ) f (z) N( σiz + ρi) . (164) −i | || − | Further, from (163),

k 2πiS (βzA 1 ) f (z)N(σiz + ρi) = ci(0) + ci(β)e i− a2ϑ1 β>0 − | k and clearly for z Vi F, f (z)N( σiz + ρi) is bounded. Moreover, since ∈k/2 ∩ k − for z Vi, N(y 1 ) < d− , we conclude from (??) that ϕ(z) is bounded in 216 Ai− h ∈ h (Vi F). Let F∗ = F (Vi F). Since F∗ is compact, ϕ(z) is bounded on i=1 ∩ − i=1 ∩ F∗. Thus there exists a constant M such that ϕ(z) M for z F. But now for α β ≤ ∈ each P = γ δ Γ, ϕ(zP) = ϕ(z), since ∈ k/2 k +k/2 k/2 f (z ) N(y ) = f (z ) N(γz + δ) − N(y) = f (z) N(y) | P | P | P || | | | = ϕ(z).

Thus, for all z h , ϕ(z) M and hence ∈ n ≤ k/2 f (z) = ϕ(z)N(y)− | | k/2 (165) MN(y)− . ≤ Now f (z) = c(0) + c(β)e2πiS (βz) ϑ1 β>0 − | and

1 1 2πiS (αz) c(α) = ... f (z)e− dx1 ... dxn 0 0 1 1 2πS (αy) 2πiS (αx) = e ... f (x + iy)e− dx1 ... dxn. 0 0 Hence, from (165), k/2 2πS (αy) c(α) MN(y)− e | | ≤ Letting y tend to (0,..., 0) and noting that k < 0, we see that c(α) = 0 for all α ϑ 1 and hence f (z) is identically zero. ∈ − Modular Functions and Algebraic Number Theory 191

Let now ψ(z) be a modular form of weight 0. For each cusp λi of F, we have 2πiS (βzA 1 ) ψ(z) = ci(0) + ci(β)e i− . a2ϑ1 β>0 i − | h Consider χ(z) = (ψ(z) ci(0)). Let, if possible, χ(z) be not identically zero i=1 − i.e. for z F, letχ(z ) = ρ , 0. Since ψ(z) c (0) tends to zero as z tends 217 0 ∈ | 0 | − i to λi in F, we can ﬁnd a neighbourhood Vi of λi such that for z Vi F, h ∈ ∩ χ(z) < (1/2)ρ. Now F∗ = F (Vi F) is compact and χ(z) attains its | | − i=1 ∩ | | maximum M at some point of F∗. We see easily that

M = max χ(z) = max χ(z) . z F | | z hn | | ∈ ∈

But his means that χ(z) attains its maximum modulus over Hn at an inner point of Hn and by the maximum principle, χ(z) is a constant. But now, since χ(z) tends to zero as z tends to the cusps of F, χ(z) = 0 identically in Hn. This h means that (ψ(z) ci(0)) = 0. Since ψ(z) is single-valued and regular in Hn, i=1 − it follows that χ(z) is a constant. Our object now is to give an explicit construction of modular forms of weight k > 2(k being integral). Our method will involve an adaptation of the well-known idea of Poincare for constructing automorphic forms belonging to a discontinuous group Γ0 of analytic automorphisms of the unit disc D deﬁned by z < 1 in the complex z-plane. Now, any analytic automorphism of D | | az + b is given by the mapping z , where a and b are complex numbers → bz + a 2 2 a b satisfying a b = 1. Let the elements of Γ0 be Mi = i i , i = 1, 2,... bi ai | | − | | aiz + bi (corresponding to the automorphisms z zMi = of D). → biz + ai Let z0 be a given point of D. Now, since Γ0 acts discontinuously on D, we can ﬁnd a neighbourhood V of z such that the images V of V for M Γ do 0 M ∈ 0 not intersect V, unless (z0)M = z0. Let n0 be the order of the isotropy group of z0 in Γ0. Then we see that

dx dy n π. " ≤ 0 MΓ0 ∈ VM Modular Functions and Algebraic Number Theory 192

= a b Γ On the other hand, for M b a 0, we have ∈ 4 dx dy = bz + a − dx dy. " " | | VM V

Therefore, 218 4 bz + a − dx dy n π. (166) " | | ≤ 0 MΓ ∈ 0 V Since a 2 b 2 = 1, a , 0 and further b/a < 1. Hence, for z D, | | − | | | | ∈ bz + a 1 z < 1 + z . (167) − | | ≤ a | | Let V be contained in the disc z r < 1 and let v denote the area of V. We have, then, in view of (166) and| (167),| ≤

4 4 4 v(1 + r)− a − bz + a − dx dy n π. (168) | | ≤ " | | ≤ 0 MΓ MΓ ∈ 0 ∈ 0 V 4 We conclude that a − is convergent. Further, from (167), we have for M Γ0 | | z V, ∈ ∈ 4 4 4 bz + a − < (1 r)− a − | | − | | MΓ MΓ ∈ 0 ∈ 0 4 Hence (bz + a)− converges absolutely, uniformly on every compact subset M Γ ∈ 0 of V. Since z0 is arbitrary, this is true even for every compact subset of D. k Using the same idea as above, we shall prove that actually (bz + a)− M Γ0 converges absolutely for k > 2. Consider ∈ dx dy I = ; l " (1 z 2)l V − | | the integral converges if and only if l < 1. Now, as before dx dy n I . " (1 z 2) l ≤ 0 l M − VM − | | But

Thus 219

4+2l Using the same argument as earlier, it is easily shown that (bz + a)− M Γ0 k ∈ and therefore (bz + a)− , (for k > 2) converges absolutely, uniformly on M Γ0 every compact subset∈ of D. Let k be an even rational integer greater than 2. Then we see that ψ(z, k) = k (bz + a)− is analytic in D. To verify that ψ(z, k) is an automorphic form M Γ ∈ 0 belonging to Γ0, we have only to show that

p q k for N = Γ0, ψ(zN , k)(qz + p)− = ψ(z, k). q p ∈ Now, for k/2 a b dzM k M = Γ0, = (bz + a)− b a ∈ dz and so dz k/2 ψ(z, k) = M dz MΓ ∈ 0 Further

k/2 dzMN ψ(zN , k) = dz MΓ N ∈ 0 k/2 k/2 dz − dz = N MN dz dz MΓ ∈ 0 = (qz + p)kψ(z, k), since MN runs over all elements of Γ0, when M does so. Thus ψ(z, k) is an automorphic form of weight k, for the group Γ0 and is a so-called Poincare´ series of weight k for Γ0. The method given above for the construction of automorphic forms for the 220 group Γ0 is due to Poincare´ and it cannot be carried over as such to the case of the Hilbert modular group, since the euclidean volume of Hn is infinite. In the following, we shall give a method of constructing Hilbert modular forms of Modular Functions and Algebraic Number Theory 194 weight k > 2, based on a slight modification of the idea of Poincare.´ We shall be interested only in the case n 2. ≥ Let λ = ρ/σ be a cusp of H , a = (ρ, σ) = a (for some i), A = ρ ξ S n i σ η ∈ associated with λ and zA 1 = z∗ = x∗ + iy∗. We had introduced inHn a sys- − 1 tem of ‘local coordinates’ relative to the cusp λ, namely, N(y∗)− 2 , Y1,..., Yn 1, − X1,..., Xn. Defining q = N(y∗), it is easily seen that the relationship between the volume element dv = dq, dY1,..., dYn 1, dX1, dX2,..., dXn and the ordi- − nary euclidean volume element dx1dy1,..., dxndyn in Hn is given by 2 N(a) 4 dv = N( σz + ρ) − dx1dy1,..., dxndyn, 2n 1R √ d | − | − 0 | | where R0 and d are respectively the regulator and the discriminant of K. Let now N(a)2 c = n 1 2 − R0 √ d 2 | | and let dω be the volume element N(y)− dx1dy1,..., dxndyn. Then dω is invariant under a (modular) substitution z z for M Γ and moreover → M ∈ 2 4 dv = c N(y∗) dω = c∆(z,λ)− dω. Now, under the transformation z z for M Γ, it may be verified that → M ∈ ∆(z,λ)4 ∆ ,λ 4 ω = , dv c (zM )− d 4 (169) ∆ 1 → (z,λM− ) 2 2 2 N(y∗) =∆(z,λ)− ∆(zM,λ)− =∆(z,λ 1 )− . → M− We can find a point z0Uaλ,d such that z0 is not a fixed point of any modular 221 transformation except the identity. Now, by Theorem 16 again, we can find a neighbourhood V of z0 contained in Uλ,d Gλ such that V VM = , except 1 0 1 0 ∩ ∩ ∅ for M = 0 1 in Γ. Let M1 = 0 1 , M2,..., be a complete system of representatives of the right cosets of Γ modulo Γλ. For i , 1, Mi takes Uλ,d into U,d , , where = λMi λ and since Uλ,d U,d = , we see that for i 1, VMi lies wholly outside the neighbourhood U∩ . Further∅ V V = unless M = M . λ,d Mi ∩ M j ∅ i j Now, by applying suitable transformations in Γλ to the images VMi or parts of the same, we might suppose that V G for all i. It is to be remarked that Mi ⊂ λ under this process, the sets VMi continue to have finite euclidean volume and to be mutually non-intersecting. Let now, for a suitable d > 0, V U U . 0 ⊂ λ,d − λ,d0 We then see that the mutually disjoint sets VMi , i = 1, 2,..., are all contained in G U . It is now easily verified that λ − λ,d0 s 2 Js = ... N(y∗) − dv Mi VMi Modular Functions and Algebraic Number Theory 195

1 1 d0 2 2 s 2 ... q − dq dY1,..., dYn 1dX1,..., dXn ≤ 0 1 1 − − 2 − 2 d0 s 2 = q − dq 0 and Js converges if and only if s > 1. α β On the other hand, we obtain by (169), for M = γ δ Γ ∈ N( σz + ρ) 4N(y)s 2 ... s 2 = ... − . N(y∗) − dv | − 2s| 2s dv N( σzM + ρ) N(γz + δ) VM V | − | | | Now, let 1 A− M j = P j = ∗ ∗ , j = 1, 2,... γ j δ j 1 Then P j runs over a complete set of matrices of the form A− M with M Γ such that for i , j, P , NP with N A 1Γ A. Equivalently, we see that∈ i j ∈ − λ (γ j,δ j) runs over a complete set of non-associated pairs of integers γ j, δ j in a such that (γ j,δ j) = a. Now

s 2 Js = ... N(y∗) − dv Mi VM

′ 2s 2s s 2 = ... N(γ jz + δ j) − N( σz + ρ) N(y∗) − dv | | | − | γ j,δ j V where on the right hand side, the summation is over a complete set of non- 222 associated pairs of integers (γ j,δ j) such that (γ j,δ j) = a. Let B be a compact set in Hn. We can then ﬁnd constants c13 and c14 depending only on B such that for all z B and real and ν, we have ∈ c N( j + ν) N(z + ν) c N( j + ν) . (170) 13| | ≤ | | ≤ 14| | Taking for B, the closure V and V in Hn, we see that for a suitable constant c15 depending only on V and for s > 1,

2s ′ N(γ i + δ ) − c J < . | j j | ≤ 15 s ∞ γ j,δ j

Let now z lie in an arbitrary compact set B in Hn. Then, in view of (170), we can ﬁnd a constant c depending only on B, such that for all z B, 16 ∈ 2s 2s ′ N(γ z + δ ) − c ′ N(γ i + δ ) − < . | j j | ≤ 16 | j j | ∞ γ j,δ j γ j,δ j Modular Functions and Algebraic Number Theory 196

2s Thus the series ′ N(γ jz + δ j)− converges absolutely, uniformly over γ ,δ j j every compact subset of Hn, for s > 1. Let a1 and a2 denote a full system of pairs of integers (γ,δ) satisfying (γ,δ) = a and not mutually associated, respectively with respect to the group of all units in K and the group of units of norm 1 in K. Let k be a rational integer greater than 2. Corresponding to an integral ideal a in K, we now deﬁne for z H , the function ∈ n k G∗(z, k, a) = N(γ∗z + δ∗)− . (γ ,δ ) a ∗ ∗ ∈ 2

We ﬁrst observe that G∗(z, k, a) is well-deﬁned and is independent of the particular choice of elements of a2. Further, since the group of units of norm 1 in 223 K is of index at most 2 in the group of all units in K. We deduce from the fore- k going considerations that the series N(γ∗z + δ∗)− converges absolutely (γ ,δ ) a ∗ ∗ ∈ 2 uniformly over any compact subset of Hn. Hence G∗(z, k, a) is regular in Hn. ρ x Moreover, for M = ν Γ, ∈ k G∗(zM, k, a) = N(γ∗zM + δ∗)− (γ ,δ ) a ∗ ∗ ∈ 2 k k = N(z + ν) N((γ∗ρ + δ∗)z + γ∗ x + δ∗ν)− . (γ ,δ ) a ∗ ∗ ∈ 2

Now (γ∗ρ + δ∗, γ∗ x + δ∗ν) again runs over a system a2 when (γ∗,δ∗) does so. Thus k G∗(zM, k, a)N(z + ν)− = G∗(z, k, a).

Therefore G∗(z, k, a) is a modular form of weight k. Let k be odd and K contain a unit ǫ0 of norm 1; for example, if the degree n is odd, then 1 has norm 1. If (γ,δ) runs− over a system a , then (γ,δ) − − 1 together with (ǫ0γ,ǫ0δ) runs over a system a2. Thus

k k G∗(z, k, a) = (1 + N(ǫ0) ) N(γz + δ)− = 0, (γ,δ) a ∈ 1 k since N(ǫ0) + 1 = 0. Thus the functon G∗(z, k, a) vanishes identically in the case when k is odd and K contains a unit of norm 1. We shall henceforth exclude this case from our consideration; in particular,− nk should necessarily be even. We have then

G∗(z, k, a) = ρG(z, k, a), Modular Functions and Algebraic Number Theory 197 where, by deﬁnition

k G(z, k, a) = N(γz + δ)− (γ,δ) a ∈ 1 and ρ is the index of the group of units of norm 1 in K in the group of all units in K. The functions G(z, k, a) are again modular forms of weight k and are the so-called Eisenstein series for the Hilbert modular group. If a = ()a for some i(= 1, 2,... h) and K, then G(z, k, a) = N() kG(z, k, a224). i ∈ − i For the sake of brevity, we may denote G(z, k, ai) by Gi(z, k) for i = 1, 2,..., h. We can obtain the Fourier expansion of Gi(z, k) at the various cusps λ j of F (cf. formula (163) above) by using the following generalized ‘Lipschitz formula’ valid for rational integral s > 2 and τ H , namely, ∈ n 2π ns

s i s 1 2πS (τ) N(τ + ζ)− = (N()) − e . (171) n √N(ϑ)(Γ(s)) 1 ζ integral ϑ− ζ K >∈ 0 ∈ But we shall not go into the details here. A modular form f (z) is said to vanish at a cusp λ = ρ/σ of Hn, if, in the Fourier expansion of f (z)N( σz + ρ)k at the cups λ (cf. formula (163)), the constant term c(0) is zero. If −f (z) vanishes at a cusp λ, it clearly vanishes at all the equivalent cusps. We shall now prove the following

Proposition 25. The Eisenstein series Gi(z, k)(k > 2) vanishes at all cusps of Hn except at those equivalent to λi.

= F = ρ j ξ j Proof. Let λ j ρ j/σ j be a base cusp of and A j σ j η j be the associated matrix in S. Then setting w = z 1 , we have A−j

k k G (z, k)N( σ z + ρ ) = ′ N((γρ + δσ )w + γξ + δη )− . i − j j j j j j (γ,δ)=ai

Let us denote by c the constant term in the Fourier expansion of G (z, k)N( σ z+ i j i − j ρ )k at the cusp λ . If we set w = ( √ 1t,... √ 1t)(t > 0), then j j − − k ci j = lim Gi(z, k)N( σ jz + ρ j) t →∞ − ′ k = lim N((γρ j + δσ j) √ 1t + γξ j + δη j)− t − →∞ (γ,δ)=ai Modular Functions and Algebraic Number Theory 198

′ k = lim N((γρ j + δσ j) √ 1t + γξ j + δη j)− , (172) t − (γ,δ)=ai →∞ since the interchange of the limit and the summation can be easily justiﬁed. 225 Now unless γρ j + δσ j = 0, the corresponding term in the series (172) is zero. If γρ j + δσ j = 0, then (γ) (ρ ) (δ) (σ ) j = j ai a j ai a j and from this, it follows that necessarily i = j. Further, if γρ j + δσ j = 0, it can be easily shown that γξ j + δη j , 0. Moreover, in the series (172), there can occur only the pair (γ,δ) for which γρ j + δσ j = 0 and indeed this happens only if = j. Therefore c = 0, unless i = j. Our contention that G (z, k) vanishes − i j i at all cusps of F except at λi is therefore established. We deduce that the h Eisenstein series Gi(z, k), i = 1, 2,..., h are linearly h independent over the ﬁeld of complex numbers. If αiGi(z, k) = 0, then letting i=1 h k z tend to cusp λ j, the relation αiGi(z, k) N( σ jz + ρ j) = 0 gives α jc j j = 0 i=1 − i.e. α j = 0. Thus αi = 0 for i = 1, 2,..., h. A modular form which vanishes at all the cusps of F is known as a cusp form. It is now easy to prove

Proposition 26. For a given modular form f (z) of integral weight k > 2, we can ﬁnd a linear combination ϕ(z) of the Eisenstein series Gi(z, k) such that f (z) ϕ(z) is a cusp form. − Proof. If c j(0) is the constant term in the Fourier expansion of f (z)N( σ jz + h − k 1 ρ j) at cusp λ j = ρ j/σ j of F, then the function ϕ(z) = ci(0)cii− Gi(z, k) i=1 satisﬁes the requirements of the proposition.

The method outlined above for constructing modular forms yields only Eisenstein series but no cusp forms. In order to construct nontrivial cusp forms of integral weight k > 2, we proceed as follows. n Let Hn∗ denote the set of z C for which z Hn. Further let w = u iv 226 H . We deﬁne for z = x + iy ∈H , integral l and− integral∈ k > 2, the function− ∈ n∗ ∈ n k l Φ(w, z) = N(γz + δ)− N(w z )− (173) − M MΓ ∈ Modular Functions and Algebraic Number Theory 199

α β where M = γ δ runs over all elements of Γ. We shall show that the series (173) converges absolutely, uniformly for z and w lying in compact sets in Hn and H respectively, provided that l > 2, k 4. n∗ ≥ To this end, we choose z0 R1 such that z0 is not a fixed point of any M 1 0 ∈ in Γ, except M = 0 1 . We can then find a neighbourhood V of z0 such that 1 0 the images VM of V for M , 0 1 in Γ, do not intersect V. We may suppose moreover that for a suitable d′ > 0, if W denotes the annular neighbourhood of the base cusp defined by 0 < d′ < ∆(z, ) < d, then V G W. It ∞ ∞ ⊂ ∞ ∩ will then follow from result (ii) on p. 199 that for M < Γ , VM lies outside the ∞ neighbourhood U ,d and moreover for M Γ , VM W. In other words, for ∞ ∈ ∞ ⊂ M Γ, the images VM of V are mutually disjoint and further if z VM, ∈ ∈ M Γ ∈ then necessarily we have ∆(z, ) d′, i.e. VM is contained in a region of ∞ ≥ M Γ the form N(y) < α, for some α > 0. ∈ Let us now consider the integral

s 2 l I(s, l, α) = ... N(y) − N(w z) − dv, | − | N(y) α ≤ where dv = dx1dy1,..., dxndyn and the integral is extended over the set in Hn deﬁned by N(y) α. If s 2, then trivially ≤ ≥

s 2 l I(s, l, α) α − ... N(w z) − dv ≤ | − | Hn i.e.

s 2 2 2 l/2 I(s, l, α) α − ... N((u x) + (v + y) )− dv ≤ − Hn

s 2 ∞ ∞ 1 l = α − ... N(v + y) − dy1 ... dyn 0 0 ∞ ∞ 2 l/2 ... N(1 + x )− dx1 ... dxn −∞ −∞ < , ∞ provided l > 2. Now, since the sets VM are mutually disjoint, we have 227

k/2 2 l k ... N(y) − N(w z) − dv I , l, α < , | − | ≤ 2 ∞ MΓ ∈ VM Modular Functions and Algebraic Number Theory 200

α β if l > 2, k 4. But, on the other hand, for M = γ δ Γ, ≥ ∈ k/2 2 l ... N(y) − (N(w z))− dv − VM

k/2 2 l k = ... N(y) − (N(w zM))− N(γz + δ)− dv − V Hence

k/2 2 l k ... N(y) − N(w z ) − N(γz + δ) − dv | − M | | | MΓ ∈ V (174) k I , l, α < . ≤ 2 ∞

Let B, B∗ be compact sets in Hn and Hn∗ respectively. Then, for z B and w B we can ﬁnd constants c and c depending only on B and B∈and on ∈ ∗ 17 18 ∗ z0, such that N(w z ) l N(γz + δ) k 0 < c | − M |− | |− c . (175) 17 ≤ N(w (z ) ) l N(γz + δ) k ≤ 18 | − 0 M − 0 |− Taking for B, the closure V of V in Hn, we obtain from (174) and (175) that l k N(w (z0)M) − N(γz0 + δ) − converges and indeed uniformly with respect 228 M Γ | − | | | to∈w B . But again from (175), we have for z B and w B , ∈ ∗ ∈ ∈ ∗ l k N(w z ) − N(γz + δ) − | − M | | | MΓ ∈ l k c N(w (z ) ) − N(γz + δ) − . ≤ 18 | − 0 M | | 0 | MΓ ∈ Hence for l > 2, k 4, the series (173) converges absolutely, uniformly when ≥ z and w lie in compact sets in Hn and Hn∗ respectively. We shall suppose hereafter that l > 2, k 4; then we see that Φ(w, z) is ≥ regular in Hn as a function of z and regular in Hn∗, as a function of w. If P = α∗ β∗ Γ, then we have γ∗ δ∗ ∈ k l Φ(w, z ) = N(γz + δ)− N(w z )− P P − MP MΓ ∈ k k l N(γ∗z + δ∗) N((γα∗ + δγ∗)z + γβ∗ + δδ∗)− N(w z )− − MP MΓ ∈ Modular Functions and Algebraic Number Theory 201 i.e. k Φ(w, zP)N(γ∗z + δ∗)− = Φ(w, z). (176) 1 Further, if T = ǫ ζǫ− Γ with a unit ǫ and integral ζ in K, then 0 ǫ 1 − ∈ k 2 l Φ(w , ζ) = N(γz + δ)− N(ǫ w + ζ z )− T − M MΓ ∈ k k l = N(ǫ) N(ǫγz + ǫδ)− N(w z 1 )− − T − M MΓ ∈ = N(ǫ)kΦ(w, z). In particular, Φ(w + ζ, z) = Φ(w, z) for integral ζ K. We shall now proceed to obtain the Fourier expansion of Φ(w, z). ∈ 1 ζ Let Γt denote the subgroup of Γ consisting of M = 0 1 with integral ζ in K. Then clearly

k l Φ(w, z) = N(γz + δ)− N(w z + ζ)− , (177) − M MΓt Γ ζ integral ∈ \ ζ K ∈ where, in the outer summation, M runs over a full set of representatives of the 229 right cosets of Γ modulo Γt. It is now easy to see that the set M M Γt Γ is in one-one correspondence with the set of all pairs of coprime{± integers| ∈ (γ,δ\ )} in K. To each coprime pair (γ,δ) of integers in K, let us assign a ﬁxed M = γ∗ ∗δ Γ. Then ∈ k l 2Φ(w, z) = N(γz + δ)− N(w z + ζ)− . − M (γ,δ)=(1) ζ integral ζ K ∈ The inner sum is obviously independent of the particular M Γ associated with the pair (γ,δ). ∈ In view of (??), we have

l N(w z + ζ)− − M ζ integral ζ K ∈ (178) ln (2πi) l 2πiS (λ(w z )) = N(λ) e− − M . √ Γ n d ( (l)) λ ϑ1,λ>0 | | ∈ − Therefore, after suitable rearrangement of the series which is certainly permis- sible, we obtain from (177) and (178) ln (2πi) l 2πiS (λw) Φ(w, z) = N(λ) e− √ Γ n × 2 d ( (l)) λ ϑ1,λ>0 | | ∈ − Modular Functions and Algebraic Number Theory 202

2πiS (λz ) k e M N(γz + δ)− . × (λ,δ)=0 We now deﬁne for k > 2 and z H , ∈ n 2πiS (λzM ) k fλ(z, k) = e N(γz + δ)− . (179) (γ,δ)=0 Then we have

ln (2πi) l 2πiS (λw) Φ(w, z) = N(λ) e− fλ(z, k). √ Γ n 2 d ( (l)) λ ϑ1,λ>0 | | ∈ − Now, the series (179) converges absolutely, uniformly over every compact 230 subset of Hn, since the series (173) does so. Therefore fλ(z, k) is regular in Hn; further it is bounded in the fundamental domain, for n = 1. Moreover, for P = α∗ β∗ Γ, we have from (176), γ∗ δ∗ ∈ k f (z , k)N(γ∗z + δ∗)− f (z, k), λ P − λ in view of the uniqueness of the Fourier expansion of Φ(w, z). Thus fλ(z, k) is a modular form of weight k and in fact, a cusp form, since it might be shown k that as z tends to the cusp λi = ρi/σi in F, fλ(z, k) N( σiz + ρi) tends to − 1 zero. A priori, it might happen that fλ(z, k) = 0 identically for all λ ϑ− with λ > 0. However, we shall presently see that this does not happen for∈ large k. The series (179) are called Poincare´ series for the Hilbert modular group. They were introduced by Poincare´ in the case n = 1 (elliptic modular group). Let A = ρi ξi S correspond to the cusp λ = ρ /σ of F and let a = i σi ηi ∈ i i i i (ρi,σi). By considering the function

k l Φ(w, z, A ) = N(γz + δ)− N(w z )− i − P P

1 where P = γ∗ ∗δ runs over all elements of S of the form A M with M Γ, we i− ∈ can obtain the Poincar e´ series

k 2πiS (λzP) fλ(z, k, Ai) = N(γz + δ)− e , (180)

(γ,δ)=ai which are again cusp forms of weight k. In (180), the summation is over all pairs of integers (γ,δ) such that (γ,δ) = ai and with each such pari (γ,δ) is 1 associated a ﬁxed P = γ∗ ∗δ S, of the form Ai− M with M Γ. Further λ > 0 2 1 ∈ ∈ is a number in ai ϑ− . It is clear that the Poincare´ series fλ(z, k, A1) coincide Modular Functions and Algebraic Number Theory 203

with fλ(z, k) introduced above. Poincare´ theta-series for a discontinuous group of automorphisms of a bounded domain in Cn have been discussed by Hua. The Fourier coefficients of the Poincare´ series fλ(z, k, Ai) have been explicitly determined by Gundlach and they are expressible as infinite series involving Kloosterman sums and Bessel functions. It has been shown by Maass that for fixed i(= 1, 2,..., h) the Poincare´ series f (z, k, A ) for λ a2ϑ 1, λ > 0 together with the h Eisenstein series λ i ∈ i − Gi(z, k) generate the vector space of modular forms of weight k(> 2) over the 231 field of complex numbers. The Poincare´ series fλ(z, k, Ai) themselves generate the subspace of all cusp forms of weight k; this is the so-called ‘completeness theorem for the Poincare´ series’. The essential idea of the proof is the in- troduction of a scalar product ( f,ϕ) for two modular forms f (z) and ϕ(z) of weight k, of which at least one is a cusp form, namely,

k 2 ( f,ϕ) = ... f (z)ϕ(z)N(y) − dx1dy1,..., dxndyn F

the integral being extended over a fundamental domain for the Hilbert modular group. This scalar product which is known as the “Petersson scalar product” was used by Petersson for n = 1, in order to prove the “completeness theorem” for the Poincare´ series of real weight k 2, belonging to even more general groups than the elliptic modular group,≥ such as the Grenzkreis groups of the ﬁrst type. They are called horocyclic groups. The “completeness theorem” for the Poincare´ series belonging to the principal congruence subgroups of the elliptic modular group was proved, independently of Petersson, by A. Selberg by a diﬀerent method. For n = 1, the scalar product ( f,ϕ) has been explicitly computed by Rankin. As we remarked earlier, it could happen a priori that all the Poincare´ series fλ(z, k) vanish identically. However, we shall presently see that this does not happen if k is large.

Proposition 27. There exist n + 1 modular forms f0(z), f1(z),..., fn(z) of large 1 1 weight k such that f1(z) f0− (z),..., fn(z) f0− (z) are analytically independent.

Proof. Let us choose a basis ω1(= 1), ω2,...,ωn of K over Q with integral ω > 0(i = 2,..., n); clearly (ω( j)) , 0. Further let λ = ω + g for a rational i | i | i i integer g > 0, to be chosen later; again we have (λ( j)) , 0. Let us, moreover, | i | denote G1(z, k), fλ1 (z, k),..., fλn (z, k) by f0(z), f1(z),..., fn(z) respectively. We shall show that for large k and g to be chosen suitably, the n functions gi(z) = 1 fi(z) f0− (z), i = 1, 2,..., n, are analytically independent. Modular Functions and Algebraic Number Theory 204

We know that for real t > 1, z0 = (it,..., it) is an inner point of q1. Hence 232 for all pairs of coprime integers (γ,δ) in K with γ , 0, we have N(γz +δ) > 1. | 0 | One can show easily as a consequence that lim G1(z0, k) = lim N(γz0 + k = →∞ (γ,δ) 1 k δ)− = 1. Thus, for large k, f0(z0) 1 can be made arbitrarily small. Similarly, 2 | − | 2πtS (λmǫ ) by taking k to be suitably large, we can ensure that fm(z0) e− | − ǫ | 2 2πtS (λmǫ ) and hence gm(z0) e− for m = 1, 2,..., n can be made as small as | − ǫ | we like; here, in the summation ǫ runs over all units of K. Since gm(z) and 2 2πiS (λmǫ z) e are regular in a neighborhood of z0 we see that ǫ 2 2πiS (λmǫ z) ∂gm(z) ∂ e , m, j = 1, 2,..., n, ∂z − ∂z j ǫ j z=z0 can be made arbitrarily small, by choosing k large. Now, if ǫ1,...,ǫn are any n totally positive units of K, then the equation n n S ((ωi + g)ǫi) = S ((ωi + g)) does not hold for large g > 0, unless we have i=1 i=1 ǫ1 = ǫ2 = ... = ǫn = 1. This result can be easily proved by using the fact that if ǫ > 0 is a unit of K, S (ǫ) > n unless ǫ = 1. (This last-mentioned fact is itself a consquence of the well-known result that the arithmetic mean of n( 2) positive numbers not all equal, strictly exceeds their geometric mean). Thus,≥ if we consider the jacobian

2 ∂ e2πiS (λmǫ z) J(λ ,...,λ , t) = 1 m  ∂z  ǫ j   z=z0     n n n ( j) 2πt as a function of t(> 1), then it contains the term (2πi) 2 (λm ) e− S (λM) | | m=1 ( j) which is not cancelled by any other term. Now, since (λ ) , 0, it follows that | m | for some t > 1, J(λ1,...,λm, t) , 0 and hence, by the foregoing, the jacobian (∂gm/∂z j) does not vanish identically if k is chosen to be large enough. Thus our| proposition| is proved. Incidentally, in the course of the proof above, we have shown that for large 233 1 k, the Poincare´ series fλ(z, k), λ ϑ− , λ > 0 cannot all vanish identically. We might sketch here two alternative∈ proofs of Proposition 27. The following proof is shorter and does not use the Fourier expansion of Φ(w, z) introduced earlier. Now, by (176) and in view of the fact that (for n = 1) lim Φ(w, it) = 0, t the function Φ(w, z) is a modular form of weight k (in z).→∞ Let now w(p) = Modular Functions and Algebraic Number Theory 205

(p) (p) (p) Φ(w , z) (w1 ,..., wn ) Hn∗, P = 1, 2,..., n and ΦP(z) = . Then, if z = ∈ G1(z, k) (it,..., it), t > 1, we see as before that

(p) 2 l lim ΦP(z) = N(w ǫ z + ζ)− , k − →∞ ǫ,ζ where, in the summation ǫ runs over all units of K and ζ over all integers in K. (p) If we suppose that wq and zq are all purely imaginary, then for integral ζ , 0 in K and for any unit ǫ, we have

2 2 w(p) ǫ(q) z < w(p) ǫ(q) z ζ(q) , p, q = 1, 2,..., n. q − q q − q − (p) If we further suppose that iwq and zq/2i, p, q = 1, 2,..., n lie close to 1 and use some well-known inequalities for elementary symmetric functions of n positive real numbers, we can show that for a unit ǫ , 1, ± N(w(p) z) < N(w(p) ǫ2z) , p = 1, 2,..., n. | − | | − | It now follows that

n (1) l (n) l ∂ΦP(z) 2 lim lim l− N(w z) ... N(w z) = l k − − ∂z  (p)  →∞ →∞ q wq zq   −    and for suitable choice of w(1),..., w(n), we can ensure that

1 , 0.  (p)  wq zq   −    Thus, for large k and l, the n functions Φ1(z),..., Φn(z) are analytically inde- 234 pendent. One can also prove Proposition 27 as follows. First, by the map z = z j θ j (z1,..., zn) w = (w1,..., wn) where w j = − and θ = (θ1, . . . , θn) F. → z j θ j ∈ − n One maps Hn analytically homeomorphically D, the unit disc in C deﬁned by w1 < 1,..., wn < 1. Now the group Γ induces a discontinuous group of |analytic| automorphisms| | of D. By using the well-known methods for a discontinuous group of analytic automorphisms of a bounded domain in Cn, we can ﬁnd n + 1 Poincare´ series f0(w), f1(w),..., fn(w) of weight k and belonging f (w) f (w) to G such that the n functions 1 ,..., n are analytically independent. f0(w) f0(w) Modular Functions and Algebraic Number Theory 206

Pulling these functions back to Hn by the inverse map, we obtain the desired n + 1. Hilbert modular forms of weight k. The modular forms of weight k form a vector space over the field of complex numbers. We shall see that this vector space is finite-dimensional and its dimension satisfies certain estimates in terms of k, for large k. In the following, we shall be interested only when k > 2.

Theorem 18. Let t(k) denote the number of linearly independent modular forms of weight k. Then for large k(> 2), t(k) 0 < c c , (181) 19 ≤ kn ≤ 20 for constants c19 and c20 depending only on K. (Note. Since for k > 2, t(k) 1, (181) may be upheld for all k > 2, by ≥ choosing c19 and c20 properly). Proof. We shall ﬁrst prove the relatively easier inequality t(k) c kn. ≥ 19 By the preceding proposition, there exist n+1 modular forms f0(z), f1(z),..., fn(z) 1 1 of weight j such that f1(z) f0− (z),..., fn(z) f0− (z) are analytically independent and therefore algebraically independent. Let us now suppose that k > nj + 2 and let m be the smallest positive integer such that

mn j + 2 k (m + 1)nj + 2. (182) ≤ ≤ 235

k1 k2 kn Consider the power-products f1 (z) f2 (z),..., fn (z), (ki = 0, 1, 2,..., m). n n mn (k1+ +kn) These are (m + 1) in number. And the (m + 1) functions ( f0(z)) − , k1 kn f1 (z),..., fn (z)(ki = 0, 1, 2,..., m) are modular forms of weight mn j and are 1 1 linearly independent, since f1(z) f0− (z),..., fn(z) f0− (z) are algebraically independent. Since k > mn j + 2, we can obtain, by multiplying these modular forms by an Eisenstein series G1(z, r) (where r may be chosen to be k mn j), (m + 1)n modular forms of weight k. Thus −

t(k) (m + 1)n. ≥ By (182), (m + 1)n ((k 2)/nj)n. Thus, for large k, we can ﬁnd a constant ≥ − n c19 > 0 such that t(k) c19k . We now proceed to≥ prove the other inequality in (181). Let a be an ideal in K and p, a positive rational integer. Then it is known that for a constant c depending only on a and K, the number of β a for 21 ∈ Modular Functions and Algebraic Number Theory 207

n which β > 0 or β = 0 and S (β) < p is at most equal to c21 p . Taking a to 1 2 1 2 1 be one of the h ideals ϑ− , a2ϑ− ,..., ahϑ− , we can ﬁnd a constant c22 > 0 2 1 depending only on K such that the number of β ai ϑ− (i = 1, 2,..., h) for n∈ which S (β) < p and β > 0 or β = 0 is at most c22 p . Let c23 = hc22. Suppose f1(z),..., fq(z) are q linearly independent modular forms of weight k(> 2). We shall be interested only in the case when k is so large as to allow the situation q > c23. Let us therefore suppose that q > c23. Let p be the largest positive rational integer such that

c pn < q c (p + 1)n. 23 ≤ 23 Now each one of the modular forms f j(z), j = 1, 2,..., q has at the cusp λi = ρi/σi, a Fourier expansion of the form

k (i) (i) 2πiS (βzA 1 ) f (z)N( σ z + ρ ) = c (0) + c (β)e i− . j − i i j j β a2ϑ 1,β>0 ∈ i −

We can ﬁnd complex numbers 1,...,q not all zero such that for f (z) = ( j) 2 1 1 f1(z) + + q fq(z), all the Fourier coeﬃcients c (β) for all β a j ϑ− 236 such that β= 0 or β > 0 and S (β) < p, in the expansion ∈

2πiS (βz ) k ( j) ( j) A 1 f (z)N( σ z + ρ ) = c (0) + c (β)e −j , (183) − j j a2ϑ1 β>0 j − | n vanish. This is possible, since it only involves solving r( c23 p ) linear equa- n ≤ tions in q(> c23 p ) unknowns 1,...,q. Clearly, f (z) is a cusp form of weight k.

1 Let z = x + iy Fλ j Gλ j and yA = (y∗,..., yn∗). Then we know that ∈ ∩ −j 1

yi∗ c7 c8 ≤ n N(y 1 ) ≤ A−j

(see p. 253) and therefore as z F tends to the cusp λ j, each one of y1∗,..., yn∗ tends to inﬁnity. Since c( j)(0) =∈0, it follows that

k/2 k/2 k N(y) f (z) = N(yA 1 ) N( σ jz + ρ j) f (z) 0, | | −j | − | | | → as z F tends to any cusp λ j of F. ∈ k/2 Let us now deﬁne for z Hn, the function g(z) = N(y) f (z) . This func- ∈ α β | | tion is a non-analytic modular function i.e. for M = γ δ Γ, ∈ k/2 k/2 k g(z ) = N(y ) f (z ) = N(y) N(γz + δ) − f (z ) = g(z). M M | M | | | | M | Modular Functions and Algebraic Number Theory 208

By the foregoing, g(z) tends to zero as z tends to the cusps of F and as a consequence, if we now make use of Proposition 22, we see that g(z) is bounded in F and attains its maximum > 0 at a point z0 = x0 + iy0 F. But, since g(z ) = g(z) for all M Γ, we obtain ∈ M ∈ g(z) g(z0) = , (184) ≤ for all z Hn. By multiplying f (z) if necessary by a number of absolute value 1, we might∈ suppose that f (z0)N(y0)k/2 = . 0 Let us assume without loss of generality that z Fλi Gλ1 (otherwise, if 0 ∈ ∩ z Fλ Gλ , j , 1, then we have only to work with the coordinates z 1 and j j A−j 0 ∈ ∩ 0 z 1 instead of z and z ). Let t1 = u1 + iv1 be a complex variable and let us A−j denote the n-tuple (t1, t1,..., t1) by t = u + iv. Then for

0 v1 < s = inf y , k=1,2,...,n k we see that z = z0 t H . 237 − ∈ n Setting r = e 2πit1 , we now deﬁne for r < e2πs, the function h(r) = f (z0 − | | − t)e2πipt1 = f (z)e2πipt1 . Taking j = 1 in (183), we obtain

p (1) 2πiS (βz0) S (β) h(r) = r− c (β)e r . (185) 1 ϑ−β>0 S (β|) p ≥ The right hand side of (185) is a power series in r containing only non-negative powers of r and is absolutely convergent for r < R = e2πs > 1. Thus h(r) is 2πv|1 | regular for r < R. Let now 1 < R1 = e < R. Then, by the maximum modulus principle,| | h(1) max h(r) , ≤ r =R1 | | | | i.e. 0 0 2πpv f (z ) max f (z t) e− 1 ≤ r =R1 | − | | | i.e. 0 k/2 0 k/2 2πpv N(y )− N(y v)− e− 1 , ≤ − in view of (184). From this, we have however k 2πpv log N(1/(1 v )/y0) 1 ≤ 2 − 1 k = v S (1/y0) + terms involving higher powers of v 2 1 1 Modular Functions and Algebraic Number Theory 209 i.e. k p S (1/y0) + terms involving v and higher powers. ≤ 4π 1 Now letting R1 tend to 1 or equivalently v1 to zero, we obtain k p S (1/y0). ≤ 4π

From p. 196, we see that S > c9 (for a constant c9 depending only on K) 238 and S (1/y0) n c . (We have similar inequalities for z0 , if z0 F G , 9 A 1 λ j λ j ≤ −j ∈ ∩ j , 1). Thus, in any case, we can ﬁnd a constant c24 depending only on K such that p c k. ≤ 24 Therefore, we see that if there exist q linearly independent modular forms of weight k, then necessarily

q c (p + 1)n c (c k + 1)n c kn ≤ 23 ≤ 23 24 ≤ 20 for large k and a suitable constant c20 independent of k. In particular, we have t(k) c kn ≤ 20 which proves the theorem completely. The idea used in the proof above is essentially the same as in Siegel’s proof of the analogous result that the maximal number of linearly independent modu- (n/2)(n+1) lar forms of degree n and weight k is at most equal to c25k for a suitable constant c25 independent of k. (See reference (28) Siegel, p. 642). We now proceed to consider the Hilbert modular functions. A function f (z) meromorphic in Hn is called a Hilbert modular function (or briefly, a modular function), if (i) for every M Γ, f (z ) = f (z); ∈ M (ii) for n = 1, f (z) has at most a pole at the infinite cusp of the fundamental domain. For n 2, no condition is imposed on the behaviour of f (z) at the cusps of F, in the definition≥ of a modular function. If n = 1, f (z) is just an elliptic modular function. Let us remark, however, that for n 2, there is no analogue of the elliptic modular invariant j(τ). For, if ≥ f (z) is a Hilbert modular function regular in Hn(n 2) then it is automatically regular at the cusps of F in view of Theorem 17 and≥ is, in fact, a modular form of weight 0 and hence a constant. Modular Functions and Algebraic Number Theory 210

The modular functions constitute a field which contains the field of com- 239 plex numbers. Our object now is to study the structure of this field. We shall not be concerned, in the sequel, with the case n = 1, since, in this case, it is well-known that the elliptic modular functions form a field of rational functions generated by j(τ) over the field of complex numbers. In the case n 2, we shall see presently that the modular functions form an algebraic function≥ field of n variables. The proof which we are going to present is mainly due to Gundlach. A major step in the direction of this result is the following Lemma. For n 2, every Hilbert modular function can be expressed as the quotient of two Hilbert≥ modular forms of the same weight k.

Proof. Let F∗ = F Uλi,d with 0 < d′ < d. Then we know that F∗ is − i=1 compact, in view of Proposition 22. Let ϕ(z) be a Hilbert modular function. Suppose we can ﬁnd a modular form χ(z) of weight k such that ψ(z) = ϕ(z)χ(z) is regular in F∗. Then, since for every M = α β Γ, ψ(z )N(γz + δ) k = ψ(z) and further since ψ(z) is γ δ ∈ M − regular in Uλi,d Uλi,d (for i = 1, 2,..., h), we can appeal to Corollary (a) (p. 214) and conclude− that ψ(z) is a modular form of weight k. Thus ϕ(z) would be expressible as the quotient of the two modular forms ψ(z) and χ(z) of weight k. Our object therefore is to construct a modular form χ(z) of weight k such that χ(z) annihilates the poles of ϕ(z) in F∗ i.e. more precisely, such that χ(z)ϕ(z) is regular in F∗. 0 To every point z F∗, we assign a neighbourhood R in Hn such that the following conditions are∈ satisﬁed

0 0 0 P(z z ) P(z1 z ,..., zn zn) (i) ϕ(z) = − = − 1 − in R where P and Q are Q(z z0) Q(z z0,..., z z0) − 1 − 1 n − n convergent power-series in z z0 and locally coprime at every point of R. −

(ii) By a suitable linear transformation, we can ﬁnd new variables w, v2,..., vn such that by the Weierstrass’ preparation theorem, we can suppose that in R, Q(z z0,..., z z0) wg+a (v)wg 1+ +a (v) where a (v),..., a (v) 240 1− 1 n− n − i − g 1 g are power-series in v2,..., vn convergent in R. If ϕ(z) is regular in R, then we might take Q tobe1in R. (iii) Further, we might assume that R is a polycylinder of the form w b, v a,..., v a where a and b are so chosen that, in addition,| | ≤ any | 2| ≤ | n| ≤ Modular Functions and Algebraic Number Theory 211

zero (w, v ,..., v ) of Q(w, v ,..., v ) for which v < a,..., v < a, 2 n 2 n | 2| | n| automatically lies in R. We shall denote the (n 1)-tuple (v2,..., vn) by v. −

0 Let R∗ R be the corresponding neighbourhood of z F∗ defined by w < b, v ⊂< a/2,..., v < a/2. Then the neighbourhoods∈R for z0 F | | | 2| | n| ∗ ∈ ∗ give an open covering of the compact set F∗ and we extract therefrom a finite (1) (c ) covering of F∗ by neighbourhoods R∗,..., R∗ of points z ,..., z ′ of F∗. 1 c′ R R (1) (c′) Let 1,..., c′ be the corresponding neighbourhoods of z ,..., z defined above. gi (i) v (i) Let in Ri, ϕ(z) = P(w, v)/Q(w, v) where Q(w, v) = av (v)w and av (v) v=0 are convergent power series in v2,..., vn in Ri. If ϕ(z) is regular in Ri, then c′ Q(w, v) = 1 in Ri and therefore gi = 0. Let us denote gi by c23. The constant i=1 c26 of course depends on ϕ(z). If M is the set of poles of ϕ(z), then M R is just the set of zeros of Q(z z0) in R. Clearly, M is invariant under the modular∩ substitutions. Let us denote− M F∗ by N; since F∗ is compact, so is N. ∩Let k > 2 be a rational integer to be chosen suitably later. From Theorem 18, we know that the number r = t(k) h of linearly independent cusp forms n − of weight k satisfies r > c27k for a constant c27 > 0, if k > c27∗ (a suitable constant). Let f1(z), f2(z),..., fr(z) be a complete set of linearly independent cusp forms of weight k. Now, let f (z) = g(w, v2,..., vn) be regular in Ri; then, performing the divi- sion of g(w, v2,..., vn) by Q(w, v2,..., vn), we can write in Ri,

g 1 i− (i) s f (z) = h(w, v2,..., vn)Q(w, v2,..., vn) + bs (v2,..., vn)w , (187) s=0

(i) where h(w, v2,..., vn) is regular in Ri and bs (v2,..., vn) are convergent power- series in v2,..., vn. Let l be the non-negative rational integer such that 241

n 1 n n 1 c l − < c k c (l + 1) − . (188) 26 27 ≤ 26 We might assume k to be so large that l 1. We now claim we can ﬁnd complex ≥ numbers α1,...,αr not all zero, such that for f (z) = α1 f1(z) + + αr fr(z), (i) the power-series bs (v2,..., vn), occurring in (187) are of order at least l. in v2,..., vn, for i = 1, 2,..., c′. This is simple to establish, for it only involves n 1 n solving q( c26l − ) linear equations in r(> c27k ) unknowns α1,...,αr and by (188), q <≤r. Modular Functions and Algebraic Number Theory 212

We shall first prove that for all sufficiently large k, the cusp form f (z) = r αi fi(z) chosen above, vanishes identically on N. Let, if possible, f (z) be i=1 not identically zero on N. Then f (z) attains its maximum modulus , 0 at some point z∗ on N, since N is compact. We shall suppose that f (z∗) = 1, after normalizing f (z) suitably, if necessary. Now z R for some i. Let w , ∗ ∈ i∗ ∗ v2∗,..., vn∗ be the coordinates of z∗, in terms of the new variables w, v2,..., vn in Ri. Let t be a complex variable. Denoting the (n 1)-tuple (v2∗t,..., vn∗t) by v t, we have (w, v t) R , for w < b, t 2. − ∗ ∗ ∈ i | | | | ≤ Figure, page 241 Let M = M R . Consider now the analytic set defined in the polycylinder i ∩ i w < b, t 2 by Q(w, v∗t) = 0. Let B be an irreducible component of this |analytic| set,| | ≤ containing (w , 1). Clearly if (w, t) B, then (w, v t) M . On B, ∗ ∈ ∗ ∈ i therefore, we have after (187), for the function f0(w, v∗t) = f (z) that

g 1 i− l 1 (i) v f ∗(w, t) = f0(w, v∗t) = t t− bv (v∗t)w . v=0

(i) l (i) Since bv (v2,..., vn) is of order at least l in v2,..., vn, it follows that t− bv (v∗t) 242 l is regular in t 2. Hence t− f ∗(w, t) is regular on B. Now it can be shown that if (w , t )| | ≤B, then B can be parametrized locally at (w , t ) by 0 0 ∈ 0 0 t t = uq, w w = R(u), − 0 − 0 where q is a positive integer and R(u) is regular in u. If we note that we can l apply the maximum modulus principle to the function t− f ∗(w, t) as a function l of u, we can show that t− f ∗(w, t) attains its maximum modulus on B at a point (w(0), t(0)) on the boundary of B and hence with t(0) = 2, necessarily. Thus | | l 1 = f (z∗) = f0(w∗, v∗) Max t− f ∗(w, t) ≤ B | | l (0) (0) = 2− f ∗(w , t ) | | l 2− Max f ∗(w, v∗t) (189) ≤ t 2, w b | | | |≤(w,t|) |≤B ∈ l 2− Max f (z) . ≤ Mi | |

(0) Let z be a point on the boundary of Mi at which max f (z) is attained. If z Mi | | (0) (0) (0) ∈ l z F∗, then z N and hence f (z ) 1. Hence we have 1 2− , which gives∈ a contradiction,∈ since l 1. Therefore| | ≤ f (z) must vanish on N≤. ≥ Modular Functions and Algebraic Number Theory 213

Suppose now z(0) < F . Then, for some M Γ, we can ensure that z(0) F. ∗ ∈ M ∈ We contend that there are only ﬁnitely many P Γ for which FP intersects c′ ∈ c′ Ri. To prove this, we ﬁrst apply Proposition 23 with Rl with C and then i=1 i=1 for a certain b = b(C) > 0, we have U C = for all cusps . It follows that ,b ∩ ∅ for all cusps λ and for all P Γ, we have (Uλ,b)P C = . In particular, for ∈ ∩ ∅ h

λ = λ1,...,λh, we see that (Uλ,b)P C = , for all P Γ. Now F Uλi,b is 243 ∩ ∅ ∈ − i=1 h c′ compact and we apply Proposition 21 with F for B and Ri for B′. Then − λi,b i=1 we see immediately that our assertion above is true. We may now suppose that z(0) F U , without loss of generality. Since M ∈ ∩ λ1,c c′ (0) α β z Ri, and since by the remark above, M = γ δ belongs to a finite set in ∈ i=1 (0) (0) Γ independent of z , there exists a constant c28 > 1 such that N(γz + δ) > c 1. Further f (z(0)) N(γz(0) + δ) k = f (z(0)) . Hence from (189),| we have | 28− | M || |− | | l (0) l k (0) 1 2− f (z ) 2− c f (z ) . ≤ | | ≤ 28| M | If z(0) F , then we have f (z(0)) 1 and therefore M ∈ ∗ | M | ≤ l k 1 2− c . ≤ 28 (0) < F (0) U F Suppose now zM ∗; then zM λ1,d′ . We shall show again that (0) ∈ ∩ f (zM ) 1. | |M ≤ M U Let ∗ be the connected component of λ1,d′ which contains the point (0) ∩ (0) (i) zM . Then f (z) is not locally constant on M∗ near zM , because of f (zM ) = 0 and the connectedness of Ri. Let sup f (z) = δ. If B = MT∗ then clearly sup f (z) = δ, again. Now, z M | | T Γλ z B | | ∈ ∗ ∈ l ∈ since f (z) tends to zero as z tends to the cusp λ1, there exists d′′ with 0 < d′′ < d′ such that f (z) < (1/2)δ in Uλ ,d . Let B1 = B (Uλ , d′ Ul , d′′) F; | | 1 ′′ ∩ 1 − λ1 ∩ U U = = then B ( λ1,d′ λ1,d′′ ) (B1)T and further sup f (z) δ, again. Since ∩ − T Γλ z B | | ∈ 1 ∈ 1 B1 is compact, f (z) attains its maximum δ (< , necessarily therefore) at a | | ∞ (i) point z′, say, in Bi. Now, there exists a sequence of points w M∗ F ∈ Ti ∩ converging to z′. Using the fact that M is closed and locally connected, it can be shown that z belongs already to M for a T Γ , which is one of the ′ T∗ ∈ λ1 T ’s above. Now f ((z(0)) ) = f (z(0)) and to show that f (z(0)) 1, we might i | M T | | M | | M | ≤ as well argue with MT∗ instead of M∗. Thus we may assume, without loss of Modular Functions and Algebraic Number Theory 214 generality, that f (z) attains its maximum δ on M at a point z . We now claim 244 | | ∗ ′ that ∆(z′,λ1) = d′. For, if ∆(z′,λ1) < d′, then z′ M∗ and we can find a (0) ∈ curve C in M∗ connecting zM with z′. Let z′′ be the first point on C such that f (z ) = δ; then f (z) is not locally constant on M near z , and the maximum | ′′ | ∗ ′′ principle gives a contradiction. Thus ∆(z′,λ1) = d′. Again, for a suitable Γ F N (0) = T λ1 , zT′ ∗ i.e. zT′ , as before. Thus f (zM ) f (z′) f (zR′ ) 1. ∈Thus, in∈ all cases ∈ | | ≤ | | | | ≤ k l 1 c 2− , ≤ 28 i.e. l c k, ≤ 29 if f (z) does not vanish identically on N. But, from this and from (188), we obtain n n n 1 n 1 0 < c l < c k c (l + 1) − c l − , 30 27 ≤ 26 ≤ 31 i.e. n k c31 0 < c30 < c27 c31. l ≤ l ≤ From the fact that (k/l)n is bounded, we see that as k tends to infinity, l also tends to infinity. But this contradicts the inequality l < c31/c30. Thus f (z) vanishes on N identically, for large k. Now taking each Ri, i = 1, 2,..., c′, we observe that since f (z) vanishes on 0 the analytic set defined by Q(z z ) = 0 in Ri, there exists an integer ni such ni − 0 ni that f (z) is divisible by Q(z z ) in Ri i.e. f (z)ϕ(z) is regular in Ri. (We − m might take ni = gi, for example). Choosing m = c26, we see that f (z)ϕ(z) m is regular on F∗. Thus the modular form f (z) of weight mk has the required property and our lemma is therefore completely proved. We are now, in a position, to prove (for n 2), the main theorem, viz. ≥ Theorem 19. The Hilbert modular functions form an algebraic function field of n variables.

Proof. We know by Proposition 27 that there exist n + 1 modular forms f0(z), 245 f1(z) fn(z) f1(z),..., fn(z) of weight j such that ,..., are algebraically inde- f0(z) f0(z) pendent over the ﬁeld of complex numbers. Since the quotient of two modular forms of the same weight is a modular function, we have n algebraically independent modular functions

f1(z) fn(z) g1(z) = ,..., gn(z) = . f0(z) f0(z) Modular Functions and Algebraic Number Theory 215

Let Ω denote the field of Hilbert modular functions and Ω0, the field generated by g1(z),..., gn(z) over the field of complex numbers. We shall first show that every element of Ω satisfies a polynomial equation of bounded degree, with coefficients in Ω0. Let ϕ(z) be any modular function. By the lemma above, ϕ(z) = ψ(z)/χ(z), where χ(z) and ψ(z) are two modular forms of the same weight k. Let us consider for fixed positive rational integers m and s, the power products

k = 0, 1, 2,..., m, k1 kn l i fk ,...,k ,l(z) = g (z),..., g (z)ϕ (z), 1 n 1 n l = 0, 1, 2,..., s.   mn s  = + Now f0 (z)χ (z) fk1,...,kn,l(z) is a modular form of weight w mn j ks. We have indeed (m + 1)n(s + 1) such modular forms of weight w. If we can choose m and s such that (m + 1)n(s + 1) > c wn( t(w)), (190) 20 ≤ then we get a nontrivial linear relation between these modular forms and hence n between the (m + 1) (s + 1) functions fk1,...,kn,l(z). But this linear relation cannot be totally devoid of terms involving powers of ϕ(z), since otherwise, this will contradict the algebraic independence of g1(z),..., gn(z). Thus ϕ(z) satisfies a polynomial equation of degree at most s and with coefficients in Ω0. Since Ω is a separable algebraic extension of Ω0 and since every element of Ω satisfies a polynomial equation of degree at most s over Ω0, it follows that Ω is a finite algebraic extension of Ω0 i.e. an algebraic function field of n variables. In order to complete the proof of the theorem, we have only to find rational integers m and s which satisfy (190), s being bounded. This is very simple, for 246 wn when m tends to infinity, tends to ( jn)n so that if we choose m large (m + 1)n n n enough and s > c20 j n , then (190) will be fulfilled. Our theorem is therefore completely proved. By adopting methods similar to the above, Siegel has recently proved that (for n 2) every modular function of degree n (i.e. a complex-valued function meromorphic≥ in the Siegel half-plane and invariant under the modular transformations) can be expressed as the quotient of two “entire” modular forms of degree n. (See reference (32), Siegel). It had been already shown by Siegel that the modular functions of degree n which are quotients of “entire” modular forms, form an algebraic function field of n(n + 1)/2 variables. Bibliography

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[27] C. L. Siegel: Uber¨ die analytische Theorie der quadratischen Formen, III, Ann. of Math. 38 (1937), 213-292.

[28] ——: Einfuhrung¨ in die Theorie der Modulfunktionen n-ten Grades, Math. Ann. 116 (1939), 617-657.

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[33] H. Weyl: On generalized Riemann matrices, Ann. of Math. 35 (1934), 714-729. Appendix

Evaluation of Zeta Functions for Integral Values of Arguments

By Carl Ludwig Siegel

Eulerpresumably enjoyed the summation of the series of reciprocals of 249 squares of natural numbers in the same way as Leibnitz did the determination of the sum of the series which was later named after him. In the nineteenth century, generalizations of these results to the L-series introduced by Dirichlet were studied; these were important for the class number formulae of cyclotomic fields; corresponding questions for the L-series of imaginary quadratic fields got elucidated by Kronecker’s limit formulae. In one of his most beautiful papers, Hecke has obtained an analogous result for the L-series of real quadratic fields, which led him to the assertion that the value of the zeta function of every ideal class of a real quadratic number field of arbitrary discriminant d is, for the values s = 2, 4, 6,..., of the variable, equal to the product of π2s √d and a rational number. He further gave a brief indication of two possible methods of proof for his assertion. The first of these has recently been carried out in different ways [1]. The second method of proof leads, after suitable modifications to the generalization, published by Klingen [2], to arbitrary totally real algebraic number fields. This second method is indeed quite clear in its underlying general ideas but, however, in the form available till now, it admits of no effective determination of the said rational number. By means of a theorem on elliptic modular forms, which in itself seems not uninteresting, we shall, in the sequel, present a proof by a construc- tive procedure meeting the usual requirement of Number Theory.

219 BIBLIOGRAPHY 220

1 Elliptic Modular Forms

Let z be a complex variable in the upper half plane and

1 k F (z) = ′(lz + m)− (k = 4, 6, 8,...), (1) k 2 l,m where l, m run over all pairs of rational integers except 0, 0. If we set 250

g 2πiz l = σg(n) (g = 0, 1, 2,...), e = q t n | then for this Eisenstein series we have the Fourier expansion

k (2πi) ∞ n Fk(z) = ζ(k) + σk 1(n)q (2) (k 1)! − − n=1 which, in view of (2πi)k B ζ(k) = k , (3) − 2 k! can also be expressed by the formulae

2k ∞ n Fk(z) = ζ(k)Gk(z), Gk(z) = 1 σk 1(n)q (k = 4, 6,...). (4) − B − k n=1 In particular, here, we have the Bernoulli numbers 1 1 1 5 7 B = , B = , B = , B = , B = ; 4 −30 6 42 8 −30 10 66 14 6 and hence

∞ ∞ G = 1 + 240 σ (n)qn, G = 1 504 σ (n)qn 4 3 6 − 5 n=1 n=1 ∞ ∞ G = 1 + 480 σ (n)qn, G = 1 264 σ (n)qn (5) 8 7 10 − 9 n=1 n=1 ∞ G = 1 24 σ (n)qn 14 − 13 n=1 with rational integral coeﬃcients and constant term 1. We set further

G0 = 1. BIBLIOGRAPHY 221

Let h be a non-negative even rational integer. If we denote the dimensionof the 251 linear space Mh of elliptic modular forms of weight h by rh, where for ﬁxed h we write, for brevity, also r for rh, then it is well known that h h rh = + 1(h . ( mod 12)), rh = (h 2( mod 12)). 12 12 ≡ We have, the particular,

2 2 G = G8, G4G6 = G10, G G6 = G14 4 4 (6) GlG14 l = G14(l = h 12rh + 12 = 0, 4, 6, 8, 10, 14) − − and for the modular form

∞ ∆ = q (1 qn)24 − n=1 of weight 12, 1728∆ = G3 G2. 4 − 6 If 3 1 1 j(z) = G ∆− = q− + (7) 4 denotes the absolute invariant, then

2 dj 2 dG4 3 d∆ 1 2 dG6 dG4 ∆ = 3G ∆ G = G G6 2G4 3G6 , dz 4 dz − 4 dz 1728 4 dz − dz and the expression in the brackets yields a modular form of weight 12 and indeed a cusp form which can therefore diﬀer from ∆ at most by a constant factor. Comparing the coeﬃcients of q in the Fourier expansions, we get

dj 1 = G ∆− . (8) d log q − 14

a b Let hereafter, h > 2 and hence rh > 0. The r isobaric power-products G4G6 where the exponents a, b run over all non-negative rational integer solutions of

4a + 6b = h, form a basis of Mh. It follows from this that, for every function M in Mh, 252 1 r 1 MGh− 12r+12 always belongs to M12r 12. Since ∆ − is a modular form of weight 12r −12, not vanishing anywhere in− the interior of the upper half-plane, − 1 1 r MG−h 12r+12∆ − = W (9) − BIBLIOGRAPHY 222 is an entire modular function and hence a polynomial in j with constant coeﬃ- cients. Let r Th = G12r h+2∆− (10) − with the Fourier expansion

r 1 T = C q− + + C q− + C + (11) h hr h1 h0 and ﬁrst coeﬃcient Chr = 1. Since

1 1 ∞ n 2n 24 ∆− = q− (1 + q + q + ) , (12) n=1 all the Fourier coeﬃcients of Th turn out to be rational integers. Theorem 1. Let M = a + a q + a q2 + 0 1 2 be the Fourier series of a modular form M of weight h. Then

C a + C a + + C a = 0 h0 0 h1 1 hr r Proof. For l = 0, 1, 2,... dj 1 djl+1 jl = dz l + 1 dz and hence, by (??), it has a Fourier series without constant term. Since the dj function W deﬁned by (9) is a polynomial in j, the product W is, similarly, dz a Fourier series without constant term.

Becauseof (6) and (8), we have 253

1 dj 1 1 r 1 r W = MGh− 12r+12∆ − G14∆− = MG12r h+2∆− = MTh −2πi dz − − from which the theorem follows on substituting the series (11) and (??) for Th and M respectively. Let us put for brevity, ch0 = ch.

It is important, for the entire sequel, to show that ch does not vanish. Theorem 2. We have ch , 0. BIBLIOGRAPHY 223

Proof. First, let h 2( mod 4), so that h 2t( mod 12) with t = 1, 3, 5. Then correspondingly≡ 12r = h 2, h + 6, h +≡2; hence 12r h + 2 = 0, 8, 4 and − − 2 G12r h+2 = G0, G4, G4. −

Since by (5), G4 has all its Fourier coeﬃcients positive and the same holds for 5 ∆− as a consequence of (12), we conclude from (10) that all the coeﬃcients in the expansion (11) are positive. Therefore in the present case, the integers ch0, ch1,..., chr are all positive and, in particular, ch = ch0 > 0 i.e. ch , 0. Let now h 0( mod 4) so that h 4t( mod 12) with t = 0, 1, 2 whence 12r = h 4t +≡12, h 12r + 12 = 4t and≡ − − t Gh 12r+12 = G4t = G4. − Further we have now

1 r 1 dj t 1 r dj T = G + + ∆ − G− = G− ∆ − h − 12r h 2 14 d log q − 4 d log q 1 t/3 3 1 r t/3 dj − = ∆ − − t 3 d log q − 3 t∆ r r 3 d(G4− − ) 3r + t 3 3 t d∆− = + − G − ; t 3 d log q (3 t)r 4 d log a − − hence Ch0 is also the constant term in the Fourier expansion of the function 254

r 3r + t 3 3 t d∆− V = − G − . h (3 t)r 4 d log q − 3 t In view of the assumption h > 2, we have 3r + t 3 > 0. The series for G4− begins with 1 and has again all its coefficients positive.− Further, by (12), the r 1 r coefficients of the negative powers q− ,..., q− of q in the derivative of ∆− with respect to log q are all negative while the constant term is absent. Hence the constant term in Vh is negative and ch = ch0 < 0 i.e. ch , 0. The proof is thus complete. A most important consequence of the two theorems is the fact that, for every modular form M of weight h, the constant term a0 in its Fourier expansion is determined by the r Fourier coefficients a1,..., ar that follow, namely by the formula 1 a c− (c a + + c a ). (14) 0 h h1 1 hr r If, in particular, a1,..., ar are rational integers, then a0 itself is rational and the denominator of a0 divides ch. BIBLIOGRAPHY 224

As a further remarkable result, we note that the vanishing of a0 follows from the vanishing of the Fourier coefficients a1,..., ar and hence the vanishing of the modular form M itself, since then the entire modular function W defined in (9) has a zero at q = 0. From (10) and (11), it follows that the numbers chl(l = 0, 1,..., r) are the l r coefficients of q− in the product of G12r h+2 with ∆− as is indeed seen by direct calculation, which can be slightly simplified− by using the formula

∞ ∞ (1 qm)3 = ( 1)n(2n + 1)qn(n+1)/2 − − m=1 n=0

Now chr = 1 always. Also we easily obtain the values

c = 240 = 24 3 5; c = 504 = 23 32 7; 4 − − 6 c = 480 = 25 3 5; c = 264 = 23 3 11; 8 − − 10 c = 196560 = 24 33 5 7 13, c = 24 = 23 3; 12 − − 12,1 255 c = 24 = 23 3; 14

c = 146880 = 26 33 5 17, c = 216 = 23 33; 16 − − 16,1 − − c = 86184 = 23 34 7 19, c = 528 = 24 3 11; 18 18,1 c = 39600 = 24 32 52 11, c = 456 = 23 3 19; 20 − − 20,1 − − c = 14904 = 23 34 23, c = 288 = 25 32; 22 22,1 c = 52416000 = 29 32 53 7 13, 24 − − c = 195660 = 22 32 5 1087, c = 48 = 24 3; 24,1 − − 24,2 c = 1224 = 23 32 17, c = 48 = 24 3. 26 26,1 From this table, we see for example, with h = 12 and h = 24, that the r numbers chl(l = 0,..., r 1) need not all have the same sign, if h is a multiple of 4. Such a change of− sign always occurs, moreover, for h > 248 since, in particular, for this h, ch,r 1 > 0 whereas Ch0 < 0. Further, since the number − ch,r 1 is 0 for h = 214 and h = 248, the assertion of Theorem 2 does not hold, without− exception, for all c (l = 1, 2,..., r 1). hl − For number-theoretic applications of theorem 1, the prime-factors of ch are of interest. We get some information regarding these prime factors if in the BIBLIOGRAPHY 225

statement of Theorem 1 we consider the Fourier coeﬃcients of Gh(h = 4, 6,...) given by (4) instead of the modular form M. Then we have from (14),

r chBh b 1 = chlσh 1(l),σh 1(l) = t − (h = 4, 6,...) (15) 2h − − l=1 t l |

Since the ﬁrst sum here is an integer, ch must be divisible by the denomina- Bh tor of ; in any case, the denominator of the Bernoulli numberB is a divisor 256 2h h of ch. By the Staudt-Clausen theorem, however, this denominator is equal to the product of those primes p, for which p 1 divides h, Thus, for example, for h = 24, one sees the reason for the presence− of the prime divisors 2, 3, 5, 7, 13 of c24. On the other hand, for h = 26, the factor 17 of c26 cannot be explained in this simple manner. By (15), we have however explicitly

2 32 17 B c B 26 26 26 = 24 3 125 + (125 + 225) = 33554481 13 − 2 26 = 3 17 657931 where thus the factor 17 appears on the right side also and we obtain 13 657931 B = 26 2 3 agreeing with the known value.

2 Modular Forms of Hecke

Let K be a totally real algebraic number field of degree g and discriminant d and let further k, be a natural number which is, inaddition, even if there exists in K a unit of norm 1. We denote the norm and trace by N and S . Let u be an − 1 ideal, d the different of K and u∗ = (ud)− , the complementary ideal to u. We assign to every one of the g fields conjugate to K, one of the variables z(1),..., z(g) in the upper half-plane and form with them the Eisenstein series

k ′ k Fk(u, z) = N(u ) N((λz + )− ) u(λ,) | where λ, run over a complete system of pairs of numbers in the ideal u, different from 0, 0 and not differing from one another by a factor which is a unit. The series converges absolutely for k > 2. For the cases k = 1, 2, in order to BIBLIOGRAPHY 226 obtain convergence, the general term of the series is multiplied, as usual, by the factor N( λz+ s), where the real part of the complex variable s is greater than | |− 2 k and F (u, z) is defined as thevalue of the analytic continuation at s = 0. 257 − k A comparison with (1) shows that Fk(u, z) gives for g = 1 and k = 4, 6,... the function denoted there as Fk(z). Let hereafter g > 1. The generalization of (2) gives for the Fourier expansion of Fk(u, z), the formula

k g (2πi) 1/2 k 2πiS (vz) Fk(u, z) = ζ(u, k) + d − σk 1(u, v)e , (16) (k 1)! − d 1 v>0 − − | 1 where v runs over all totally positive numbers in d− and

k k ζ(u, k) = N(u ) N(− ), u() | (17) k k 1 σk 1(u, v) = sign(N(α ))N((α)ud) − . − d 1(α)u v − | | Here the summation is over principal ideals (), (α) under the conditions given. For even k, (17) can be put in the form

k 1 σk 1(u, v) = N(t − )(k = 2, 4,...) (18) − udt (v)d ∼ | where t runs over all integral ideals in the class of ud dividing (v)d. As a departure from the case g = 1, formula (??) is valid also for k = 2, while for k = 1, an additional term is to be tagged on. For this reason, it is provisionally assumed that k > 1. The function Fk(u, z) is a modular form of Hecke of weight k, since under the substitutions αz + β z → γz + d of the Hilbert modular group, it takes the required factor N((γz + d)k). If we set all the g independent variables z(1),..., z(g) equal to the same value z in the upper half-plane, then form Fk(u, z) one obtains an elliptic modular form, which we denote by Φk(u, z). Its weight is clearly h = kg. Let its Fourier expansion in powers of q = e2πiz be

∞ n Φk(u, z) = bnq n=0 BIBLIOGRAPHY 227

wherenow the Fourier coeﬃcients b0, b1,... are determined, in view of (??) by 258 the formulae

k g (2πi) 1/2 k b0 = ζ(u, k), bn = d − σk 1(u, v)(n = 1, 2,...) (19) (k 1)! − − S(v)=n 1 Here v runs over all totally positive numbers with trace n in the ideal d− and σk 1(u, v) is deﬁned by (17). For brevity, let us put −

σk 1(u, v) = sn(u, k)(n = 1, 2,...); (20) − S(v)=n this is a rational integer. Explicitly,

k k 1 sn(u, k) = sign(N(α ))N((α)ud) − (21)

u∗ (α),u β αβ>0|,S (αβ| )=n where the summation is over the principal ideals (α) and the numbers β in the ﬁeld, subject to the given conditions. On applying (14) with M = Φk(u, z), it now follows

r g h k 1/2 1 ((k 1)!) (2πi)− d − ζ(u, k) = c− c S (u, k); (22) − − h h l l=1 g here we have to take h = kg and r = rh. We note further that N( 1) = ( 1) and therefore h is always an even number. These formulae permit− the calculat− ion of ζ(u, k), if the ideal u and the natural number k are given. They show, in particular, that the value

kg k 1/2 π− d − ζ(u, k) = ψ(u, k) is rational and more precisely, its denominator divides ((k 1)!)gc . Itistobe − h stressed here that ch depends only on h = kg but not on the other properties of the field K. In the special case g = 2 and thus for a real quadratic field, it was already known that the number on the left side of (22) is rational and that its denominator divides 4((2k)!)2v2k where however v still depends on K, being given by the smallest positive solution y = v of Pell’s equationx2 dy2 = 4. On the 259 other hand, with the help of a computing machine, Lang has− listed the values of ψ(u, 2) in 50 different examples for k = g = 2 and remarked, in addition, that in every one of the cases considered, only a divisor of 15 appears as the denominator. It is now easy to explain this phenomenon since 2 4c = 15. − − 4 BIBLIOGRAPHY 228

For every ﬁxed totally real algebraic number ﬁeld K, we can, following (3), look upon the positive rational numbers ψ(u, k)(k = 2, 4,...) as the ana- k 1 1 logues of the values 2 − (k!)− Bk formed with the Bernoulli numbers. Unfor- tunately, however, one cannot| expect| that for this generalisation, corresponding recursion-formulae and divisibility theorems will exist as for the Bernoulli numbers themselves. For, they are connected with the properties of the func- 2πz 1 tion (e 1)− whose partial fraction decomposition also leads to (3). Indeed, similar to− (??) we have even the simpler formule

k g k (2πi) 1/2 1 k 1 2πiS (vz) N((z+)− ) = d− N(u− ) N(v − )e (k = 2, 3,...); (k 1)! u uv>0 | − ∗| but in contrast with the case of one variable, the analytic function appearing here for g > 1 does not have, any more, important properties such as the addition theorem and the diﬀerential equation enjoyed by the exponential function. Further by an important result of Hecke, it is always singular for all real values of any one of the g variables, so that one cannot as in the case g = 1 obtain the numbers ζ(u, k) by expansion in powers of z. We get these numbers by going to the modular forms Fk(u, z) of Hecke, which in spite of their complicated construction are really connected with one another by algebraic relations. To the result (22) we may add a remark which relates to a lower bound for d discovered by Minkowski. The left side of (22) is clearly diﬀerent from 0 for even k, since then ζ(u, k) is positive. Consequently among the r sums sl(u, k) given for l = 1, 2,..., r by (21), at least one must be non-empty and hence there always exist two numbers α, β in K which satisfy the four conditions

u∗ α, u β, αβ > 0, S (αβ) r | | ≤ with r = rh = rkg. For this assertion, it is optimal to choose k = 2 and hence 260 h = 2g; accordingly g g r = + 1(g . 1( mod 6)), r = (g 1( mod 6)) 6 6 ≡ so that in every case g r + 1. (23) ≤ 6 From the decompositions

(α) = u∗t, (β) = ub with integral ideals t, b, it follows that there exist two integral ideals t and b (in every ideal class and its complementary class), such that 1 tbd− = (v), v > 0, S (v) r (24) ≤ BIBLIOGRAPHY 229

We therefore have g 1 1 1 g r d− N(tb) = N(tbd− ) = N(v) (g− S (v)) ≤ ≤ g (25) r g N(tb) d ≤ g and hence r g/2 Min(N(t), N(b)) √d ≤ g Since especially for every real quadratic field, the ideal d = ( √d) belongs to the principal class, the classes of t and b are inverses of each other because of 1 the first formula (24) and hence the conjugate ideal b′ = (N(b))b− is equivalent to t; further, g = 2 and so r = 1. Since N(b) = N(b′), there exists, in every ideal 1 class of a real quadratic field, an integral ideal of norm √d always. ≤ 2 For arbitrary totally real fields of degree g 2, we always have by (25) ≥ r g 1 N(t) d, ≤ ≤ g from which, by (23) the inequality 261

g g g g 6 − g 6 d 6 1 + > 6 e− (26) ≥ r ≥ g follows. Since in the cases g = 2, 3, 4, 5, 7 the number r = 1, we have then

d gg ≥ and this is a new result for ﬁeld degrees 5 and 7. Since e2 > 6, the estimate

gg 2 d ≥ g! found by Minkowski is better than (??) for all suﬃciently large g, but not for a few small g to which set the given values 5 and 7 belong. For g = 4, it is well-known that 725 is the smallest discriminant of a totally real ﬁeld while

48 7 44 = 256, = 113 + . (4!)2 9 BIBLIOGRAPHY 230

The special case g = 2 yields also a connection with the reduction theory of indeﬁnite quadratic forms due to Gauss. Namely, let [κ,λ] be a basis of the ideal u and hence

λ b + √d 2 = , d = b 4ac, N(u) √d = κλ′ λκ′ κ 2a − | − | with rational integers

1 1 1 a = N(κ)N(u− ), b = S (κλ′)N(u− ), c = N(λ)N(u− ).

1 Then the conjugate ideal is u′ = (κ′,λ′) and (κλ′ λκ′)− [λ′, κ′] gives the 1 − − basis of u∗ = (ud)− complementary to [κ,λ]. If we now put κ x + λ y α = ′ 2 ′ 2 , β = κx + λy κλ λκ 1 1 ′ − ′ withrational integers x1, y1, x2, y2, then 262 2κκ x x + (κλ + λκ )(x y + x y ) + 2λλ y y x y x y αβ = ′ 1 2 ′ ′ 1 2 2 1 ′ 1 2 + 1 2 − 2 1 2(κλ λκ ) 2 ′ − ′ 2ax x + b(x y + x y ) + 2cy y x y x y = 1 2 1 2 2 1 1 2 + 1 2 − 2 1 2 sign(κλ λκ ) √d 2 ′ − ′ and the two conditions S (αβ) = 1, αβ > 0 go over into

x y x y = 1, 2ax x + b(x y + x y ) + 2cy y < √d. 1 2 − 2 1 | 1 2 1 2 2 1 1 2| But this precisely means that the binary quadratic form au2 + buv + cv2 of discriminant d goes over, under the linear substitution u x u+x v, v y u+ → 1 2 → 1 y2v of determinant 1, into a properly equivalent form whose middle coeﬃcient is smaller than √d in absolute value. This means again that in the transformed form both the outer coeﬃcients are of opposite signs and this is just the essence of reduction theory. We have still to consider the case k = 1 excluded hitherto. In this case [3], in contrast to (??), the constant term is

b0 = ζ(u, 1) + ζ(u∗, 1) where we have to deﬁne

1 s ζ(u, 1) = N(u)lim N(− )N( − ) s 0 | | → u() | BIBLIOGRAPHY 231 and also change (22) correspondingly. This way we do not obtain the individual numbers ζ(u, 1) and ζ(u∗, 1) but only their sum. By the way, this is always 0 if either g = 2 or u2d is a principal ideal (γ) and N(γ) < 0.

3 Examples

In conclusion, we give three simple examples for evaluating ζ(u, k) using(22), 263 where the result can also be checked in another way. With k = g = 2, we have h = 4, r = 1 and

4 4 2 2 (2π) π N(u ) N(− ) = s (u, 2) = s (u, 2); −c d3/2 1 1 u() 4 15d √d | here sn(u, k) for k = 2, 4,... and n = 1, 2,... is given by (18) and (20). If a + δ we take, in particular, d = 4.79, u = b = (1), then d = (2δ), v = with 2δ δ = √79 and a = 0, 1,..., 8, N(a + δ) = 79, 2 3 13, 3 52, 2 5 7, 32 7, 2 33, 43, 2 3 5,± 3 5. We± now− note that 2 = N(9 +δ), while the cube of the prime ideal p = (3 , 1 + δ) gives a principal ideal. For the determination of the principal ideal divisors (τ)of(a + δ), we have to consider besides the trivial decomposition (a + δ) = (1) (a + δ), only a + δ (a + δ) = (9 + δ) 9 + δ with odd a; hence, for

σ1(o, v) = o(a) = N((τ)), o (τ) (a+δ) | | we have the values

σ(0) = 1 + 79 = 80, σ( 1) = 1 + 78 + 2 + 39 = 120, ± σ( 2) = 1 + 75 = 76, σ( 3) = 1 + 70 + 2 + 35 = 108, ± ± σ( 4) = 1 + 63 = 64, σ( 5) = 1 + 54 + 2 + 27 = 84, ± ± σ( 6) = 1 + 43 = 44, σ( 7) = 1 + 30 + 2 + 15 = 48, ± ± σ( 8) = 1 + 15 = 16. ± Accordingly, in the present case,

s1(o, 2) = 80 + 2(120 + 76 + 108 + 64 + 84 + 44 + 48 + 16) = 1200 BIBLIOGRAPHY 232

4 2 10π N(− ) = o () 79 √79 | inagreement with the value found by me elsewhere. For u = p = (3, 1 + δ), we 264 obtain correspondingly

s1(p, 2) = 0 + (3 + 6 + 13 + 26) + (3 + 5 + 15 + 25)+ + (5 + 10 + 7 + 14) + (3 + 7 + 21 + 9)+ + (3 + 6 + 9 + 18) + 0 + (3 + 6 + 5 + 10) + (3 + 5) = 240 4 2 2 2π N(p ) N(− ) = p() 79 √79 | and the same for u = p = (3, 1 δ). Since the 3 ideal classes of the ﬁeld ′ − are represented by 1, p, p′, we have therefore for the zeta function ζK(s) of the ﬁeld, π4 14π4 ζK(2) = (10 + 2 + 2) = , 79 √79 79 √79 while from the law of decomposition, we get, likewise, on using the formula

k d d k (2πi) d l n− = pk , n −1 k! √d l d n=1 l=1 k k k l Pk(x) = Bl x − (k = 2, 4,...) l l=0 that

∞ 316 2 ζK(2) = ζ(2) n− n n=1 π2 (2π)2 316 316 l = P2 6 2 2! √316 l 316 l=1 π4 1 39 79 2l 1 14π4 = 1 − = . 12 √79 2 2l 1 − 79 79 √79 l=1 − We now take k = 12, g = 2; then h = 24, r = 3 and

12 12 N(u ) N(− ) u() | BIBLIOGRAPHY 233

265 (2π)24 = u, + u, + u, 2 23/2 (c24,1 s1( 12) c24,2 s2( 12) s3( 12)) −(11!) c24d π24 = ( 195660s1(u, 12) + 48s2(u, 12) + s3(u, 12)). 2 310 57 73 112 13d11 √d − In the special case d = 5, there exists only one class d = ( √5) and the totally √ 1 1 + 5 positive numbers v of the ideal d− with trace 1, 2, 3 are given by ± , 2 √5 a + √5 a + 3 √5 1 + √5 (a = 0, 1, 2), (a + 1, 3, 5). Since ± , 2 + √5 √5 ± ± 2 √5 ± ± ± 2 ± 1 + 3 √5 3 + 3 √5 5 + 3 √5 are units and also √5, 1 + √5, ± , ± , ± are inde- ± 2 2 2 composable, we have s (o, 12) = 2 111 = 2, s (b, 12) = 111 + 511 + 2(111 + 411) + 2 111 1 2 = 57216738, 11 11 11 11 11 11 s3(o, 12) = 2(1 + 11 ) + 2(1 + 9 ) + 2(1 + 5 ) = 633483116696 and thus 4 24 12 2 691 110921π N(− ) = 10 16 3 2 o() 3 5 7 11 13 √5 | | On the other hand, in the present case too, we have

12 N(− ) = ζK(12) o () | 24 4 ∞ 5 12 (2π) B12 5 l = ζ(12) n− = P12 n (2 12!)2 √5 l 5 n=1 l=1 with the same result, where this time the factor 691 comes in through the numerator of B12. Finallylet k = 2, g = 3 so that h = 6, r = 1 and 266

6 6 2 2 (2π) 8π N(u ) N(− ) = s (u, 2) = s (u, 2). c d3/2 1 1 u() 6 63d √d | In particular, let d = 49 and hence K, the totally real cubic subﬁeld of the ﬁeld of 7th roots of unity. It is generated by

1 1 ρ = 2 ǫ ǫ− = (1 ǫ)(1 ǫ− ) − − − − BIBLIOGRAPHY 234 with ǫ = e2πi/7 and has class number 1, as is also evident from (25). Since [1, ρ, ρ2] is a basis of o and N(ρ) = 7, we have (ρ3) = 7 and so d = (ρ2). We have hence to determine for

1 2 v = x + yρ− + zρ− , the solutions of S (v) = 1, v > 0 with rational integers x, y, z. A short computation gives

3 2 2 1 2 ρ = 7(ρ 1) , S (ρ) = 7, S (ρ ) = 21, S (ρ− ) = 2, S (ρ− ) = 2, − 3 3 S (ρ− ) = 2 + , 7 S (v) = 3x + 2y + 2z = 1

By the substitution x = 1 + 2u with integral u, it follows that y = 1 y 3u and the condition v > 0, and consequently ρ2v > 0, goes over into − − −

(2ρ2 3ρ)u + (1 ρ)z + ρ2 ρ > 0. (27) − − − If we set again

k k 2kπ ǫ + ǫ− = 2 cos = λ (k = 1, 2, 3), 7 k then π π 3 1 = 2 cos <λ < 2 cos = √2 < , 3 1 4 2 1 π π π < < 2sin = λ < 2 cos = 0 − 2 − 7 − 14 2 2 π π 3 2 = 2 cos π<λ = 2 cos < 2 cos = √3 < . − 3 − 7 − 6 − −2 267 With this, we obtain from (27) by a simple modiﬁcation, the three conditions

z + λ 2 > (λ + 5)u, z + λ 2 < (λ + 5)u, 1 − 2 2 − 3 z + λ 2 < (λ + 5)u 3 − 1 whence, it follows, in particular, that

λ3 < u <λ1 BIBLIOGRAPHY 235

Therefore, for u, only the values 0, 1 have to be considered. Of these however, u = 1 leads to the contradiction ±

5 <λ λ + 7 < z <λ λ + 7 < 6. 2 − 1 3 − 2 For u = 0, we have 2 λ < z < 2 λ < 2 λ , − 1 − 2 − 3 and hence either z = 1, y = 2, x = 1 or z = 2, y = 3, x = 1 and for u = 1, − − − λ 2 < z <λ 2 <λ 2 3 − 2 − 1 − and hence z = 3, y = 5, x = 1. The ﬁrst of the three solutions formed above, gives − − ρ ρ2v = (ρ 1)2, v = . − 7 and the other two v must therefore be conjugate to this, as we can also check directly. In all the three cases, ρ2v proves to be a unit and hence (v)d = o, t = o, s1(o, 2) = 3 and

6 3 6 2 8π 3 2 π N(− ) = = . 3 74 o () 63 49 √49 | In order to verify this result in another way, we note from the law of decomposition for the cyclotomic ﬁeld, that

2 2 2 1 l l N(− ) = ζ (2) = (1 7− )− (w + η w + η− w ) K − 1 2 3 o () l=0 | 72 = (w3 + w3 + w3 3w w w ) 24 3 1 2 3 − 1 2 3 with 268 ∞ 2 2πi/3 wk = (7n + k)− (k = 1, 2, 3),η = e . n= −∞ But now 2 2 2 π 2π k k 1 2π 1 wk =   = (2 ǫ ǫ− )− = ρ−  k  7 − − 7 7sin   7    6 6 3 3 3 2π 3 2π 3 w + w + w = S (ρ− ) = 2 + 1 2 3 7 7 7 BIBLIOGRAPHY 236

6 6 2π 1 2π 1 w1w2w3 = N(ρ− ) = 7 7 7 and indeed we have again

2 6 3 6 2 7 2π 2 π N(− ) = 2 = 24 3 7 3 74 o () | Bibliography

[1] Meyer, C: Uber¨ die Bildung von elementar-arithmetischen Klassenin- varianten in reell-quadratischen Zahlkorpern¨ Berichte. math. Forsch-Ist. Oberwalfach Z, Mannheim 1966, 165-215. Lang, H: Uber¨ eine Gattung elementar-arithmetischer Klasseninvarianten reell-quadratischer Zahkkorper.´ Inaug Diss. Koln.¨ 1967 VII u. 84 S. Siegel, C. L.: Bernoullische Polynome und quadratische Zahlkorper.¨ Nachr. Akad. Wiss. Gottigen, Math-Phys. K1. 1968, 7-38. Barner, K: Uber¨ die Werte der Ringklassen-L-Funktionen reell- quadratischer Zahlkorper¨ an naturlichen¨ Argumentstellen. J. Numbr. Th. 1. 28-64 (1969).

[2] Klingen, H: Uber¨ die Werte der Dedekindschen Zetafunktionen, Math. Ann. 145, 265-272 (1962).

[3] Gundlach, K. B: Poincaresche´ und Eisensteinsche Reihen zur Hilbertschen Modulgruppe. Math. Zeitschr. 64, 339-352 (1956); insbes 350-351.

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