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Modular Forms and Dirichlet Series MODULAR FORMS AND DIRICHLET SERIES ANDREW OGG Preface These are the official notes for a course given at Berkeley during the fall and winter quarters of 1967–68 on Hecke’s theory of modular forms and Dirichlet series. The reader who is conversant with Hecke’s Werke will find nothing new here, except I have taken the liberty of including a recent paper of Weil, which stimulated my interest in this field. The prerequisites for reading these notes are the theory of analytic functions of one complex variable and some number theory. Unattributed theorems are generally due to Hecke. A. P. Ogg Berkeley, California March, 1968 TEX edition. This is a re-issue of Ogg’s book [8] published in 1969, typeset with TEX. In particular, the numbering system (theorems, propositions) differs from [8]. Marked text in [8] is emphasized here and included in the index. Detected typos in [8] have been corrected; they are listed at the end. c 2018 TEX version Berndt E. Schwerdtfeger, v1.0, 29th August 2018 Contents Preface 1 TEX edition 1 Introduction2 1. Dirichlet series with functional equation4 2. Hecke operators for the full modular group 23 3. The Petersson inner product 29 4. Congruence subgroups of the modular group 34 5. A theorem of Weil 50 6. Quadratic forms 56 Corrected typos in Ogg’s book 66 References 66 Index 68 2010 Mathematics Subject Classification. Primary 11F11; Secondary 11F25, 11F66. Key words and phrases. modular forms, Dirichlet series. 1 2 ANDREW OGG Introduction The simplest and most famous series is the Riemann zeta-function ζ(s), defined for Re(s) > 1 by 1 X Y ζ(s) = n−s = (1 − p−s)−1; n=1 p the product being over all primes p; the equality of the two expressions is just an analytic statement of the fundamental theorem of arithmetic. Riemann [10, VII.] proved in 1859 that ζ(s) has an analytic continuation to the whole s-plane except for a simple pole of residue 1 at s = 1 and satisfies the functional equation: s Z(s) = π−s=2Γ( )ζ(s) 2 is invariant under s 7! 1 − s. In fact, this functional equation almost character- izes ζ(s), for Hamburger [4] showed in 1921 that any Dirichlet series satisfying this functional equation and suitable regularity conditions is necessarily a constant multiple of ζ(s). However, the situation did not become clear until greatly general- ized by Hecke in his paper “Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung”, published in 1936 [5, 33.]. Let us sketch that proof of the functional equation for ζ(s) which leads naturally to Hecke’s generalization. Starting from Z 1 Γ(s) = ts−1e−tdt (Re (s) > 0) 0 we find 1 X Z 1 π−sΓ(s)ζ(2s) = (πn2)−sts−1e−tdt (Re (s) > 1) n=1 0 1 Z 1 X 2 = ts−1e−πn tdt n=1 0 Z 1 1 = ts−1(#(it) − )dt 0 2 where 1 1 X 2 #(τ) = e+πin τ (Im τ > 0) 2 −∞ 1 1 X 2 = + eπin τ 2 n=1 is the basic theta-function. Now #(τ) is holomorphic on the upper half plane, and satisfies #(τ + 2) = #(τ) τ #(−1/τ) = ( )1=2#(τ); i where the square root is defined on Re (z) > 0 to be real on the real axis. These 1 two equations say that #(τ) is a modular form of dimension − 2 for the group G(2) generated by τ 7! τ + 2, τ 7! −1/τ, and # is up to a constant multiple the only solution of these equations. (These facts have been known for ages, and will be MODULAR FORMS AND DIRICHLET SERIES 3 proved later in these notes.) The functional equation for ζ(s) is now a consequence of that for #(τ): Z 1 s Z 1 −s s−1 1 1 t 1 s−1 π Γ(s)ζ(2s) = t (#(it) − )dt − j0 + t #(it)dt 1 2 2 s 0 Z 1 1 1 Z 1 i = ts−1(#(it) − )dt − + t−s−1#( )dt 1 2 2s 1 t Z 1 1 1 1 = (ts−1 + t1=2−s−1)(#(it) − )dt − − ; 1 2 2s 1 − 2s 1 1 visibly invariant under s 7! 2 −s; furthermore, the integral is entire, since #(it)− 2 = O(e−ct), for some c > 0. On the other hand, by Mellin inversion we have 1 1 Z #(ix) − = x−s(π−sΓ(s)ζ(2s))ds 2 2πi Re (s)=c for sufficiently large c > 0, and by similar reasoning (carried out in detail in a more general situation) the functional equation for #(τ) can be derived from that for ζ(s); this is Hecke’s proof that ζ(s) is determined by its functional equation. The above proof generalizes directly, as follows. Given a sequence of complex c numbers a0; a1; a2; : : : ; an = O(n ) for some c > 0, and given λ > 0, k > 0, C = ±1, form 1 1 X 2π X '(s) = a n−s Φ(s) = ( )−sΓ(s)'(s) f(τ) = a e2πinτ/λ n λ n n=1 n=0 (the O-condition ensures that '(s) converges somewhere, and f(s) is holomorphic in the upper half plane.) Theorem. The following two conditions are equivalent: a0 Ca0 (A) Φ(s) + s + k−s is entire and bounded in every vertical strip (henceforth abbreviated to EBV ) and satisfies Φ(k − s) = CΦ(s); τ k (B) f(−1/τ) = C( i ) f(τ). τ τ k k log i (( i ) = e , where log is real on the real axis.) 1 Note for '(s) = ζ(2s), f(τ) = #(τ), we have C = 1; λ = 2; k = 2 . Generally, let G(λ) be the group of substitutions of the upper half plane gener- ated by τ 7! τ + λ, τ 7! −1/τ.A modular form of dimension −k and multiplier C for G(λ) is a holomorphic function f(τ) on the upper half plane satisfying (1) f(τ + λ) = f(τ) τ k (2) f(−1/τ) = C( i ) f(τ) (3) the expansion of f(τ) in a Laurent series in e2πiτ/λ (from (1)) has no neg- P1 2πinτ=λ ative terms: f(τ) = n=0 ane , i.e. f is “holomorphic at 1”. We denote the space of such f by M(λ, k; C). We also denote by M0(λ, k; C) the subspace of those f which satisfy the additional condition that the Fourier c coefficients an satisfy an = O(n ) for some c > 0. The theorem then says there is a one-one correspondence between the elements of M0(λ, k; C) and Dirichlet series satisfying (A); note that '(s) is regular at s = k if and only if a0 = 0, i.e. f(τ) “vanishes at 1”. We say '(s) has signature (λ, k; C) if (A) holds. 4 ANDREW OGG Remark. If '(s) = ζ(K; s) is the zeta-function of an algebraic number field K, its functional equation is that s jdjs=2((2π)−sΓ(s))r2 (π−s=2Γ( ))r1 '(s) 2 is invariant under s 7! 1 − s, where d is the discriminant of K and r1 resp. r2 is the number of real resp. complex primes of K. Note this falls within the scope of the the- orem only when there is only one Γ-function, i.e. K is rational or imaginary quad- ratic. If K is imaginary quadratic, then '(s) has signature (λ, k; C) = (pjdj; 1; 1); it turns out that '(s) is determined by its signature when d = −3; −4 but not for d < −4. The other part of Hecke’s theory concerns the question of whether '(s) has an Q −s Euler product, i.e. '(s) = p 'p(s), where 'p(s) is a power series in p . Suppose for concreteness that λ = 1, so G(λ) = Γ is the modular group. It turns out that M(1; k; C) = M0(1; k; C), and this space is 0 unless k is an even integer ≥ 4, and C = ik, and the only possible Euler product for '(s), of signature (1; k; ik), is 1 X −s Y −s k−1−2s −1 '(s) = ann = (1 − app + p ) ; n=1 p and '(s) has this Euler product if and only if the associated modular form f is an eigenfunction for a certain ring of operators on M(1; k; ik), the Hecke operat- ors. The question of the existence of Euler products is of course fundamental for number theory, since in practice the numbers an will be the number of solutions of some number-theoretic problem and the knowledge of an Euler product reduces knowledge of all the an to knowledge of the ap for primes p. 1. Dirichlet series with functional equation c All that we need about Dirichlet series is that if an = O(n ), then '(s) = P1 −s n=1 ann converges absolutely and uniformly in Re (s) ≥ c + 1 + ", since it is P1 −1−" dominated term-by-term uniformly by n=1 n < 1, and hence '(s) defines a holomorphic function in (at least) the half plane Re (s) > c + 1. Conversely, if '(s) σ0 −s converges at s0 = σ0 + it, we see an = O(n ) since the general term ann tends P1 −s c to 0. Thus '(s) = n=1 ann converges somewhere if and only if an = O(n ). As to the gamma-function, we need: R 1 s−1 −t (a) Γ(s) = 0 t e dt; for Re (s) > 0 1 p (b) Γ(s + 1) = sΓ(s); Γ(1) = 1; Γ( 2 ) = π.
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