Cubic Theta Functions

Home , Nome (mathematics), Theta function

Daniel Schultz

July 13, 2011

Abstract P m2+mn+n2 m n The theory of the cubic theta function q z1 z2 is developed analogously to the theory of the classical Jacobian theta functions. Many inversion formulas, addition formulas, and modular equations are derived, and these results extend previous work done on cubic theta functions.

Introduction

For any positive integer n, the rising factorial is deﬁned by

Γ(a + n) (a) := = a(a + 1) ··· (a + n − 1), n Γ(a) and the Gaussian hypergeometric function is deﬁned by

∞ X (a)n(b)n F (a, b; c; z) := zn. 2 1 (c) (1) n=0 n n In his paper [20] on modular equations and approximations to π, Ramanujan gives two series for π that “belong to the theory of q2” where

1 2 2! 2π 2F1 , ; 1; 1 − k q (k2) = exp −√ 3 3 . (0.1) 2 1 2 2 3 2F1 3 , 3 ; 1; k Such an expression is an analogue of the classical relation between the nome q and modulus k2 of the elliptic curve y2 = x(1 − x)(1 − k2x) given by

1 1 2! 2F1 , ; 1; 1 − k q = exp −π 2 2 (0.2) 1 1 2 2F1 2 , 2 ; 1; k in which the hypergeometric function

1 1 K (z) := F , ; 1; z (0.3) 2 2 1 2 2 has been replaced by 1 2 K (z) := F , ; 1; z . (0.4) 3 2 1 3 3

1 Ramanujan gives several results in the theory of q2, and among the most interesting are several modular equations and an analogue of a inversion formula for the classical Jacobian elliptic functions. The latter (Theorem 8.1 in [3]) gives the solution for φ when √ Z φ cos 1 sin−1 ( m sin φ) u = 3 dφ (0.5) p 2 0 1 − m sin φ by ∞ dφ 3 X xn = , (0.6) du K (m) 1 + qn + q−n 3 n=−∞ where x = exp 2iu , and q = q (m) is given by (0.1). Here it seems more natural to use m K3(m) 2 2 to denote the modulus instead of using Ramanujan’s k , and Ramanujan’s “theory of q2” is more aptly called “the theory of cubic elliptic functions” or “the theory of signature 3”. A proof of this inversion formula is given in [3] or [4], and a simpler derivation is given in [22]. Ramanujan deﬁned a cubic modular equation of degree p as a relation between two moduli m1 and mp induced by the relation

K (1 − m ) K (1 − m ) 3 p = p 3 1 , (0.7) K3(mp) K3(m1) and gave the following modular equations of degree 2, 5, and 11 respectively:

x4 + y4 = 1, (0.8) x4 + 3x2y2 + y4 = 1, (0.9) √ 2 4x2y2 + 3 3xy x2 + y2 + x2 + y2 = 1, (0.10)

1/12 1/12 1/12 1/12 where x = m1 mp and y = (1 − m1) (1 − mp) in all cases. Ramanujan’s modular equations may be found in Chapter 33 of [3]. An example of a modular equation of degree 23 that will be proved in this paper is

√ 2 44x2y2 + 15 3xy x2 + y2 + 2 x2 + y2 = √ √ √ q √ √ (0.11) 2−3 xy 3x2 + 4xy + 3y2 4 3x2 + 13xy + 4 3y2.

Thus we see that the theory of the cubic elliptic functions is strikingly similar to the classical theory of elliptic functions; there are modular equations and elliptic functions that arise from the hypergeometric function K3(m) in place of the hypergeometric function K2(m). In fact, Ramanujan also developed results for the hypergeometric functions

1 3 K (z) := F , ; 1; z (0.12) 4 2 1 4 4 and 1 5 K (z) := F , ; 1; z , (0.13) 6 2 1 6 6 but the case of K3(z) is the one that gives the most interesting analogue to the classical theory of elliptic functions, and this is the case that is developed here.

2 Two central objects in the theory of elliptic functions (see, for example, [16] for a treatment of elliptic functions) are the elliptic integral

1 Z x dt u = p (0.14) πK2(m) 0 2 t(1 − t)(1 − mt) and the group of four theta functions

1/2 Θ (u; q) = Θ (u; q), 1 1/2 1/2 Θ (u; q) = Θ (u; q), 2 0 0 Θ (u; q) = Θ (u; q), 3 0 0 Θ (u; q) = Θ (u; q), 4 1/2 where ∞ b X 2 Θ (u; q) = q(n+b) e2(n+b)πi(u+a). (0.15) a n=−∞ −1 The periods of the integral in (0.14) (without the factor (πK2(m)) ), are πK2(m) and iπK2(1 − m), hence the inversion of this integral gives rise to a doubly periodic function. More speciﬁcally, when u and x are related by (0.14) and q is given by (0.2) (and k2 = m), we have the fundamental inversion formulas which relate ratios of theta functions to x: Θ (0; q) Θ (u; q) √ − 3 1 = x, Θ2(0; q) Θ4(u; q) Θ (0; q) Θ (u; q) √ 4 2 = 1 − x, (0.16) Θ2(0; q) Θ4(u; q) Θ (0; q) Θ (u; q) √ 4 3 = 1 − mx. Θ3(0; q) Θ4(u; q) These functions are the the Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m) respectively. There is another important set of inversion formulas which relate m and K2(m) to the theta constants: 2 Θ1(0; q) = 0, 2 1/2 Θ2(0; q) = m K2(m), 2 (0.17) Θ3(0; q) = K2(m), 2 1/2 Θ4(0; q) = (1 − m) K2(m). With the last set of inversion formulas, the relation

K (1 − m ) K (1 − m ) 2 p = p 2 1 , (0.18) K2(mp) K2(m1) which deﬁnes a modular equation of degree p in the classical theory, becomes (where q is

3 deﬁned by (0.2))

4 Θ1(0; q) m1 = 4 , Θ1(0; q) p 4 Θ1(0; q ) mp = p 4 , Θ1(0; q ) so that a modular equation is equivalent to an identity involving theta constants at q and qp. This modular equation is actually only part of the general pth order transformation for elliptic integrals, which asks for the relationship between x1 and xp induced by the equation

Z x1 Z xp pK2(mp) dt dt p = p . (0.19) K2(m1) 0 2 t(1 − t)(1 − m1t) 0 2 t(1 − t)(1 − mpt)

The quantity M = K2(m1) is known as the multiplier of degree p in the classical theory, and p K2(mp) the case p = 2 of this relationship is known as the Gauss or Landen transformation. In terms of a parameter t, we have t(t + 2) m = , 1 (t + 1)2 t2 m = , 2 (t + 2)2 (0.20) 2 t + 2 = , M t + 1 2 √ √ √ 2 x1 1 − x1 x2 = √ , M2 1 − m1x1 and this is a complete statement of the transformation of degree 2. The theta constants responsible for inverting K3(m) analogously to (0.17) were ﬁrst identi- ﬁed by the Borweins in [6] and Hirschhorn, Garvan, and Borwein in [15], where they introduced the notations (set ζ = exp 2πi/3 through out this paper)

∞ X 2 2 a(q) = qm +mn+n , m,n=−∞ ∞ X 2 2 b(q) = ζm−nqm +mn+n , m,n=−∞ ∞ 2 2 X m+ 1 + m+ 1 n+ 1 + n+ 1 c(q) = q( 3 ) ( 3 )( 3 ) ( 3 ) . m,n=−∞ They proved that these functions satisfy the identity

a(q)3 = b(q)3 + c(q)3 (0.21) and the inversion formula c(q)3 K = a(q), (0.22) 3 a(q)3 hence these functions should be called “cubic theta functions”. In order to produce satisfactory cubic analogues of the Landen transformation for elliptic integrals and diﬀerential equations and addition theorems satisﬁed by the Jacobian elliptic

4 functions, it seems that a group of nine theta functions in two analytic variables and the complex parameter q is necessary. As an example, we will deﬁne cubic theta functions of two 2πiu complex variables u = (u1, u2), four of which are (set zi = e i )

∞ X 1 2 1 1 1 2 m+ 1 n+ 1 1+m−n (m+ 3 ) +(m+ 3 )(n+ 3 )+(n+ 3 ) 3 3 Θ1(u; q) := ζ q z1 z2 , m,n=−∞ ∞ X 1 2 1 1 1 2 m+ 1 n+ 1 (m+ 3 ) +(m+ 3 )(n+ 3 )+(n+ 3 ) 3 3 Θ2(u; q) := q z1 z2 , m,n=−∞ ∞ X m2+mn+n2 m n Θ3(u; q) := q z1 z2 , m,n=−∞ ∞ X m−n m2+mn+n2 m n Θ4(u; q) := ζ q z1 z2 , m,n=−∞ that satisfy 3 3 3 3 Θ1(u; q) + Θ3(u; q) = Θ2(u; q) + Θ4(u; q) , which gives a two variable generalization of (0.21), and is an analogue of the the equations

sn(u)2 + cn(u)2 = 1, k2sn(u)2 + dn(u)2 = 1.

The other ﬁve cubic theta functions are denoted by Θ11,Θ12,Θ14,Θ22,Θ44 and are deﬁned in Section 2.3. We also have a triplication formula,

b(q)c(q)2 Θ (3u; q3) = Θ (u; q)Θ (u; q)Θ (u; q)+ b(q3) 4 1 12 44

Θ2(u; q)Θ4(u; q)Θ22(u; q) + Θ3(u; q)Θ11(u; q)Θ14(u; q), that is analogous to the classical theta function identity

2 2 Θ4 0; q Θ4 2u; q = Θ3(u; q)Θ4(u; q), which is a key component in deriving the Landen transformation. Just as the classical theta functions are intimately related to the elliptic curve

y2 = x(1 − x)(1 − mx) the nine cubic theta functions are intimately related to the genus 2 hyperelliptic curve

y3 = x(1 − x)2(1 − mx). (0.23)

Formulas of the type (0.22) or (0.17) are known as a Thomae-type formulas for the curves (0.23) and y2 = x(1 − x)(1 − mx) respectively, and these identities were established in [11]. In fact, we cite Example 6.3 of [11] for proofs of the following fundamental inversion formulas:

a(q) = K3(m), 1/3 b(q) = (1 − m) K3(m), 1/3 c(q) = m K3(m),

5 K (1−m) where q = exp − √2π 3 . 3 K3(m) Section 1 discusses the basics of the hyperelliptic curve (0.23), while Sections 2 and 3 collect the necessary facts about theta functions and Riemann surfaces. Inversion formulas for the cubic theta functions that are direct analogues of (0.16) are given in Section 4, and some examples of the addition theorems satisﬁed by the cubic theta functions are presented in Section 5. In Section 6, Ramanujan’s inversion formula in (0.6) is deduced in an entirely diﬀerent fashion than in [3] or [22] along with an elegant series-product identity which follows from an addition theorem for the cubic theta functions. Finally Sections 7 and 8 are devoted to modular transformations of the cubic theta functions, and many cubic modular equations are derived, including (0.11).

1 The Theta Function and its Corresponding Riemann Surface

Here we will attempt to give a full treatment of the properties of the function ∞ X m2+mn+n2 m n q z1 z2 , (1.1) m,n=−∞ which is in many ways completely analogous to the theory of the Jacobian elliptic theta functions. As mentioned in the introduction, the algebraic curve X associated to this theta function is y3 = x(1 − x)2(1 − mx). (1.2) That this is the corresponding algebraic curve will be clear when its period matrix is computed in (2.2). Much of the theory of theta functions on Riemann surfaces will be utilized in studying the theta function (1.1). In particular, we will establish connections between this theta function and the integrals that it inverts, addition formulas, and modular transformations. General references for this theory are [1], [19], and [13]. The curve (1.2) can be viewed as a three sheeted covering of the Riemann sphere with four branch points at x = 0, x = 1, x = 1/m, and x = ∞, which we also refer to as b1, b2, b3 and b4. We can choose a branch of the function y so that its only discontinuities as a function of x are on the intervals [0, 1] and [1/m, ∞]. For concreteness, this branch will be denoted y(0) and its value will be fixed to y(0) := (−1)2/3x1/3(x − 1)2/3(1 − mx)1/3, where the principle value of ab is taken. The y values on the other two sheets y(1) and y(2) differ from y(0) by a factor of a cube root of unity so that y(i) = ζiy(0).Gluing the three sheets y(0),y(1),y(2) together gives a genus 2 compact Riemann surface, and the standard homotopy basis of loops A1,A2,B1,B2 is shown in Figure 1. The loop A1 (resp. A2) lies entirely in the first sheet (resp. second sheet) and loops around the branch cut [0, 1]. The loop B1 (resp. B2) goes around the second and third branch cuts by starting in the zeroth sheet and passing to the first sheet (resp. second sheet) via the branch cuts. A point p on X will be denoted by giving its x and y coordinates as p = (x, y), and the y coordinate will be omitted for brevity when the context is clear. We will now fix local coordinates centered at each point on X, and this will give X the structure of a Riemann surface. At a non-branch point (x0, y0), y is locally a holomorphic function of x, so we have the coordinates y0 1 m 2 2 (x, y) = x0 + t, y0 + − − t + O(t ) . 3 x0 1 − mx0 1 − x0

6 Figure 1:

A2 B2

A1 B1

0 1 1 ¥ m

At the branch point (0, 0), we have coordinates (x, y) = t3, t + O(t2) , at (1, 0), (x, y) = 1 − t3, (1 − m)1/3t2 + O(t3) , at (1/m, 0), ! 1 − t3 1 1/3 1 1/3 (x, y) = , 1 − t + O(t2) , m m m and at the inﬁnite point on X, (x, y) = t−3, −m1/3t−4 + O(t−3) .

As X has genus 2 and every curve of genus 2 is hyperelliptic, there is a function of order 2 on X. Indeed, the function y x˙ = 1 − x can be veriﬁed by the charts constructed above to have zeros {0, 1/m} and poles {1, ∞}. This functionx ˙ plays an important role in the theory of X. The curve X also possesses a number of symmetries described in the automorphism group of X, Aut(X). For general m, Aut(X) has order 12, generated by the automorphisms σ(x, y) = (x, ζy), 1 − mx (m − 1)y γ(x, y) = , , m(1 − x) m(1 − x)2 (1.3) 1 1 − x 1 δ(x, y) = , 1 − , mx y mx where, as before, ζ = exp 2πi/3. The automorphism σ cycles the three sheets, while γ is the hyperelliptic involution. We need the following proposition describing the basic structure of meromorphic functions on X.

7 Proposition 1.1. The ﬁeld of meromorphic functions on X, M(X), is generated by x and y, and each meromorphic function with poles only at inﬁnity can be written uniquely as

y2 p (x) + p (x)y + p (x) , 0 1 2 1 − x where each pi(x) is a polynomial in x. Proof. The first statement, which is true of any algebraic curve, follows from Proposition 1.21 in Chapter 6 of [18], where it is shown that the index of a function in M(X) is equal to its order. Since the function x ∈ M(X) has order 3 and C[x] has index 3 in C[x, y] by the defining equation for X, we see that M(x) = C[x, y]. We now know that each meromorphic function can be written uniquely as y2 r (x) + r (x)y + r (x) 0 1 2 1 − x where each ri(x) is a rational function in x. Calculations using the local coordinates given above show that each ri(x) can have poles only at infinity, i.e. each function is a polynomial.

Proposition 1.2. Let f be a function of order 2 with a pole at p = (x0, y0) and a zero at q = (x1, y1). Then the other pole and the other zero of f are given by γp and γq, respectively, and f is constant multiple of x˙ − x˙ 1 . x˙ − x˙0 Proof. This proposition is a consequence of the fact that a function f of order two on a hyperelliptic surface must be ﬁxed by the hyperelliptic involution. Hence, γp must also be a pole of f if p is. This fact may be seen easily by bringing the equation deﬁning X into the form 2 5 4 3 2 w = z + a4z + a3z + a2z + a1z + a0 ax˙+b by a suitable bi-rational transformation in which z = cx˙+d for some constants a, b, c and d. The hyperellipitic involution γ now corresponds to the transformation

γ(z, w) = (z, −w).

At the inﬁnite point in these variables, the local coordinates are

(z, w) = t−2, t−5 + O t−4 .

Let (z1, w1), (z2, w2) be the poles of f. Then (z − z1)(z − z2)f(z, w) is a function of order at most four and has poles only at inﬁnity. Thus, there are polynomials p0(z) and p1(z) with

(z − z1)(z − z2)f(z, w) = p0(z) + p1(z)w.

If p1(z) is non-zero, then the right hand side has a pole at inﬁnity of order at least 5, which is a contradiction. Thus f(z, w) = f(z, −w), that is, f is ﬁxed by the hyperelliptic involution.

8 2 Deﬁnitions

2.1 Differentials of the First, Second and Third Kinds on X The differentials of the first kind are 1-forms on X that are everywhere holomorphic. It is well known that the space of such functions has the same dimension as the genus of the curve. For the genus 2 curve X, a basis is given by

dx ! du duu = 1 = 3y . du (1−x)dx 2 3y2

The generalized incomplete integrals of the ﬁrst kind are the integrals of these two diﬀerentials, which we put together into a vector-valued function

x1 R x1 Z du1 duu = x0 . R x1 du x0 x0 2 Implicit in this notation is the use of the same path of integration in both of the integrals on the right hand side. These have associated period matrices

2πi ζ 1 2πi −1 ζ ω = K (m) and ω = K (1 − m) ,. 1 3 3 −ζ2 −1 2 3 3 −1 ζ2

It follows from Euler’s integral representation

Γ(c) Z 1 tb−1(1 − t)c−b−1 2F1(a, b; c; z) = a dt (2.1) Γ(b)Γ(c − b) 0 (1 − tz) of the hypergeometric function that √ 3 Z 1 dx K (m) = 3 2/3 1/3 2π 0 x1/3(1 − x) (1 − mx) √ 3 Z 1 dx = , 1/3 2/3 2π 0 x2/3(1 − x) (1 − mx)

R 1 th and these two integrals are directly related to 0 duu. The (i, j) component of ω1 (or ω2) is the integral of dui around the loop Aj (or Bj). Since the loops Aj and Bj form a homotopy R x basis of X, 0 duu will change only by integral multiples of the four columns of ω1 and ω2 as the path of integration from 0 to x changes. The ratio of these two period matrices is −1 i K3(1 − m) 2 1 2τ τ ω1 ω2 = √ = , (2.2) 3 K3(m) 1 2 τ 2τ

K (1−m) which depends only on the single complex number τ = √i 3 . This matrix is the B-period 3 K3(m) matrix for the normalized basis of diﬀerentials of the ﬁrst kind given by

−1 duˆ = ω1 · duuu.

We now need a diﬀerential that is holomorphic everywhere except at an arbitrary point p1 = (x1, y1) where it has a residue of 1 and at ∞ where it has a residue of −1. Such a

9 differential is called a differential of the third kind for the points p1 and ∞. Following the general method in Section 45 of [1], we find that the differential

2 y1 (1 − x)y1 dx hx|x1idx := 1 + + 2 y (1 − x1)y 3 (x − x1) has the required properties. The differential hx|x1, x2idx = hx|x1idx − hx|x2idx is then a differential of the third kind for the points (x1, y1) and (x2, y2), for it has residues 1 and −1 as these two points respectively. Finally, we seek a vector of differentials dvv(x) such that

∂zhx|zidxdz + duu(x) · dvv(z) is a symmetric bi-meromorphic diﬀerential in x and z. With the choice

(1−mx)(1−x)dx ! dv1 − 3y2 dvv = = (1+m−2mx)dx , dv2 − 3y the above expression is indeed symmetric in x and z. These differentials are called differentials of the second kind associated with the basis of the differentials of the first kind given above. As with the integrals of the first kind, the integrals of these differentials are the generalized elliptic integrals of the second kind, and we use the notation

x1 R x1 Z dv1 dvv = x0 . R x1 dv x0 x0 2 These integrals have associated period matrices

2πi ζ2 1 2πi 1 −ζ2 η = E (m) and η = (K (1 − m) − E (1 − m)) , 1 3 3 −ζ −1 2 3 3 3 1 −ζ where 1 1 E (m) : = F , − ; 1; m 3 2 1 3 3 √ 3 Z 1 (1 − mx)1/3dx = 1/3 2π 0 x2/3(1 − x) is the generalized complete elliptic integral of the second kind. The integral representation follows from (2.1). Also, −1 E3(m) 0 1 η1ω1 = − . K3(m) 1 0

The relations between the period matrices ω1, ω2, η1, and η2 derived from the bilinear relations (see Section 140 of [1]) may be stated as

−1 1. ω1 ω2 is symmetric with positive deﬁnite imaginary part, −1 2. η1ω1 is symmetric, T T T T 3. ω1η2 − ω2η1 = 2πiI = η2ω1 − η1ω2 .

10 The first two are easily confirmed by the calculations given above while the last one is equivalent to the important identity √ E (m) E (1 − m) 3 3 3 + 3 = 1 + , (2.3) K3(m) K3(1 − m) 2πK3(m)K3(1 − m) which we refer to as the generalized Legendre identity. This is a special case of an identity due to Elliott [10]. We also list the system of differential equations satisfied by K3(m) and E3(m) for convenience: d 3m(1 − m) K3(m) = E3(m) − (1 − m)K3(m), dm (2.4) d 3m E (m) = E (m) − K (m). dm 3 3 3

2.2 Riemann Theta Function The Riemann theta function associated to X is

 T    T   a −1 a a u1 ∞ iπ ·(ω ω2)· 2πi · u X  b  1  b   b   u  Θ(u|τ) := Θ 1 τ = e e 2 u2 a,b=−∞ ∞ X a2+ab+b2 a b = q z1 z2 a,b=−∞

2πiu 2πiτ where zi = e i , q = e . It will be convenient to have notation for the shift of this function by certain vectors. Just as there are four elliptic theta functions corresponding to shifts of a certain fixed theta function by half-periods, there are nine cubic theta functions corresponding to shifts of Θ (u|τ) by certain third-periods. Without defining these vectors immediately, the n1 n2 2×2 Riemann theta function with an arbitrary characteristic ∈ R is defined as m1 m2

∞ 2 2 n1 n2 X (a+n1) +(a+n1)(b+n2)+(b+n2) 2πim1 a+n1 2πim2 b+n2 Θ (u|τ) = q e z1 e z2 m1 m2 a,b=−∞ −n2−n n −n2 −2πi(n (m +u )+n (m +u )) m1 + 2n1τ + n2τ = q 1 1 2 2 e 1 1 1 2 2 2 Θ u + τ . m2 + n1τ + 2n2τ (2.5) The nine characteristics are derived from the integrals of duu between any two of the four branch points 0, 1, 1/m and ∞. Using the deﬁnitions of the loops Aj and Bj, one can compute that

Z 1 1/3 0 duu ≡ ω1 · + ω2 · , 0 2/3 0 Z ∞ 0 1/3 duu ≡ ω1 · + ω2 · , 0 0 1/3 Z 1/m 1/3 1/3 duu ≡ ω1 · + ω2 · , 0 2/3 1/3

11 where ≡ is taken modulo the period lattice, and the characteristic symbols for these three 2 vectors in C are 0 0 := , 4 1/3 2/3 1/3 1/3 := , 2 0 0 1/3 1/3 := , 1 1/3 2/3

respectively. Since the four columns of the period matrices ω1 and ω2 are linearly independent 2 over R, every vector in C has a unique corresponding characteristic symbol with entries in R. Furthermore, two vectors are congruent modulo the period lattice if and only if the characteristics diﬀer by an integer characteristic. The group of nine characteristics is then deﬁned as the group of characteristics generated by the integrals of duu between the branch points modulo the period lattice. Since 1 = 2 + 4 and 34 ≡ 32 ≡ 00, we see that this is a group of order nine generated by 2 and 4. We use the following notation for the elements of this group:

3 = 02 + 04, 4 = 02 + 14, 44 = 02 + 24,

2 = 12 + 04, 1 = 12 + 14, 14 = 12 + 24,

22 = 22 + 04, 12= 22 + 14, 11 = 22 + 24.

2.3 Theta, Sigma, Zeta and Weierstrass Functions Deﬁne the ϑ and σ functions for an arbitrary characteristic by

−1 ϑ [](u|m) := Θ [](ω1 · u|τ(m)), − 1 uT ·η ω−1·u σ [](u|m) := e 2 1 1 ϑ [](u),

−1 Where Θ [](ω1 ·u|τ(m)) is deﬁned by (2.5). We now deﬁne the nine ϑ functions ϑa(u), nine σ functions σa(u), two ζ functions ζi(u) (1 ≤ i ≤ 2), and three ℘ functions ℘ij(u) (1 ≤ i ≤ j ≤ 2) by

ϑa(u|m) := ϑ [a](u),

σa(u|m) := caσ [a](u), ∂ ζi(u|m) := − log σ4(u), ∂ui ∂2 ℘ij(u|m) := − log σ4(u), ∂ui∂uj where ca is chosen so that the first non zero differential coefficient of σa(u) is 1. For brevity, ζ(u) will denote the gradient of − log σ4(u) (vector of ζi functions), and ℘(u) will denote the Hessian of − log σ4(u) (matrix of ℘ij functions). Whenever the arguments u, τ, or m are omitted from the functions, they assume the default values of 00, τ, and m, respectively. For

12 example,

Θa(u) = Θa(u|τ),

Θa(2u) = Θa(2u|τ),

Θa(2τ) = Θa(00|2τ),

Θa = Θa(00|τ), while the Borweins’ functions a(q), b(q), and c(q) have the more systematic notation

2πiτ a(e ) = Θ3(τ), 2πiτ c(e ) = Θ2(τ), 2πiτ b(e ) = Θ4(τ).

3 Basic Properties

In this section, the quasi-periodicity properties of the theta functions are stated as well as general theorems that hold on any compact Riemann surface.

3.1 Periodicity Properties of the Theta Functions 2 Proposition 3.1. Let a1 and a2 denote column vectors in R , and 1 and 2 denote row 2 2πi( ·a − ·a ) vectors in R . Then with C = e 2 1 1 2 , T T T −1 T + a 2 −1 −πi(2a2 ·u+a2 ·ω1 ω2·a2)−2πiaa1 ·a2 −2πi2·a1 2 2 Θ u + a1 + ω1 ω2 · a2 = Ce e Θ T (u) , 1 1 + a1 T T −1 T −1 T + a 2 −πi(2a2 ·ω1 ·u+a2 ·ω1 ω2·a2)−2πiaa1 ·a2 −2πi2·a1 2 2 ϑ (u + ω1 · a1 + ω2 · a2) = Ce e ϑ T (u) , 1 1 + a1 T 1 T T + a 2 − (η1·a1+η2·a2) ·(2u+ω1·a1+ω2·a2)−πiaa1 ·a2 −2πi2·a1 2 2 σ (u + ω1 · a1 + ω2 · a2) = Ce 2 e σ T (u) . 1 1 + a1 2 If a1 and a2 are integer vectors, then the expressions in brackets are equal to Θ (u), 1 ϑ 2 (u), and σ 2 (u), respectively. Furthermore, in this case, 1 1

ζ (u + ω1 · a1 + ω2 · a2) = ζ(u) + η1 · a1 + η2 · a2,

℘ (u + ω1 · a1 + ω2 · a2) = ℘(u). The ϑ functions increase by a non-vanishing exponential factor when the argument is increased by any of the four columns of the matrices ω1 and ω2, while the ζ function increases by the corresponding column of the matrices η1 and η2 and the ℘ function is genuinely periodic 2 2 with respect to the lattice ω1Z + ω2Z . If a holomorphic function f(u) satisﬁes

T T −1 T r −1 2πi(2·a1−1·a2) −πi(2a2 ·u+a2 ·ω1 ω2·a2)−2πiaa1 ·a2 f u + a1 + ω1 ω2 · a2 = e e f(u) for integer vectors a1 and a2, then it is called a theta function of order r with characteristic 2 . 1

13 3 For example, the functions ϑ1(u) and ϑ1(u) are theta functions of order 1 and 3, respectively, with characteristics 1 and 3, respectively. The space of all theta functions with order r and a given characteristic has dimension r2 (see pg. 124 of [19] for a proof of this basic but very useful fact). Proposition 3.2. If ϑ [0](u)ϑ [00](u) ··· ϑ (n) (u) f(u) = ϑ [δ0](u)ϑ [δ00](u) ··· ϑ δ(n) (u) and 0 + 00 + ··· + (n) ≡ δ0 + δ00 + ··· + δ(n), then

f (u + ω1 · a1 + ω2 · a2) = f(u) for integer vectors a1 and a2. These propositions enable us to calculate the eﬀect of shifting the characteristics of the theta functions and reﬂecting them, as well as to calculate periodicity factors for ratios of theta functions, as the next three examples illustrate.

Example 3.3. Shifting u by b in ϑa(u) gives ϑc(u), where c is the characteristic such that c ≡ a + b, but only up to some factor of exponentials, which we calculate here for a specific case. First by the definition of the characteristics, and then directly from Proposition 3.1, we find that

2/3 2/3 1/3 2τ τ 1/3 Θ (u + ) = Θ u + + · 11 1 2/3 4/3 2/3 τ 2τ 1/3 − πi − πi − 2πi (u +u +τ) − 4πi 1 1 = e 3 e 3 e 3 1 2 e 3 Θ (u) 1 2 − 2πi (u +u +τ) 0 0 = e 3 1 2 Θ (u) 0 0 2πi − (u1+u2+τ) = e 3 Θ3(u).

Similar calculations show that, regardless of the starting characteristic a, shifting by the characteristics 3, 4, or 44 gives no exponential factor, shifting by 2, 1, or 14 gives the factor 2πi 4iπ − (u1+u2+τ) − (u1+u2−τ) e 3 , and shifting by 22, 12, or 11 gives the factor e 3 .

Example 3.4. Reﬂecting u in ϑa(u) gives ϑb(u), where b is the characteristic such that b ≡ −a. By the deﬁnition of the Riemann theta function with characteristics, the fact that Θ(u) is an even function, and Proposition 3.1,

1/3 Θ (−u) = Θ −u + 4 2/3 −1/3 = Θ u + −2/3 2/3 = Θ u + 4/3 0 0 = Θ (u) 2/3 4/3

= Θ44(u).

14 Similar calculations show that

Θ3(−u) = Θ3(u),

Θ2(−u) = Θ22(u),

Θ4(−u) = Θ44(u),

Θ1(−u) = Θ11(u),

Θ12(−u) = Θ14(u).

Example 3.5. The theta function ratio Θ1(u) has the factors ζ, ζ, ζ2, ζ associated with the four Θ4(u) periods. That is, with Proposition 3.1 we ﬁnd that

Θ 1 Θ (u) 1 u + = ζ 1 , Θ4 0 Θ4(u) Θ 0 Θ (u) 1 u + = ζ 1 , Θ4 1 Θ4(u) Θ 2τ Θ (u) 1 u + = ζ2 1 , Θ4 τ Θ4(u) Θ τ Θ (u) 1 u + = ζ 1 . Θ4 2τ Θ4(u)

3.2 Abel’s Theorem and Properties of the Integrals of the First Kind Abel’s theorem gives an exact criterion for the pole and zero set of a meromorphic function on X in terms of the integrals of the ﬁrst kind. A proof of this important theorem can be found in Chapter 8 of [18] or Chapter 2 of [19].

Proposition 3.6. The points {p1, ..., pn} and {q1, ..., qn} are the poles and zeros of a meromorphic function on X if any only if

Z q1 Z qn duu + ··· + duu ≡ 0. p1 pn R γx When the automorphisms σ, γ and δ act on the integrals of the first kind (to give 0 duu, etc.), the result is, of course, still an integral of the first kind that can be written in terms R x of the original integrals of the first kind ( 0 duu). The precise effect they have is given in the following proposition whose proof is by direct calculation.

Proposition 3.7.

Z σx ζ2 0 Z x duu = · duu, 0 0 ζ 0 Z γx −1 0 Z x duu = · duu, 0 0 −1 1/m Z δx 0 1 Z x duu = · duu. 0 1 0 ∞

15 3.3 Zeros of the Theta Functions We will now explicitly describe the zeros of the theta functions when they are composed with the integrals of the ﬁrst kind.

2 R x Proposition 3.8. For any vector e ∈ C , the function Θ 0 duˆ + e has two zeros z1 and z2 as a function of x that satisfy

Z z1 Z z2 duˆ + duˆ ≡ ∆ − e 0 0 2 for some fixed vector ∆ ∈ C provided this function does not vanish identically. Also, here ∆ satisfies ∆ ≡ −1. Proof. The first assertion is identical to Theorem 3.1 in [19]. The only thing left to do is to compute the vector of Riemann constants by − 1 ω−1ω Z Z x Z Z x ∆ = 2 1 2 11 + duˆ(x) duˆ (z) + duˆ(x) duˆ (z) − 1 ω−1ω 1 2 2 1 2 22 x∈A1 z=0 x∈A2 z=0 2/3 − τ ≡ 1/3 − τ 1/3 2τ τ 1/3 ≡ − − · 2/3 τ 2τ 1/3

≡ −1.

R x1 R x2 Proposition 3.9. For i = 1, 2, 3, 4, the zeros of ϑi 0 duu + 0 duu with respect to x1 are th the i branch point bi and γx2.

Proof. For general x2, the function does not vanish identically, so its two zeros z1 and z2 are determined from Z x2 Z z1 Z z2 i + duu + duu + duu ≡ −1. 0 0 0 Since Z 1/m i = duu, bi and by Proposition 3.7, Z γx Z x Z 1/m duu = − duu + duu, 0 0 0 it easily seen that this equation is satisﬁed with the choices z1 = bi and z2 = γx2.

4 Inversion Formulas

In this section we will obtain algebraic expressions for ratios of the theta functions ϑa(u), the zeta functions ζ(u), and the Weierstrass functions ℘(u) when their argument is set to R (x1,y1) R (x2,y2) u = 0 duu + 0 duu. These algebraic expressions will be rational functions of x1 and x2 and the corresponding y coordinates on the curve y1 and y2. Also, for brevity, we have

yi x˙ i = . 1 − xi

16 As a consequence of these inversion formulas we will be able to evaluate all differential coefficients up to the second order of each of the nine theta functions ϑa(u), thus completing the definitions of the sigma functions σa(u). The formulas are also useful in that they give parametric representations of the functions, from which many identities may be found.

R x1 R x2 Theorem 4.1. With u = 0 duu + 0 duu, the ϑ functions with simple characteristics satisfy

ϑ3 ϑ1(u) 1/3 1/3 σ1(u) ζ = x1 x2 = − , ϑ2 ϑ4(u) σ4(u) ϑ4 ϑ2(u) 1/3 1/3 σ2(u) = (1 − x1) (1 − x2) = , ϑ2 ϑ4(u) σ4(u) ϑ4 ϑ3(u) 1/3 1/3 σ3(u) = (1 − mx1) (1 − mx2) = , ϑ3 ϑ4(u) σ4(u) ϑ1(u)ϑ3(u) σ1(u)σ3(u) ζ =x ˙ 1x˙ 2 = − ; ϑ2(u)ϑ4(u) σ2(u)σ4(u) the ϑ functions with compound characteristics satisfy

1 x1

ϑ3ϑ4 ϑ2(u)ϑ11(u) 1 x2 σ2(u)σ11(u) ζ 2 2 = = 2 , ϑ ϑ4(u) 1x ˙ 1 σ4(u) 2 1x ˙ 2

x1 x˙ 1 2 ϑ3 ϑ1(u)ϑ12(u) x2 x˙ 2 σ1(u)σ12(u) 2 2 = = 2 , ϑ ϑ4(u) 1x ˙ 1 σ4(u) 2 1x ˙ 2

x1 y1 2 ϑ3ϑ4 ϑ1(u)ϑ2(u)ϑ14(u) x2 y2 σ1(u)σ2(u)σ14(u) − 3 3 = = 3 , ϑ ϑ4(u) 1x ˙ 1 σ4(u) 2 1x ˙ 2

1 − mx1 y1 3 ϑ4 ϑ2(u)ϑ3(u)ϑ22(u) 1 − mx2 y2 σ2(u)σ3(u)σ22(u) 2 3 = = 3 , ϑ ϑ3 ϑ4(u) 1x ˙ 1 σ4(u) 2 1x ˙ 2

1 − mx1 x˙ 1

ϑ4 ϑ3(u)ϑ44(u) 1 − mx2 x˙ 2 σ3(u)σ44(u) 2 = = 2 ; ϑ3 ϑ4(u) 1x ˙ 1 σ4(u)

1x ˙ 2 the ℘ functions satisfy 1 mσ2(u)σ11(u) −σ3(u)σ44(u) ℘(u) = 2 ; σ4(u) −σ3(u)σ44(u) mσ22(u)σ14(u)

R xi and if the same path of integration is used in the 0 dvv, the ζ functions satisfy Z x1 Z x2 m 0 ζ(u) = dvv + dvv + 2 . 0 0 σ4(u) σ2(u)σ11(u)

17 The integral of the third kind has the evaluation

R x1 R x2 Z z2 σ duu σ duu Z z2 Z x2 1 z2 1 z1 hx|x1, x2idx = log + duu · dvv. R x1 R x2 z1 σ duu σ duu z1 x1 1 z1 1 z2

Corollary 4.2. We also have the following table of diﬀerential coeﬃcients of the nine ϑ functions.

2 2 2 a ϑ (000) ∂ϑa (000) ∂ϑa (000) ∂ ϑa (000) ∂ ϑa (000) ∂ ϑa (000) a ∂u1 ∂u2 ∂u1∂u1 ∂u1∂u2 ∂u2∂u2

3 K3(m) 0 0 0 (1 − m)K3 − E3 0 1/3 1/3 2 m K3(m) 0 0 0 −m E3 0 1/3 1/3 22 m K3(m) 0 0 0 −m E3 0 1/3 1/3 4 (1 − m) K3(m) 0 0 0 (1 − m) (K3 − E3) 0 1/3 1/3 44 (1 − m) K3(m) 0 0 0 (1 − m) (K3 − E3) 0 2 2 1 0 −ζ ϑ2ϑ4/ϑ3 0 0 0 ζ ϑ2ϑ4/ϑ3 2 2 11 0 ζ ϑ2ϑ4/ϑ3 0 0 0 ζ ϑ2ϑ4/ϑ3 12 0 0 −ζϑ2ϑ4/ϑ3 ζϑ2ϑ4/ϑ3 0 0 14 0 0 ζϑ2ϑ4/ϑ3 ζϑ2ϑ4/ϑ3 0 0

Proof of Corollary 4.2. The evaluations of the non-vanishing theta constants are a special case of the formulas derived in [11] (see Example 6.3 there). They are also equivalent to Theorem 2.10 on page 99 of [3]. The fact that the remaining four theta constants vanish follows from R x Proposition 3.9; the function ϑ1 0 duu vanishes identically, and setting x to any of the four branch points gives the results. The first order differential coefficients of ϑ3 are zero since it is an even function. The rest of the entries will be established after the inversion formulas have been established.

3 ϑ1(u) 2 2 Proof of Theorem 4.1. The function f(u) = 3 is periodic on ω1 + ω2 by Proposi- ϑ4(u) Z Z R x1 R x2 tion 3.2, hence the function F (x1, x2) = f( 0 duu + 0 duu) is single valued and hence a meromorphic function of x1 and x2. By Proposition 3.9, it has a triple zero at 0 and triple pole at ∞ with respect to either variable. Therefore, F (x1, x2) = cx1x2, where c is independent R 1 of x1 and x2. By setting x1 = x2 = 1 and using the fact that 0 duu = 4, we deduce that

3 3 ϑ1(24) ϑ2 c = f(24) = 3 = 3 . ϑ4(24) ϑ3

Substituting this value for c, taking the cube root, and rearranging gives the formula for ϑ1(u) . ϑ4(u) The cube roots are indeed arbitrary as the calculations in Example 3.5 show that as x1 goes around any of the period loops, this ratio is multiplied by cube roots of unity. The formulas for the remaining simple characteristics follow from the same type of argument.

The evaluations of the theta functions with compound characteristics are more involved as we do not have direct information on the zeros of these functions. Despite this, all of the formulas may be deduced from the same type of argument, which we illustrate for the ﬁrst ϑ2(u)ϑ11(u) 2 2 function ϑ11(u). The function f(u) = 2 is periodic on ω1 +ω2 by Proposition 3.2, ϑ4(u) Z Z R x1 R x2 and hence the function F (x1, x2) = f( 0 duu + 0 duu) is a meromorphic function of x1 and

18 x2. To calculate this function, we need to know the zeros of ϑ11 with respect to x1, say. By Proposition 3.8, the zeros z1 and z2 of ϑ11(u) with respect to x1 satisfy

Z x2 Z z1 Z z2 21 + duu + duu + duu ≡ −1. 0 0 0 R ∞ Since 31 ≡ 0 and 3 0 duu ≡ 00, this is equivalent to

Z x2 Z z1 Z z2 duu + duu + duu ≡ 00. ∞ ∞ ∞

That is, there is a meromorphic function on X with zeros {x1, z1, z2} and a triple pole at ∞. By Proposition 1.1, such a function must be of the form A + Bx1. Since x2 is a zero of this function, it must be a constant multiple of x1 − x2. Now, with respect to x1 the zeros and poles of F (x , x ) are {1, z , z } and {∞, ∞, γx }. However, the function x1−x2 has exactly 1 2 1 2 2 x˙ 1−x˙ 2 the same set of zeros and poles since the zeros and poles ofx ˙ 1 − x˙ 2 are {x2, γx2} and {1, ∞} respectively, and the zeros and poles of x1 − x2 are {x2, z1, z2} and {∞, ∞, ∞} respectively. Therefore, x1 − x2 F (x1, x2) = c , x˙ 1 − x˙ 2 where c is independent of x1. Since both sides are symmetric in x1 and x2, c is independent of both x1 and x2. Unfortunately F (x1, x2) is zero or inﬁnite when x1 and x2 are at any of the branch points 0, 1, 1/m and ∞, so a simple substitution will not determine c. By combining ϑ4 ϑ2(u) 1/3 1/3 ϑ2(u)ϑ11(u) x1−x2 the formulas = (1 − x1) (1 − x2) and 2 = c we deduce that ϑ2 ϑ4(u) ϑ4(u) x˙ 1−x˙ 2

ϑ ϑ (u) 1 x − x 2 11 = c 1 2 . 1/3 1/3 ϑ4 ϑ4(u) (1 − x1) (1 − x2) x˙ 1 − x˙ 2

2/3 (1−1/m) ϑ3ϑ4 As x2 tends to 1 and x1 tends to 1/m, the right hand side tends to c 1/3 = cζ 2 and (1−m) ϑ2 ϑ2 ϑ11(1+4) 1 ϑ3ϑ4 the left hand side tends to = 1 . Thus = ζ 2 and the result is the stated ϑ4 ϑ4(1+4) c ϑ2 formula

1 x1

ϑ3ϑ4 ϑ2(u)ϑ11(u) 1 x2 ζ 2 2 = . ϑ ϑ4(u) 1x ˙ 1 2 1x ˙ 2 The formulas for the remaining compound characteristics follow from the same type of argument. The formulas for ℘ will be calculated using a general formula of Fay (see Corollary 2.12 of [13]). This formula in the notation used here is

T duu(x) · ℘(u) · duu(z) = ∂zhx|zidxdz + duu(x) · dvv(z)

Since tx + ··· duu(x) = 1+2m 3 dtx, 1 + 3 tx + ···

19 where tx is the local coordinate at x = 0, and 1 + 1+2m t3 + ··· duu(x) = m1/3 3m4/3 x dt , 1 t + ··· x m2/3 x where tx is the local coordinate at x = ∞, ℘11 will be isolated by expanding both sides at 0 0 x = ∞ and z = ∞ and extracting the coeﬃcients of txtzdtxdtz. Similarly, ℘12 and ℘22 will be extracted with the choices x = ∞, z = 0 and x = 0, z = 0, respectively. Carrying out these plans and expanding the right hand side gives

1/3 −2/3 0 0 m (x1 − 1) (x1 − x2)(x2 − 1) 0 0 m ℘11(u)txtzdtxdtz + ··· = − txtzdtxdtz + ··· , x2y1 − x1y2 − y1 + y2 (x − 1) y (mx − 1) − (x − 1) y (mx − 1) m−1/3℘ (u)t0t0dt dt + ··· = 2 1 2 1 2 1 t0t0dt dt + ··· , 12 x z x z 1/3 x z x z m (x2y1 − x1y2 − y1 + y2) 2 2 2 2 m (x1y2 − x2y1) (x1 − 1) x1y − (x2 − 1) x2y 0 0 2 1 0 0 ℘22(u)txtzdtxdtz + ··· = 2 txtzdtxdtz + ··· . x1x2(−x2y1 + x1y2 + y1 − y2)

With some easy manipulations, these expressions can be brought into the forms given in the σ2(u)σ11(u) σ3(u)σ44(u) σ22(u)σ14(u) stated formula for ℘(u) using formulas for the functions 2 , 2 , and 2 σ4(u) σ4(u) σ4(u) obtained form the formulas stated in the first and second parts of the theorem. At this point it is possible to finish deriving all the entries of the table. At x = 0 the integrals of the first kind have the expansion Z x 1 2 m+2 5 2 tx + 15 tx + ··· duu = 1+2m 4 . 0 tx + 12 tx + ··· Expanding, for example, the identity

ϑ3(000) ϑ3(u) 1/3 1/3 = (1 − mx1) (1 − mx2) ϑ4(000) ϑ4(u) at x = x = 0, substituting the known values ∂ϑ3 (000) = ∂ϑ3 (000) = 0, and equating coeﬃcients 1 2 ∂u1 ∂u2 1 0 1 1 2 0 of tx1 tx2 , tx1 tx2 , and tx1 tx2 gives, respectively, ∂ϑ4 (000) 0 = − ∂u2 , ϑ4(000) 2 2 2 ∂ϑ4 (000) ∂ ϑ3 (000) ∂ ϑ4 (000) ∂u2 ∂u2∂u2 ∂u2∂u2 0 = 2 2 + − , ϑ4(000) ϑ3(000) ϑ4(000) 2 2 2 ∂ϑ4 (000) ∂ ϑ3 (000) ∂ ϑ4 (000) ∂ϑ4 (000) ∂u2 1 ∂u2∂u2 1 ∂u2∂u2 1 ∂u1 0 = 2 + − − . ϑ4(000) 2 ϑ3(000) 2 ϑ4(000) 2 ϑ4(000)

From these equations the values ∂ϑ4 (000) = ∂ϑ4 (000) = 0 easily follow. In this way, all of the ∂u1 ∂u2 first order differential coefficients can be computed, resulting in the second and third columns of the table. To compute the second order differential coefficients, it is simplest to use the equations for ℘(u). Recall,

∂ϑ ∂ϑ ∂2ϑ ∂2 4 (u) 4 (u) 4 (u) u u −1 ∂ui ∂uj ∂ui∂uj ℘ij(u) = − log σ4(u) = η1ω1 ij + 2 − . ∂ui∂uj ϑ4(u) ϑ4(u)

20 Combining these with the algebraic evaluations of the ℘ functions gives

2 ∂ϑ4 (u) ∂ϑ4 (u) ∂ ϑ4 (u) ∂u1 ∂u1 ∂u1∂u1 m (x1 − 1) (x1 − x2)(x2 − 1) 2 − = − , ϑ4(u) ϑ4(u) x2y1 − x1y2 − y1 + y2 2 ∂ϑ4 (u) ∂ϑ4 (u) ∂ ϑ4 (u) E3(m) ∂u1 ∂u2 ∂u1∂u2 (x2 − 1) y1 (mx2 − 1) − (x1 − 1) y2 (mx1 − 1) − + 2 − = , K3(m) ϑ4(u) ϑ4(u) x2y1 − x1y2 − y1 + y2) ∂ϑ ∂ϑ ∂2ϑ 2 2 2 2 4 (u) 4 (u) 4 (u) m (x1y2 − x2y1) (x1 − 1) x1y2 − (x2 − 1) x2y1 ∂u2 ∂u2 ∂u2∂u2 2 − = 2 , ϑ4(u) ϑ4(u) x1x2(−x2y1 + x1y2 + y1 − y2) and expanding these identities at x = x = 0, using the known values ∂ϑ4 (000) = ∂ϑ4 (000) = 0, 1 2 ∂u1 ∂u2 0 0 and equating coefficients of tx1 tx2 gives, respectively, 2 ∂ ϑ4 (000) 0 = − ∂u1∂u1 , ϑ4(000) 2 E (m) ∂ ϑ4 (000) −1 = − 3 − ∂u1∂u2 , K3(m) ϑ4(000) 2 ∂ ϑ4 (000) 0 = − ∂u2∂u2 , ϑ4(000) from which the remainder of the differential coefficients of ϑ4 follow. Once all of the zeroth and first order differential coefficients are known and the second order ones are known for ϑ4, the second order coefficients for the rest of the theta functions may be easily deduced by continuing with the series expansion approach. Now there remains only the formula for ζ(u). The function

Z x1 Z x2 F (x1, x2) = ζ(u) − dvv + dvv 0 0 is actually a single valued function of x1 and x2, hence a vector of meromorphic functions of x1 and x2. The reason for this is that the quasi-periodicity of the ζ function exactly matches up with the periods of the integrals of the second kind. More precisely, as x1 goes around the period loop A1, u increases by the first column of ω1 and thus ζ(u) increases by the first R x1 column of η1 by Proposition 3.1 while 0 dvv also increases by the first column of η1 by the definition of the period matrices. Thus the difference is unaltered when x1 goes around A1 or any other period loop, hence the function is single-valued when the same path of integration is used in corresponding integrals. Whereas the method used to evaluate quotients of theta functions was to match up zeros and poles, the functions

Z x1 Z x2 F1(x1, x2) = ζ1(u) − dv1 + dv1 , 0 0 Z x1 Z x2 F2(x1, x2) = ζ2(u) − dv2 + dv2 0 0 will be evaluated by matching up poles and residues at these poles, leaving an additive constant that needs to be evaluated. First, the only poles of the diﬀerentials of the second kind are at ∞, and expanding the deﬁnition of dvv at x = ∞ gives ! m1/3t−2 + m1− t + ··· dvv = x 3m2/3 x dt , 2m2/3t−3 + m−1 + ··· x x 3m1/3

21 so that Z x 1/3 −1 −m tx 0 dvv = 2/3 −2 + O(tx). 0 −m tx Now,

∂ du1 du2 ζ1(u) = ℘11(u) + ℘12(u) ∂x1 dx1 dx1 1 m (x − 1) (x − x )(x − 1) = − 1 1 2 2 3y1 x2y1 − x1y2 − y1 + y2 1 − x1 (x2 − 1) y1 (mx2 − 1) − (x1 − 1) y2 (mx1 − 1) + 2 3y1 x2y1 − x1y2 − y1 + y2) m1/3 1 − m = − t2 + t5 + ··· , 3 x1 9m2/3 x1 where tx1 is the local coordinate at x1 = ∞, and thus integrating in x1 gives

u 1/3 −1 ζ1(u) = −m tx1 + ··· .

R x1 Upon comparing the principal parts of ζ1(u) and 0 v1 at x1 = ∞, we see that F1(x1, x2) does not have a pole at x1 = ∞. Since the only other possibility of a pole is at x1 = γx2, and this would give at most a simple pole, F1(x1, x2) is a function of order at most one in x1. Since there are no functions of order one on X, F1(x1, x2) must be constant in x1. Since it is symmetric in x1 and x2, it must be an absolute constant. Therefore, F1(x1, x2) = F1(0, 0) = 0. R x1 Since ζ2(u) has poles at most {∞, γx2} and 0 dvv has a double pole at ∞, we see that F2(x1, x2) has poles at most {∞, ∞, γx2} as a function of x1. The expansion of ζ2(u) at x1 = ∞ may be derived exactly as the expansion for ζ1(u) was, and is

u 1/3 −1 ζ2(u) = m x˙ 2tx1 + ··· .

R x1 By combining this expansion and the expansion of 0 v2 at x1 = ∞, we deduce that

2/3 −2 1/3 −1 F2(x1, x2) = m tx1 + m x˙ 2tx1 + ··· , where tx1 is the local variable at x1 = ∞. Coincidentally,

x1 − x2 2/3 −2 1/3 −1 m = m tx1 + m x˙ 2tx1 + ··· x˙ 1 − x˙ 2 has exactly the same principal part. As noted earlier, x1−x2 has poles {∞, ∞, γx } as a x˙1−x˙2 2 function of x1. Therefore, x1 − x2 F2(x1, x2) = m + c, x˙1 − x˙2 where c is an absolute constant. Upon letting x1 and x2 tend to 0, we deduce that c = 0. The formula for the integral of the third kind is standard (see Section 187 of [1]). It is proved by noting that the logarithm of the quotient of theta functions is also an integral of the third kind, and thus the diﬀerence between the two is an integral of the ﬁrst kind.

The inversion formulas stated above imply a homogeneous polynomial identity between any four of the theta functions. One of these can be stated as a sum of cubes identity.

22 Corollary 4.3. We have

3 3 3 3 ϑ1(u) + ϑ3(u) = ϑ2(u) + ϑ4(u) .

Proof. Elimination of x1 and x2 from the ﬁrst three identities in Theorem 4.1 along with the values of the theta constants given in Corollary 4.2 gives this identity.

R x1 R x2 Corollary 4.4. When u = 0 duu + 0 duu, the points (x1, y1) and (x2, y2) may be determined uniquely up to a permutation of the pair from the following formulas, except when u ≡ 1 in which case the inﬁnite solution set is parametrized by x1 = x, x2 = γx for any point x:

3 σ1(u) − 3 = x1x2, σ4(u) 3 σ2(u) 3 = (1 − x1) (1 − x2) , σ4(u) 3 σ3(u) 3 = (1 − mx1) (1 − mx2) , σ4(u)

σ (u)σ (u) y y − 1 3 = 1 2 , σ2(u)σ4(u) 1 − x1 1 − x2 σ (u)σ (u) σ (u)σ (u) y y m 12 22 + (1 − m) 14 44 = 1 + 2 . σ2(u)σ4(u) σ2(u)σ4(u) 1 − x1 1 − x2 Proof. The stated formulas follow by combining the various inversion formulas in Theorem 4.1. Any two of the ﬁrst three give x1 and x2 up to a permutation. The corresponding y coordinates y1 and y2 are then determined up to a cube root of unity which can be determined uniquely so that the last two equations hold. Clearly, from the formulas for x1x2,(1 − x1) (1 − x2), and (1 − mx1) (1 − mx2), a u such that x1 and x2 are not uniquely determined must be a common zero of σ4(u) and at least two of σ1(u), σ2(u), and σ3(u). The only characteristic with this R x1 R x2 property is 1. To show that this is indeed the only such u, suppose that u = 0 duu+ 0 duu ≡ R z1 duu + R z2 duu. Then R z1 duu + R z2 duu ≡ 00, and thus there is a meromorphic function with 0 0 x1 x2 poles {x1, x2} and zeros {z1, z2}. By Proposition 1.2, we must have x2 = γx1 and z2 = γz1, R x1 R γx1 2 2 2 n and so u = 0 duu+ 0 duu = 1 is the only such point in C /(ω1Z +ω2Z ). If O (u ) denotes a remainder in which all terms have total degree at least n, it is also clear from the expansions

σ −1 u + u2 + O(u)3 1 ( + u) = 1 2 , 1 1/3 2 3 σ4 m u2 + u1 + O(u) σ (1 − m)1/3 −u + u2 + O(u)3 2 ( + u) = 2 1 , 1 1/3 2 3 σ4 m u2 + u1 + O(u) σ −u + u2 + O(u)3 3 u 1/3 1 2 (1 + u) = (1 − m) 2 3 , σ4 u2 + u1 + O(u) that the expressions for x1x2 and (1 − mx1) (1 − mx2) do not have unique limits as u tends m−1 to 1 but that (1 − x1) (1 − x2) tends to m . This is in agreement with what was obtained 1−mx m−1 above since (1 − x)(1 − γx) = (1 − x) 1 − m(1−x) = m .

23 In Weierstrass’s approach to elliptic functions, the x and y coordinates on an elliptic curve are given by ℘(u) and ℘0(u). Since the field of meromorphic functions on the elliptic curve is generated by the coordinates x and y, every elliptic function on the associated period lattice is a rational function of ℘(u) and ℘0(u). A similar list of generators can be given for the field 2 2 of hyperelliptic functions on ω1Z + ω2Z , but the situation is not as simple, as each point 2 2 2 2 2 u ∈ C /(ω1Z + ω2Z ) is associated with two points on X. If f(u) is periodic on ω1Z + ω2Z , R x1 R x2 then f( 0 duu + 0 duu) is a symmetric bi-meromorphic function of x1 and x2 on X. While x1 and x2 cannot individually be solved uniquely as meromorphic functions of u, any symmetric function of x1 and x2 is single valued, and hence a meromorphic function of u. In this way, 2 2 the field of hyperelliptic functions on Z ω1 + Z ω2 corresponds to the field of symmetric bi- meromorphic functions on X. Combining this with Proposition 1.1, we then have the following theorem.

Theorem 4.5. The ﬁeld of invariants of C [x1, y1, x2, y2] under the action of g :(x1, y1, x2, y2) → (x2, y2, x1, y1) is generated by the ﬁve elements

x1 + x2, x1x2, y1 + y2, y1y2, y1/x1 + y2/x2,

2 2 and thus the ﬁeld of hyperelliptic functions on ω1Z + ω2Z is generated by

3 3 2 σ2(u)σ11(u) σ1(u)σ12(u) σ1(u) σ2(u) σ1(u)σ2(u) σ3(u) 2 , 2 , 3 , 3 , 4 . σ4(u) σ4(u) σ4(u) σ4(u) σ4(u) Proof. A theorem due to Noether (see Theorem 5 of Chapter 7 in [9]) guarantees that the ring of invariants is generated by homogeneous polynomials derived from the monomials of degree not more than the order of the group. That is, since g generates a group of order two here, the ring of invariants is generated by

x1 + x2, y1 + y2, 2 2 2 2 x1 + x2, x1x2, y1 + y2, y1y2, x1y1 + x2y2, x2y1 + x1y2, which can be reduced to

x1 + x2, y1 + y2,

x1x2, y1y2, x2y1 + x1y2.

N Every element in the field of invariants can be written as D where N and D are ring elements N D with no common factors. If this ratio is fixed by g, we should have gN = gD , and since N and N D D have no common factors, gN = gD = c where c is a constant. Since g has order 2, c must be 1 or −1. If c = 1, both N and D are in the ring of invariants, and we are done. If c = −1, write N = (x1−x2)N , and the numerator and denominator are in the ring of invariants. So we D (x1−x2)D see that the same five elements generate the field of invariants as well. It now suffices to establish that each of the five elements x1 + x2, x1x2, y1y2, y1 + y2, and y1/x1 + y2/x2 can be written rationally in the five displayed sigma function quotients R1, ..., R5. From the first four equations stated in Theorem 4.4, x1 + x2, x1x2, and y1y2 are

24 rational in R3, R4, and R5. For y1/x1 + y2/x2 we have (1 − mx ) y1 − (1 − mx ) y2 y1 + y2 y y 2 1−x1 1 1−x2 x1 x2 1 + 2 = x x (1 − mx ) y1 − (1 − mx ) y2 1 2 2 1−x1 1 1−x2 2 2 (1−mx1x2)y1y2 (1−mx2)y1 (1−mx1)y2 (x1 − x2) + − = x1x2(1−x1)(1−x2) (1−x1)x1 (1−x2)x2 (1 − mx ) y1 − (1 − mx ) y2 2 1−x1 1 1−x2 ( ) (1 − mx x ) y y x − x = 1 2 1 2 1 2 x x (1 − x ) (1 − x ) (1 − mx ) y1 − (1 − mx ) y2 1 2 1 2 2 1−x1 1 1−x2 ( y y ) (1 − x ) (1 − x ) (1 − mx ) (1 − mx ) 1 − 2 − 1 2 1 2 1−x1 1−x2 , y y (1 − mx ) y1 − (1 − mx ) y2 1 2 2 1−x1 1 1−x2 and the ﬁrst and third terms in brackets are rational in R3, R4, and R5, while the second and fourth are R1 and 1 respectively (from the inversion formulas). Similarly, 1−mR2 1−mR2

y1 y2 (1 − mx2) 1−x − (1 − mx1) 1−x (y1 + y2) y + y = 1 2 1 2 (1 − mx ) y1 − (1 − mx ) y2 2 1−x1 1 1−x2 ( ) (1 + m − m (x + x )) y y x − x = 1 2 1 2 1 2 (1 − x ) (1 − x ) (1 − mx ) y1 − (1 − mx ) y2 1 2 2 1−x1 1 1−x2 ( y1 y2 ) (1 − mx ) (1 − mx ) (1 − x ) (1 − x ) x2 − x1 − 1 2 1 2 1−x1 1−x2 , y y (1 − mx ) y1 − (1 − mx ) y2 1 2 2 1−x1 1 1−x2 from which y1 + y2 is rational in the Ri as before.

5 Addition Formulas

The classical addition formula for the elliptic integral, Z x1 dt Z x1 dt Z x3 dt p + p = p (5.1) 0 2 t(1 − t)(1 − mt) 0 2 t(1 − t)(1 − mt) 0 2 t(1 − t)(1 − mt) where √ √ √ √ √ √ √ x1 1 − x2 1 − mx2 + x2 1 − x1 1 − mx1 x3 = , 1 − mx1x2 is most easily derived via the inversion formulas (0.16) by deriving an addition formula for the various products Θa(u + v)Θb(u − v) (as in Section 1.4 of [16]) and then taking quotients. The cubic analogue we are interested in is

Z x1 Z x2 Z x3 Z x4 Z x5 duu + duu + duu = duu + duu, (5.2) 0 0 0 0 0 for if the sum of three integrals can be reduced to a sum of two, the sum of any number of integrals can be reduced to a sum of two. With the the inversion formulas in Theorem 4.1, we σa(u+v) will have the ability to solve for x4 and x5 if we have addition formulas for the ratios . σ4(u+v) Although there are 81 expressions of the form

ϑa(u + v)ϑb(u − v),

25 it suﬃces to consider only the expression

ϑ3(u + v)ϑ3(u − v) since the shifts u → u − a − b, and v → v − a + b convert this expression to the general form.

2 Theorem 5.1. For uu,vv ∈ C , the ϑ functions satisfy the addition formula

ϑ2(u)ϑ22(u)ϑ2(v)ϑ22(v) ϑ4(u)ϑ44(u)ϑ4(v)ϑ44(v) ϑ3(u + v)ϑ3(u − v) = + ϑ2ϑ3 ϑ3ϑ4 ϑ (u)ϑ (u)ϑ (v)ϑ (v) ϑ (u)ϑ (u)ϑ (v)ϑ (v) + 1 11 12 14 + 12 14 1 11 ϑ2ϑ4 ϑ2ϑ4 2 2 2 2 ϑ3(u) ϑ3(v) ϑ3(u) ϑ4(v)ϑ44(v) ϑ4(u)ϑ44(u)ϑ3(v) = 3 − 3 − 3 ϑ2/ϑ3 ϑ2/ϑ4 ϑ2/ϑ4 ϑ4(v)ϑ44(v)ϑ4(u)ϑ44(u) ϑ1(u)ϑ11(u)ϑ12(v)ϑ14(v) ϑ12(u)ϑ14(u)ϑ1(v)ϑ11(v) + 3 2 + + , ϑ2ϑ4/ϑ3 ϑ2ϑ4 ϑ2ϑ4 Proof. Fix v, and consider the ﬁrst identity as a function of u. Every term is a second order theta function with zero characteristic (3). But the space of all such functions is a vector space of dimension 22. In particular, if the four functions

ϑ2(u)ϑ22(u), ϑ4(u)ϑ44(u), ϑ1(u)ϑ11(u), ϑ12(u)ϑ14(u) are linearly independent, the left hand side is such a linear combination. Now at each of the four points 2, 4, 1, and 12 exactly three of the functions vanish and the other becomes a product of non-vanishing theta constants. Furthermore, at each of these four points it is a diﬀerent product that does not vanish, so indeed the four functions are linearly independent, and we have A, B, C, and D, independent of u, such that

ϑ3(u + v)ϑ3(u − v) = Aϑ2(u)ϑ22(u) + Bϑ4(u)ϑ44(u) + Cϑ1(u)ϑ11(u) + Dϑ12(u)ϑ14(u).

By setting u in turn to 2, 4, 1, and 12, we immediately obtain A, B, C, and D and the result is the stated formula. The remaining formula is derived using another linearly independent set 2 ϑ3(u) , ϑ4(u)ϑ44(u), ϑ1(u)ϑ11(u), ϑ12(u)ϑ14(u) in exactly the same way.

The addition formulas just obtained may be exploited by expanding both sides as power series in u1 and u2. The series expansions of the σ functions are much cleaner than the 1 T −1 − u ·η1ω ·u expansions of the ϑ functions since the introduction of factor cae 2 1 eliminates the appearance of the transcendental functions K3(m) and E3(m) from the series coeﬃcients. The

26 expansions up to the third order are listed here for convenience. 4 σ3(u) = 1 + (1 − m)u1u2 + O u , 3 3 4 σ2(u) = 1 − (1 − m)u1/3 + (1 − m)u2/3 + O u , 3 3 4 σ22(u) = 1 + (1 − m)u1/3 − (1 − m)u2/3 + O u , 3 3 4 σ4(u) = 1 + u1u2 − mu1/3 − mu2/3 + O u , 3 3 4 σ44(u) = 1 + u1u2 + mu1/3 + mu2/3 + O u , (5.3) 2 2 4 σ1(u) = u1 − u2/2 + u1u2/2 + O u , 2 2 4 σ11(u) = u1 + u2/2 + u1u2/2 + O u , 2 2 4 σ12(u) = u2 − u1/2 + u1u2/2 + O u , 2 2 4 σ14(u) = u2 + u1/2 + u1u2/2 + O u . Armed with these series expansions and addition formulas, we can easily evaluate all derivatives of ratios of σ functions. These formulas are the cubic analogues of the diﬀerential equations sn0(u) = cn(u)dn(u), cn0(u) = −sn(u)dn(u), dn0(u) = −msn(u)cn(u). Corollary 5.2. With the notation ∇f(u) = ∂f(u1,u2) , ∂f(u1,u2) , ∂u1 ∂u2 σ1(u) σ2(u)σ44(u) σ3(u)σ14(u) ∇ = 2 , − 2 , σ4(u) σ4(u) σ4(u) σ2(u) σ12(u)σ22(u) σ14(u)σ44(u) σ1(u)σ3(u) ∇ = −m 2 − (1 − m) 2 , − 2 , σ4(u) σ4(u) σ4(u) σ4(u) σ3(u) σ2(u)σ12(u) σ1(u)σ22(u) ∇ = −m 2 , −m 2 , σ4(u) σ4(u) σ4(u) σ11(u) σ1(u)σ14(u) σ3(u)σ22(u) σ12(u)σ44(u) ∇ = m 2 + 2 , 2 , σ4(u) σ4(u) σ4(u) σ4(u) σ12(u) σ3(u)σ11(u) σ22(u)σ44(u) ∇ = − 2 , 2 , σ4(u) σ4(u) σ4(u) σ14(u) σ1(u)σ44(u) σ2(u)σ3(u) σ11(u)σ12(u) ∇ = 2 , 2 + m 2 , σ4(u) σ4(u) σ4(u) σ4(u) σ22(u) σ3(u)σ12(u) σ11(u)σ44(u) σ1(u)σ2(u) ∇ = − 2 , −(1 − m) 2 − m 2 , σ4(u) σ4(u) σ4(u) σ4(u) σ44(u) σ1(u)σ11(u) σ12(u)σ14(u) ∇ = m 2 , m 2 . σ4(u) σ4(u) σ4(u)

σa(u+v)σb(u−v) Proof. Use the addition formula to get a formula for 2 , and expand both sides up σb(u) to the ﬁrst order in v1 and v2. The left hand side is σa(u) σa(u) σa(u) 2 + ∂u1 v1 + ∂u2 v2 + O(v ). σb(u) σb(u) σb(u) The expansion of the right hand side may be found using (5.3), and equating the coeﬃcients of v1 and v2 in each case gives the identities.

27 Recall the addition formula for elliptic integral of the second kind. That is, when (5.1) holds, a similar equation holds for the integrals of the second kind but with an “error” term. This equation (see pg. 65 of [16]) is: √ √ √ Z x1 1 − mt Z x2 1 − mt √ √ √ Z x3 1 − mt p dt + p dt = m x1 x2 x3 + p dt. 0 2 t(1 − t) 0 2 t(1 − t) 0 2 t(1 − t) The error term we wish to evaluate is F in the equation

Z x1 Z x2 Z x3 Z x4 Z x5 dvv + dvv + dvv = F (x1, x2, x3, x4, x5) + dvv + dvv 0 0 0 0 0 when (5.2) holds. By the inversion formulas, this is equivalent to an addition formula for the ζ function. 2 Theorem 5.3. For uu,vv ∈ C , the ζ function satisﬁes the addition formula m σ (u)σ (v)σ (u + v) + σ (u)σ (v)σ (u + v) ζ(u+v)−ζ(u)−ζ(v) = 11 11 2 12 12 1 . σ4(u)σ4(v)σ4(u + v) σ14(u)σ14(v)σ22(u + v) + σ1(u)σ1(v)σ12(u + v)

Proof. We will only show how the formula for ζ1 may be proved as the proof of the formula for ζ2 is essentially the same. We would like to show that

σ11(u)σ11(v)σ2(u + v) σ12(u)σ12(v)σ1(u + v) ζ1(u + v) − ζ1(u) − ζ1(v) = m + m . σ4(u)σ4(v)σ4(u + v) σ4(u)σ4(v)σ4(u + v)

By multiplying both sides by σ4(u)σ4(v)σ4(u +v) and using the deﬁnition of ζ1, this is seen to be equivalent to a formula for the order 2 theta function with characteristic 44 + v given by 1 1 1 σ4(u)σ4(v)σ4(u + v) + σ4(u)σ4(v)σ4(u + v) − σ4(u)σ4(v)σ4(u + v) (5.4) where f ij···(u) denotes ∂f(u1,u2) . We will represent this as a linear combination of 4 linearly ∂ui∂uj ··· independent theta functions (of u) of the same type as

Aσ11(u)σ11(v)σ2(u + v) + Bσ12(u)σ12(v)σ1(u + v) (5.5) +Cσ3(u)σ3(v)σ44(u + v) + Dσ4(u)σ4(v)σ4(u + v) Since (5.4) is symmetric in u and v, A, B, C, and D are independent of v as well. By a direct calculation, the series expansion of (5.4) side is found to be

2 2 2 σ4(v) ℘11(v)u1 + σ4(v) (1 + ℘12(v)) u2 + O(u ) (5.6) while the series expansion of (5.5) is

Cσ3(v)σ44(v) + Dσ4(v)σ4(v) 1 1 + Cσ3(v)σ44(v) + Dσ4(v)σ4(v) + Aσ2(v)σ11(v) u1 (5.7) 2 2 2 + Cσ3(v)σ44(v) + Dσ4(v)σ4(v) + Bσ1(v)σ12(v) u2 + O(u ). Thus, C = D = 0 and A and B are then given by σ (v)2℘ (v) A = 4 11 = m, σ2(v)σ11(v) σ (v)2 (1 + ℘ (v)) B = 4 12 = m. σ1(v)σ12(v)

28 6 Cubic Elliptic Functions

The main aim of this section is a derivation of Ramanujan’s inversion formula for (0.5) as a special case of Theorem 4.1 and obtaining an elegant cubic analogue of the addition theorem for elliptic integrals of the second kind (see pg. 65 of [16]), namely, Z(u + v) = Z(u) + Z(v) − k2sn(u)sn(v)sn(u + v). where Z(u) is Jacobi’s zeta function. This addition theorem is given as a series-product identity via ∞ 2 2 2 (q; q)2 X qn q T x, q T y, q T xy, q ∞ (xnyn − xn − yn) = − , (6.1) (q2; q2)4 1 − q2n xy T (qx, q2) T (qy, q2) T (qxy, q2) ∞ n=−∞ n6=0 where, for |q| < 1, the q-product symbol is defined as n Y k (x; q)n := 1 − xq k=0 and the generic genus 1 theta function T (x; q) is defined by ∞ X n n(n−1)/2 n T (x; q) := (x; q)∞(q/x; q)∞(q; q)∞ = (−1) q x . n=−∞ The equality of the two representations is the Jacobi triple product identity (see Entry 19 in Ch. 16 of [2]). We will also make extensive use of the Dedekind η function defined by ∞ 2πiτ/24 Y 2kπiτ 1/24 η(τ) = e 1 − e = q (q; q)∞, k=1 where q = e2πiτ . We have the following cubic analogue of (6.1). Theorem 6.1. For |q| < |x|, |y|, |xy| < |q|−1,

∞ n 3 3 3 3 3 3 3 (q; q)∞ X 1 + q T q x, q T q y, q T qxy, q T q xy, q qn (xnyn − yn − xn) = qxy 3 3 3 1 − q3n T (q2x, q3) T (q2y, q3) T (xy, q3) T (q2xy, q3) (q ; q )∞ n=−∞ n6=0 q T x, q3 T y, q3 T xy, q3 T q2xy, q3 − . xy T (qx, q3) T (qy, q3) T (qxy, q3) T (q3xy, q3) In order to deduce this identity as a special case of Theorem 5.3, we need to develop a set of cubic elliptic functions and give Fourier series and product representations of these functions. Recall the period matrix of the differentials of the first kind given in Section 2. The periods of a related basis (and the corresponding differentials of the second kind) are given in the table below.

A1 A2 B1 B2

du − du 2 πiK (m) − 4 πiK (m) 0 √2π K (1 − m) 2 1 3 3 3 3 3 3 du + du − √2π K (m) 0 − 4 πiK (1 − m) − 2 πiK (1 − m) 2 1 3 3 3 3 3 3 dv − dv 2 πiE (m) − 4 πiE (m) 0 √2π (K (1 − m) − E (1 − m)) 2 1 3 3 3 3 3 3 3 dv + dv √2 πE (m) 0 4 πi (K (1 − m) − E (1 − m)) 2 πi (K (1 − m) − E (1 − m)) 2 1 3 3 3 3 3 3 3 3

29 Thus the differentials du2 − du1 and du2 + du1, while still a basis for the differentials of the first kind on x, are only doubly periodic. This phenomenon is related to the following facts:

1. The equation deﬁning X can written in the Rosenhain normal form w2 = z(1 − z)(1 − λ1z)(1 − λ2z)(1 − λ3z) in which λ3 = λ1λ2. 2. X can be given as a two sheeted covering of two tori. The expressions for the moduli of these two elliptic curves in terms of the modulus m is quite complicated, but the interested reader can ﬁnd the equations in [12].

2τ τ 3. The normalized period matrix can be transformed by a transformation of τ 2τ order 2 to a diagonal matrix. This results in expressions for the cubic theta functions as sums of products of genus 1 theta functions. The explicit factorization can be given as:

∞ X a2+ab+b2 a b X a2+ab+b2 a b X a2+ab+b2 a b q z1 z2 = q z1 z2 + q z1 z2 a,b=−∞ b even b odd ∞ ∞ X 2 2 X 1 2 1 2 (6.2) A +3B A−B 2B (A+ 2 ) +3(B+ 2 ) A−B 2B+1 = q z1 z2 + q z1 z2 A,B=−∞ A,B=−∞ 2 3 2 6 2 2 6 2 6 = T −qz1; q T −q z2/z1; q + qz2T −q z1; q T −q z2/z1, q .

Integrals of both the diﬀerentials du2 − du1 and du2 + du1 can be inverted easily with specializations of the cubic theta functions ϑa(u). Since the formulas for du2 − du1 seem to come out nicer and there would been no new ideas introduced by considering du2 + du1 as well, only the inversion of the integral of du2 − du1 will be treated in detail here. Deﬁne the one variable cubic theta functions for a characteristic a in terms of the two variable functions by

u θ˙ (u) = Θ , a a 0 ζu ϑ˙ (u) = ϑ , a a −ζ2u ζu σ˙ (u) = d σ , a a a −ζ2u ∂ ζ˙(u) = − log σ (u), ∂u 4 ∂2 ℘˙(u) = − log σ (u), ∂u2 4 where da is chosen so that the first non-zero differential coefficient ofσ ˙ a(u) is one. Since ϑa(u)

30 −1 −1 ζu 3 u was deﬁned as Θa ω · u and ω · = , we see that 1 1 −ζ2u 2πiK3(m) 0

∞ 2 2 3au X a +ab+b K (m) ϑ˙3(u) = q e 3 , a,b=−∞ 1 E3(m) 2 − K (m) u σ˙ 3(u) = e 3 ϑ˙3(u), K3(m) ∞ X a2+ab+b2 2aπiu θ˙3(u) = q e , a,b=−∞ 3u ϑ˙a(u) = θ˙a . 2πiK3(m) Also define the differentials of the first and second kind as 1 − x 1 du˙ = du − du = − dx, 2 1 3y2 3y (1 − mx)(1 − x) (1 + m − 2mx) dv˙ = dv − dv = − dx. 2 1 3y2 3y

These have periods 2πi −2π ω˙ 1 = K3(m),ω ˙ 2 = √ K3(1 − m), 3 3 2πi −2π η˙1 = E3(m),η ˙2 = √ (K3(1 − m) − E3(1 − m)) , 3 3 respectively. As we shall see shortly, integrals of these diﬀerentials are inverted by the functions ˙ ˙ ˙ deﬁned above. For the functions ϑa,σ ˙ a, ζ, and℘ ˙, the period lattice isω ˙ 1Z +ω ˙ 2Z. The θa(u) ˙ ˙ have period lattice Z + 3τZ. The functions ϑa(u), θa(u) all have two zeros in the fundamental parallelogram (they are theta functions of order two). The generalized Legendre identity becomesω ˙ 1η˙2 − ω˙ 2η˙1 = 2πi, and the quasi-periodicity of ζ˙(u) is expressed by

ζ˙(u + aω˙ 1 + bω˙2) = ζ˙(u) + 2aη˙1 + 2bη˙2 for integers a and b, while θ˙3(u) satisﬁes

θ˙3(u + 1) = θ˙3(u), −2πi(2u+3τ) θ˙3(u + 3τ) = e θ˙3(u).

Theorem 6.2. For the elliptic sigma functions σ˙ a(u) we have the following inversion formula. Let m−1/2 denote one of the ﬁxed points of the automorphism δ (the point on the second sheet with x coordinate m−1/2). Then with √ Z x 2i Z φ cos 1 sin−1 ( m sin φ) u = du˙ ≡ 3 dφ, p 2 m−1/2 3 0 1 − m sin φ

31 we have σ˙ (u) 1 x−1/3m1/3 1 = 1 + , σ˙ 4(u) x˙ σ˙ (u) −x1/3m2/3 11 = 1 +x ˙, σ˙ 4(u) σ˙ (u) x˙ 1 −x−1/3m1/3 2 = 1 + + , σ˙ 4(u) x x˙ σ˙ (u) mx −x1/3m2/3 22 = 1 +x ˙ + , σ˙ 4(u) x˙ σ˙ (u) 1 − 3 = 1 +x ˙ + , σ˙ 4(u) x˙

σ˙ 14(u) =σ ˙ 1(u),

σ˙ 12(u) =σ ˙ 11(u),

σ˙ 44(u) =σ ˙ 4(u), and therefore σ˙ (u) + σ (u) σ˙ (u) 1 2 = e2iφ/3 = 11 , σ˙ 4(u) σ˙ 1(u) σ˙ (u) − σ˙ (u) σ˙ (u) 22 11 = e−2iφ/3 = 1 , σ˙ 4(u) σ˙ 11(u) σ˙ (u) 2 √ 2i dφ − 3 = 1 − 2 cos sin−1 m sin φ = − . σ˙ 4(u) 3 3 du Also, √ 1 − 2mx Z x ζ˙(u) = 2 m − 2 − x˙ − + 2 dv˙, x˙ m−1/2 σ˙ 2(u)σ ˙ 11(u) σ˙ 1(u)σ ˙ 22(u) σ˙ 3(u) ℘˙(u) = m 2 − m 2 + 2 , σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) 1 − mx2 1 +x ˙ = 2 + . 1 − x x˙ 2 2 Proof. In Theorem 4.1, set x1 = δσx, x2 = γσ x. Then from Proposition 3.7, Z δσx Z γσ2x u = duu + duu 0 0 0 1 ζ2 0 Z x −1 0 ζ 0 Z x = duu + 2 duu 1 0 0 ζ ∞ 0 −1 0 ζ 1/m −ζ ζ Z x 0 1 ζ2 0 Z 0 −1 0 ζ 0 Z 0 = 2 2 duu + duu + 2 duu ζ −ζ 0 1 0 0 ζ ∞ 0 −1 0 ζ 1/m −ζ ζ Z x 0 1 ζ2 0 −1 0 ζ 0 = 2 2 duu + (−2) + 2 (−1) ζ −ζ 0 1 0 0 ζ 0 −1 0 ζ ζu ≡ + . −ζ2u 1

32 −1/2 R m ω˙ 2 This last equality uses the fact that 0 du ≡ 3 , an identity that follows easily from the fact that m−1/2 is a fixed point of δ (see Proposition 3.7). Performing these substitutions in Theorem 4.1, shifting the characteristics by 1 and simplifying gives the first group of equations. The next three follow from the relationship between φ and the point (x, y) on X given by 1 − mx2 sin(φ) = √ √ √ √ , 2 m x 1 − x 1 − mx 1 − 2mx + mx2 cos(φ) = i √ √ √ √ , 2 m x 1 − x 1 − mx dφ dx = √ √ √ , p1 − m sin(φ)2 x 1 − x 1 − mx and the previous equations. The equation for℘ ˙(u) follows from the equation for ℘(u) given in Theorem 4.1 and the chain rule. The equation for ζ˙(u) follows from the fact that by the periodicity properties of ζ˙(u), Z x F (x) = ζ˙(u) − 2 dv˙ m−1/2 dF is a meromorphic function of x (on X). With the help of the equation for℘ ˙(u) we can find dx , integrate this, and use the known values at x = m−1/2 to evaluate the constant of integration. The result is the stated formula.

By comparing the poles and zeros of the meromorphic functions on the right hand sides of Theorem 6.2 with the theta function expressions on the left, we can deduce the location of the two zeros of θ˙a(u). In each case we must convert poles or zeros on X to points in the fundamental parallelogram using the integral in Theorem 6.2. This integral maps X surjectively and generically two-to-one onto the fundamental parallelogram. For example the two zeros of θ˙3(u) are determined from the four solutions of 1 +x ˙ +x ˙ −1 = 0, √ thus the x-coordinates of the zeros are all 1 ± 1 − m−1. Care must be taken in choosing the correct y coordinate, and the result of simpliﬁng the integrals obtained for the zeros of the four functions θ˙4(u), θ˙1(u), θ˙11(u), and θ˙3(u) is given in the following table.

a zeros of θ˙a(u) 4 τ,2τ 1 0,2τ 11 0,τ 1 1 3 2 + τ − c3, 2 + 2τ + c3 where, at least for 0 < m < 1,

√1 1 Z 1+ 1−m 1 1 c = c (m) = − dt 3 3 1/3 2/3 1/3 2/3 1/3 2/3 4πiK3(m) 0 t (1 − t) (1 − mt) t (1 − t) (1 − mt) i log 2 im 11im2 = − − + O(m3). 2π 24π 576π (6.3) The zeros of θ˙2 and θ˙22 can also be given explicitly, but the formulas are much too complicated to write down. Product representation of the cubic theta functions of one variable easily follow from this information on their zeros.

33 2πiu 2πiτ −2π K3(1−m) Proposition 6.3. With x = e , and q = e = exp √ and c3 as in (6.3), 3 K3(m)

˙ (q; q)∞ 3 2 3 θ4(u) = 3 3 T qx; q T q x; q , (q ; q ) ∞ ˙ 1/3 1/3 (q; q)∞ 3 3 3 θ1(u) = −x q 3 3 T q x; q T qx; q , (q ; q ) ∞ ˙ −1/3 1/3 (q; q)∞ 3 2 3 θ11(u) = −x q 3 3 T x; q T q x; q , (q ; q ) ∞ T q; q2 T q3; q6 −2πic3 3 2 2πic3 3 θ˙3(u) = T −qe x; q T −q e x; q . T (qe−2πic3 ; q3) 2

Proof. The function on the right hand side of the ﬁrst equation is a meromorphic function of u with the same zeros as θ˙4 and the same quasi-periodicity relations. Therefore, the two functions diﬀer by a multiplicative constant. Since

u + 1/3 θ˙ (u) = Θ , 4 2/3 we have, from the factorization of Θ(u) into genus 1 sums by (6.2),

2 3 −1 6 2 2 2 6 −1 6 θ˙4(u) = T −qxζ; q T −q x ; q + qζ T −q xζ; q T −q x ; q = CT qx; q3 T q2x; q3 .

Upon setting x = −ζ2, the second term on the left vanishes, and we are left with

2 3 6 T q; q T ζq ; q (q; q)∞ C = 2 3 2 2 3 = 3 3 T (−ζ q; q ) T (−ζ q ; q ) (q ; q )∞ after some manipulation of the q-products. The proof of the remaining three formulas is exactly the same.

Proposition 6.4.

σ˙ 1(u) σ˙ 2(u) σ˙ 3(u)σ ˙ 1(u) ∂u = − 2 , σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) 2 σ˙ 11(u) σ˙ 1(u) σ˙ 11(u) σ˙ 3(u)σ ˙ 22(u) ∂u = −m 2 + + 2 , σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) σ˙ 2(u) σ˙ 11(u)σ ˙ 22(u) σ˙ 1(u) σ˙ 1(u)σ ˙ 3(u) ∂u = m 2 + (1 − m) + 2 , σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) σ˙ 22(u) σ˙ 1(u)σ ˙ 2(u) σ˙ 11(u)σ ˙ 4(u) σ˙ 3(u)σ ˙ 11(u) ∂u = m 2 + (1 − m) 2 + 2 , σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) σ˙ 3(u) σ˙ 2(u)σ ˙ 11(u) σ˙ 1(u)σ ˙ 22(u) ∂u = m 2 + m 2 , σ˙ 4(u) σ˙ 4(u) σ˙ 4(u) and z(u) = σ˙ 3(u) satisﬁes σ˙ 4(u)

z02 = (z − 1) z3 + 3z2 − 4(1 − m) .

34 Proof. Use the deﬁnition ζu σ˙ (u) = d σ a a a −ζ2u along with Corollary 5.2 and the chain rule. The diﬀerential equation for z(u) may be computed by combining the formula for z0 stated above and the inversion formulas. In the notation of Theorem 6.2, 2 02 σ˙ 2(u)σ ˙ 11(u) σ˙ 1(u)σ ˙ 22(u) z = m 2 + m 2 σ˙ 4(u) σ˙ 4(u) !2 (1 +x ˙) x˙ 3 − mx2 = xx˙ 2 (1 +x ˙)2 = 1 + 2(1 − 2m)x ˙ 3 +x ˙ 6 x˙ 4 = (z − 1) z3 + 3z2 − 4(1 − m)

3 x(1−mx) 1 The third equality follows fromx ˙ = 1−x and the fourth from −z = 1 +x ˙ + x˙ . By the transformation 4(1 − z(u))℘(u) = 3 − 4m − 3z(u) the diﬀerential equation for z(u) 0 2 3 stated in Theorem 6.4 above can be converted to the Weierstrass form (℘ ) = 4℘ − g2℘ − g3 in which 3 (9 − 8m) −27 + 36m − 8m2 g = , g = . 2 4 3 8 This leads to the following identity where ℘ denotes the ordinary Weierstrass ℘ function with the invariants as above: (3 − 4m)σ ˙ (u) − 3σ ˙ (u) 4℘(u) = 4 3 . σ˙ 4(u) − σ˙ 3(u) 3 If e1, e2 and e3 denote the roots of the cubic 4℘ − g2℘ − g3 = 0, then the period lattice of ℘(u) is generated by the periods of Z dt p 2 (t − e1)(t − e2)(t − e3)

With the substitution s = (e3−e1)(t−e2) , the integral is reduced directly to the elliptic integral (e3−e2)(t−e1) in (0.14), and we deduce that π e2 − e3 πi e1 − e2 √ K2 , √ K2 e1 − e3 e1 − e3 e1 − e3 e1 − e3 generate the period lattice of ℘(u). Since the period lattice of z(u) is generated byω ˙ 1 andω ˙ 2, we obtain an algebraic relationship between K2(m) and K3(m) as a corollary of the diﬀerential equation in Proposition 6.4. To rationalize this relationship, let us introduce a parameter t deﬁned by t2(t + 3)2(t + 6)2 m = . (t2 + 6t + 12)3 In Section 8.1 we will see that η(τ)2η 3τ η(6τ)3 t = 12 2 τ 3 2 η 2 η(2τ)η(3τ)

35 3 Θ2(τ) when m = 3 . The three roots are then rational in t and we obtain the following formal Θ3(τ) identities. √ t2(t + 3)2(t + 6)2 √ t2 + 6t + 12 t(t + 4)3 −3K3 1 − = −3 K2 1 − , (t2 + 6t + 12)3 (t + 2)3/2(t + 6)1/2 (t + 2)3(t + 6) t2(t + 3)2(t + 6)2 1 t2 + 6t + 12 t(t + 4)3 K3 = √ K2 . (t2 + 6t + 12)3 3 (t + 2)3/2(t + 6)1/2 (t + 2)3(t + 6) (6.4) These identities are formal in the sense that they are each valid only in some subset of C. For general t, the most that can be said is that the lattice generated by the two expressions on the left will equal the lattice generated by the two expressions on the right. These are equivalent −t to Theorem 5.6 and Corollary 5.7 in Chapter 33 of [3] where the parameter p = 2(t+3) was used.

K (1−m) Proposition 6.5. With x = exp 3u and q = e2πiτ = exp −√2π 3 , the following Fourier K3(m) 3 K3(m) series and partial fraction expansions are valid where they converge. A sum of the form Σ f(k) Z+a PN × denotes the symmetric limit limN→∞ k=−N f(k + a) in all cases, and Z = Z − {0}

2n n K3(m) σ˙ 1(u) X q x = for |q| < |x| < |q|−2, −2/3 3n 3m σ˙ 4(u) 1 − q Z−1/3 X qk = x1/3 , 1 − xq3k Z−1/3 n n K3(m) σ˙ 11(u) X q x = for |q|2 < |x| < |q|−1, −2/3 3n 3m σ˙ 4(u) 1 − q Z−1/3 X q2k = x2/3 , 1 − xq3k Z+1/3

1/3 n K3(m) σ˙ 2(u) X m − q = qnxn for |q| < |x| < |q|−1, −2/3 3n 3m σ˙ 4(u) 1 − q Z+1/3 X qk X qk = m1/3x1/3 − x1/3 , 1 − xq3k 1 − xq3k Z+1/3 Z−1/3 1/3 n K3(m) σ˙ 22(u) X 1 − m q = qnxn for |q| < |x| < |q|−1, −2/3 3n 3m σ˙ 4(u) 1 − q Z−1/3 X q2k X q2k = x2/3 − m1/3x2/3 , 1 − xq3k 1 − xq3k Z+1/3 Z−1/3

36 n K3(m) σ˙ 3(u) X 1 − q n n = 3n q x , 3 σ˙ 4(u) 1 − q Z 1 1 X 1 + xq3k 1 X 1 + xq3k = + − , 3 2 1 − xq3k 2 1 − xq3k Z+1/3 Z−1/3 n K3(m) 2E3(m) X 1 + q ζ˙(u) − u = qnxn, 3 3 1 − q3n Z× 1 X 1 + xq3k 1 X 1 + xq3k = + . 2 1 − xq3k 2 1 − xq3k Z+1/3 Z−1/3 Proof. Calculations using the relations in Proposition 3.1 show that

θ˙ (u + 1) θ˙ (u) 1 = ζ 1 , θ˙4(u + 1) θ˙4(u) θ˙ (u + 3τ) θ˙ (u) 1 = 1 . θ˙4(u + 3τ) θ˙4(u)

2 θ1(u) ∞ nπiu Thus has a period of 3 and can be represented in the form P a e 3 where θ4(u) n=−∞ n

Z 3 ˙ − 2 nπiu θ1(u) 3an = e 3 du. 0 θ˙4(u)

To evaluate this integral, we need to ﬁrst calculate the residue of θ1(u) at its simple pole at θ4(u) u = τ. Again by Proposition 3.1,

− 2 πiτ θ˙1(τ) = e 3 θ4, − 2 iπ(τ+2u) θ˙4(u + τ) = e 3 θ˙12(u) − 2 iπ(τ+2u) 2πi ζu = e 3 ϑ K (m) 12 3 3 −ζ2u − 2 iπ(τ+2u) 2πi 2 2 = e 3 (−ζθ θ /θ ) − K (m)ζ u + O(u ) . 2 4 3 3 3

˙ From this it is clear that the residue of θ1(u) at u = τ is 3 . It follows that the residue of θ4(u) 2πiθ2 ˙ −2nπiu/3 θ1(u) i(1−n) −2nπiτ/3 e at the pole at u = τ + i is 3ζ e / (2πiθ2) for any integer i. Now θ˙4(u)

2 −2nπiτ/3 Z 3 Z 3+3τ Z 3τ Z 0 ˙ X e θ1(u) 3ζi(1−n) = + + + e−2nπiu/3 du θ ˙ i=0 2 0 3 3+3τ 3τ θ4(u) Z 3 Z 3τ θ˙ (u) = + e−2nπiu/3 1 du 0 3+3τ θ˙4(u) Z 3 θ˙ (u) = 1 − e−2nπiτ e−2nπiu/3 1 du 0 θ˙4(u) −2nπiτ = 1 − e 3an,

37 −2nπiτ/3 and the sum on the left is zero unless n ≡ 1 mod 3 in which case it is 9e /θ2. Substitut- θ1(u) ing this formula for an and noticing that a0 = 0 by the periodicity properties of and then θ4(u) converting to theσ ˙ and q notation completes the proof of the ﬁrst Fourier series expansion. The proofs of the remaining Fourier series for σ˙ a(u) are similar; the method only fails for the σ˙ 4(u) constant term in σ˙ 3(u) , but this can be deduced using the inversion formulas. Writing σ˙ 4(u)

∞ σ˙ 3(u) X = a xn σ˙ (u) n 4 n=−∞ gives Z ω˙ 1 Z φ0+π σ˙ 3(u) 2i ω˙ 1a0 = du = dφ 0 σ˙ 4(u) φ0 3

As u ranges from 0 toω ˙ 1, x goes around an A loop and consequently φ increases by π from some φ0 to φ0 + π. Thusω ˙1a0 = 2πi/3 and so a0 = 1/K3(m), which is consistent with the qn(1−qn) formula since 1−q3n approaches 1/3 when n = 0 The partial fraction expansions are analytic continuations of the Fourier series and follow by series manipulations. The formula for ζ˙(u) follows from the deﬁnitions

∂ ζu˙ ζ˙(u) = − logσ ˙ ∂u 4 −ζ˙2u ˙ ˙ ∂ 1 −1 ζu ζu ˙ 3u = η1ω1 · ˙2 · ˙2 − log θ4 ∂u 2 −ζ u −ζ u 2πiK3(m) ∂ E3(m) 2 3u = u − log θ˙4 , ∂u K3(m) 2πiK3(m) along with the product formula for θ˙4 given in Proposition 6.3. Corollary 6.6. The theta constants are given by

η(3τ)3 Θ (τ) = 3 , 2 η(τ) η(τ)3 Θ (τ) = , 4 η(3τ) η (τ/3)3 η(3τ)3 Θ (τ) = + 3 . 3 η(τ) η(τ)

Proof. The second equality follows from letting u = 0 in Theorem 6.3. The ﬁrst then follows from the second by replacing τ with −1/3τ. The last equality follows from the identity

Θ3(3τ) = Θ2(3τ) + Θ4(τ).

Which we may prove as follows. If we let, for i = 0, 1, 2, A = b + 2a + i, B = b − a, then each lattice point (A, B) correspond to exactly one lattice point (a, b) with i = A − B mod 3. Also, i 2 i i i 2 A2 + AB + B2 = 3 a + + 3 a + b + + 3 b + 3 3 3 3

38 so that ∞ X A−B A2+AB+B2 Θ3(τ) = ζ q A,B=−∞ 2 ∞ 2 2 X X i 3 b+ i +3 b+ i a+ i +3 a+ i = ζ q ( 3 ) ( 3 )( 3 ) ( 3 ) i=0 a,b=−∞ 2 = Θ3(3τ) + ζΘ2(3τ) + ζ Θ22(3τ)

= Θ3(3τ) − Θ2(3τ).

We are now in a position to easily deduce Theorem 6.1. The addition formula for ζ˙(u) may be obtained from the addition formula for ζ(u) by using the relation

ζu ζ ζ˙(u) = ζ · . −ζ2u −ζ2

This gives σ˙ (u)σ ˙ (v) σ˙ (u + v) +σ ˙ (u + v) ζ˙(u + v) − ζ˙(u) − ζ˙(v) = m 11 11 1 2 σ˙ 4(u)σ ˙ 4(v) σ˙ 4(u + v) σ˙ (u)σ ˙ (v) σ˙ (u + v) − σ˙ (u + v) − m 1 1 22 11 (6.5) σ˙ 4(u)σ ˙ 4(v) σ˙ 4(u + v) σ˙ (u)σ ˙ (v) σ˙ (u + v) σ˙ (u)σ ˙ (v) σ˙ (u + v) = m 11 11 11 − m 1 1 1 , σ˙ 4(u)σ ˙ 4(v) σ˙ 1(u + v) σ˙ 4(u)σ ˙ 4(v) σ˙ 11(u + v) where the last equality follows from the identities σ˙ 1(u)+σ ˙ 2(u) = σ˙ 11(u) and σ˙ 22(u)−σ˙ 11(u) = σ˙ 1(u) σ˙ 4(u) σ˙ 1(u) σ˙ 4(u) σ˙ 11(u) K (1−m) given in Theorem 6.2. By setting x = exp 3u , y = exp 3v and q = exp −√2π 3 in K3(m) K3(m) 3 K3(m) (6.5), and using the series representation of ζ˙(u) given in Proposition 6.5 and the product representations given in Proposition 6.3, we produce Theorem 6.1.

7 Modular Transformations

In this section we will consider how the theta functions Θa(u|τ) are aﬀected by the transformations τ → −1/3τ and τ → pτ. Since

i K (1 − m) τ = √ 3 , 3 K3(m) the transformation m → 1−m corresponds to τ → −1/3τ. Such a transformation corresponds to switching the period matrices ω1 and ω2, and is a direct analogue of the imaginary transformation of Jacobi where sn(u) is transformed into sn(iu), but here, as we will see, the role of the fourth root of unity is played by the matrix

0 −1 1 0 whose fourth power also is the identity (see Theorem 7.2).

39 For small p (2 or 3), the the full eﬀect of the transformation τ → pτ on the theta functions, modulus, and period matrices can be given explicitly, (see Theorem 7.3 and Theorem 7.5). For larger p it seems that the theta functions do not satisfy simple transformations, and we are only able to describe the transformations of the modulus, and period matrices. Let mp denote the modulus associated with pτ, let

K3(mp) Mp = p K3(m1) be the multiplier associated with the periods of the diﬀerentials of the ﬁrst kind, and let

2E3 (mp) − K3 (mp) 2E3 (m1) − K3 (m1) Np = p − K3 (m1) K3 (mp) be the multiplier associated with the periods of the diﬀerentials of the second kind. For rational p, m1 and mp are algebraically related, while Mp and Np are algebraic functions of m1 (and mp). We refer loosely to these four quantities as the modular quantities associated with the transformation τ → pτ. The eﬀect of the this transformation on the modulus m and the four period matrices is completely determined once the modular quantities are known.

Proposition 7.1. The various modular quantities are given in terms of Θ functions by:

1/3 Θ2(pτ) 1/3 Θ4(pτ) mp = , (1 − mp) = , Θ3(pτ) Θ3(pτ) d log Θ2(pτ)Θ4(pτ) 3 dτ Θ2(τ)Θ4(τ) Np = , 2πi Θ3(τ)Θ3(pτ) Θ3(pτ) Mp = p , Θ3(τ) and Mp and Np can be calculated by:

2 m1 (1 − m1) dmp Mp = p , (7.1) mp (1 − mp) dm1

p dMp Np = (1 − 2mp) Mp − (1 − 2m1) + 6mp (1 − mp) . (7.2) Mp dmp

Proof. The formulas for mp and Mp in terms of Θ functions follow directly from Corollary 4.2, so only the formula for Np needs to be proved. First we note that by the generalized Legendre identity (2.3) and (2.4),

dτ d 1 i K (1 − m ) = √ 3 p dmp dmp p 3 K3(mp) 1 1 = 2 . p 2πimp(1 − mp)K3(mp)

40 Therefore, by a direct calculation,

d log Θ2(pτ)Θ4(pτ) 3 dτ Θ2(τ)Θ4(τ) 2πi Θ3(τ)Θ3(pτ) 1/3 dmp d 1/3 2 dm1 d 1/3 1/3 2 3 dτ dm {log mp (1 − mp) K3(mp) } 3 {log m (1 − m1) K3(m1) } = p − dτ dm1 1 2πi K3(m1)K3(mp) 2πi K3(m1)K3(mp) 2E (m ) − K (m ) 2E (m ) − K (m ) = p 3 p 3 p − 3 1 3 1 K3 (m1) K3 (mp)

= Np. Next,

dm dmp 2pπim (1 − m )K (m )2 p = dτ = p p 3 p , dm dm1 2πim (1 − m )K (m )2 1 dτ 1 1 3 1 and so the second formula for Mp is clear. The formula for Np follows by diﬀerentiating the identity K3(m1)Mp = pK3(mp) with respect to mp and rearranging.

7.1 Transformations of the Theta Functions Theorem 7.2. Under the complementary transformation, the theta functions satisfy 1 √ 2 2 u1 2πiτ(u −u1u2+u ) τu1 − 2τu2 Θa − = −cτ −3e 1 2 Θb τ , u2 3τ −τu1 − τu2 √ K (1 − m) 3 3 u1u2 u1 3 − 2π K (m)K (1−m) −u2 ϑa 1 − m = c e 3 3 ϑb m , u2 K3(m) u1 u1 u1u2 −u2 σa 1 − m = de σb m , u2 u1 for the following values of a, b, c and d: a b c d

3 3 1 1 4 22 1 1 44 2 1 1 2 4 1 1 22 44 1 1 1 12 ζ ζ2 11 14 ζ ζ2 12 11 ζ2 −ζ 14 1 ζ2 −ζ Proof. These can be obtained by applying the general modular transformation for theta functions with arbitrary period matrices (given in Ch. 5 of [19]), but here we will prove that 2πi 2 2 1 1 u1 − (u −u1u2+u ) u1 − 2u2 Θ3 τ = Ce 3τ 1 2 Θ3 − − u2 3τ −u1 − u2 3τ

41 2 −1 2 for some constant C simply by noting that both sides are theta functions (on Z + ω1 ω2Z ) of order 1 with zero characteristic. That is, they satisfy the relations

1 f u + = f(u), 0 0 f u + = f(u), 1 2τ f u + = e−2πi(u1+τ)f(u), τ τ f u + = e−2πi(u2+τ)f(u). 2τ

The value of C may be deduced by setting u = 0 to obtain

Θ(00|τ) = CΘ(00| − 1/3τ).

Thus, by Corollary 4.2,

Θ(00|τ) K (m) p−1/3 C = = 3 = . Θ(00| − 1/3τ) K3(1 − m) τ Replacing τ with −1/3τ gives the ﬁrst identity stated. The transformations for the remaining characteristics follow by shifting this identity; save for an exponential factor, the complementary transformation just permutes the nine theta functions. In translating the identity to ϑa(u) and σa(u), we only need the evaluations given in Corollary 4.2 and the generalized Legendre identity (2.3).

Thus the transformation τ → −1/3pτ induces the transformations m1 → 1 − mp, mp → 1 − m1, Mp → p/Mp Np → Np on the modular quantities. To produce cubic analogues of the general pth order tranformation of the classical incomplete elliptic integral of the ﬁrst kind (see (0.19)), we need formulas relating the Θa(puu|pτ) to polynomials in the Θa(u|τ). Armed with these formulas and the inversion formulas in Theorem 4.1, we can provide the explicit relationship between the pair x1 and x2 and the pair x3 and x4 induced by

Z x1 Z x2 Z x3 Z x4 Mp duu + duu = duu + duu, 0 0 0 0 and this constitues a cubic analogue of the classical pth order transformation for elliptic integrals in (0.19).

Theorem 7.3. The theta functions satisfy the degree 2 modular identities

Θ2(2τ) Θ4(2τ) Θ3(2u|2τ) = Θ2(u)Θ22(u) + Θ4(u)Θ44(u) Θ2(τ)Θ3(τ) Θ4(τ)Θ3(τ) Θ (2τ) Θ (2τ)Θ (2τ) 4 u 2 2 4 u u = 2 Θ3(u) − 2 2 Θ2(u)Θ22(u) Θ4(τ) Θ2(τ)Θ4(τ) Θ2(2τ) 2 Θ2(2τ)Θ4(2τ) = 2 Θ3(u) + 2 2 Θ4(u)Θ44(u). Θ2(τ) Θ2(τ) Θ4(τ)

42 Formulas for Θ[a2 + b4](2u|2τ) may be obtained with the shift u → u + a2 − b4. For some parameter t, the various modular quantities are related by t(t + 9)2 t2(t + 9) m = , m = , 1 (t + 6)3 2 (t + 12)3 t + 12 M = , N = 1, 2 t + 6 2 and the theta function identities may be recast for the sigma functions as

9(t+8) 2 − u1u2 (t + 6) 9(t + 8) e (t+6)2 σ (M u|m ) = σ (u)2 + σ (u)σ (u) 3 2 2 (t + 9)(t + 12) 3 (t + 9)(t + 12) 4 44 (t + 6)2 t(t + 9) = σ (u)2 − σ (u)σ (u) 3(t + 12) 3 3(t + 12) 2 22 t(t + 9) 9(t + 8) = σ (u)σ (u) + σ (u)σ (u). (t + 6)(t + 12) 2 22 (t + 6)(t + 12) 4 44

Proof. The substitution u → u + a2 − b4 turns Θ3 (2u|2τ) into

Θ3 (2 (u + a2) − 2b4|2τ) = Θ3 (2 (u + a2) + b4|2τ), which is Θ[a2 + b4](2u|2τ) up to an exponential factor. Thus it suﬃces to give formulas for 2 −1 2 Θ3 (2u|2τ) only. On the period lattice Z + ω1 ω2Z , the theta function Θ3(2u|2τ) has order 2 and zero characteristic, and the ﬁve functions

2 Θ3(u|τ) ,Θ2(u|τ)Θ22(u|τ), Θ4(u|τ)Θ44(u|τ), Θ1(u|τ)Θ11(u|τ), Θ12(u|τ)Θ14(u|τ) have the same property. Furthermore, as used in the proof of the ﬁrst addition formula, the four functions

Θ2(u)Θ22(u), Θ4(u)Θ44(u), Θ1(u)Θ11(u), Θ12(u)Θ14(u) are linearly independent, and Θ3(2u|2τ) can be written as linear combinations of these. The coefficients may be isolated by evaluating both sides at u = 2, 4, 1, 12. This gives the first formula for Θ3(2u|2τ) exactly as it is stated. The next two follow similarly by using the sets of functions 2 Θ3(u) ,Θ4(u)Θ44(u), Θ1(u)Θ11(u), Θ12(u)Θ14(u) and 2 Θ3(u) ,Θ2(u)Θ22(u), Θ1(u)Θ11(u), Θ12(u)Θ14(u). However, in both cases the parameterizations of the modular quantities are needed to simplify the coefficients of Θ2(u)Θ22(u) and Θ4(u)Θ44(u) to a single term, so the proof of these last two formulas is complete once the modular equations have been established. These parameterizations are established in Section 8 where modular equation are studied in more detail, so the proof of the theta function transformations is complete. The formulas for the σ functions follow by converting those for the Θ functions along with the help of the three modular equations and the generalized Legendre identity (2.3).

The sum of cubes identity

3 3 3 3 ϑ1(u) + ϑ3(u) = ϑ2(u) + ϑ4(u) . from Corollary 4.3 is interesting for methods one might employ to prove it. We can prove it:

43 R x1 R x2 1. algebraically by setting u = 0 duu + 0 duu 2. by actually multiplying out and combining the series

3. by verifying that both sides agree when written in terms of some basis of the theta functions of order 3

4. by verifying that both sides agree up to a certain ﬁnite order at a ﬁnite set of points

While this identity was obtained by the first method, the calculations involved in this approach easily become irksome as the identity becomes more complicated. The second method, while possibly providing the most satisfying proof, relies on possibly ingenious manipulations of infinite series and already looks troublesome, as the series defining the ϑ functions are already double sums. The third method, while being the method used to prove theta function identities of order 2, is actually not feasible for identities of higher order. The reason for this is that the set of functions spanned by {ϑa(u)ϑb(u)ϑc(u)}a,b,c (where a+b+c is some fixed characteristic) has dimension 6 instead of the required dimension 9 = 32. Hence, we cannot be sure that a given theta function of order 3 is a linear combination of functions of the form ϑa(u)ϑb(u)ϑc(u). The method employed here to obtain and prove cubic theta function identities of order 3 or higher is that of the fourth. A precise statement of to what order the series expansions must be calculated is given in Proposition 7.4. This Proposition is a genus 2 analogue of the fact that an elliptic (genus 1) theta function or order r has exactly r zeros counted according to multiplicity insides its fundamental parallelogram unless it vanishes identically.

Proposition 7.4. Let f(u) be a cubic theta function of order r, and suppose that at each 2r+1 of the 9 characteristics f(u) has a zero of order at least n = 4 (that is, all diﬀerential coeﬃcients of f(u) of order less than n vanish). In this case, f(u) vanishes identically.

Proof. Recall that a cubic theta function of order r is a holomorphic function f(u) satisfying

T T −1 −1 2πi(2·a1−1·a2) −πir(2a2 ·u+a2 ·ω1 ω2·a2) f u + a1 + ω1 ω2 · a2 = e e f(u)

2 R x for integer vectors a1 and a2. Fix e ∈ C , and suppose that g(x) = f(e + 0 duˆ) does not vanish identically. The number of zeros of g(x) on X is given by

1 I d log g(x) 2πi ∂X where ∂X is obtained by dissecting X along its homotopy basis A1, A2, B1, B2. The path obtained is:

44 -A1

-B1 B1

A2 X A1

B2 -B2

-A2 and log g(x) is a single-valued function on this dissection of X. The integral may be evaluated as 2 1 I 1 X Z Z Z Z d log g(x) = + + + d log g(x) 2πi 2πi ∂X i=1 Ai Bi −Ai −Bi 2 1 X Z Z = + d log g(x) 2πi i=1 Ai −Ai 2 1 X Z = 2πirduˆ 2πi i i=1 Ai = 2r.

The second line follows since d log g(x) is the same on B1 and B2 by the quasi-periodicity of f. Hence the two integrals cancel completely. The third line follows from the quasi-periodicity of f since for x ∈ Ai, d log g(x + Bi) = 2πirduˆ i + d log g(x) where x + Bi is the point on −Ai after traversing Bi. Finally, by deﬁnition of the diﬀerentials R duˆ i (see Section 2.1), duˆ i = 1. Ai Thus we see that g(x) has 2r zeros counted according to multiplicity unless it vanishes identically. The hypothesis of the theorem implies that, for each a, the function Z x1 f a + duˆ 0 has zeros at the four branch points and that total number of zeros counted according to multiplicity is greater than 2r. Hence this function vanishes identically for each a. It follows that, for each x1, Z x1 Z x2 f duˆ + duˆ 0 0 has zeros when x2 is any of the four branch points, and the total number of zeros is greater than 2r. Thus, for each x1, this expression vanishes identically as a function of x2, and f(u) must then vanish identically as a function of u.

45 Theorem 7.5. The theta functions satisfy the degree 3 modular identities

2 Θ3(τ)Θ4(τ) Θ3(3τ) Θ3(3u|3τ) = Θ3(u)Θ4(u)Θ44(u) + Θ1(u)Θ2(u)Θ14(u) + Θ11(u)Θ12(u)Θ22(u), Θ2(3τ) Θ2(3τ) 2 Θ4(τ)Θ2(τ) Θ4(3u|3τ) = Θ1(u)Θ12(u)Θ44(u) + Θ2(u)Θ4(u)Θ22(u) + Θ3(u)Θ11(u)Θ14(u). Θ4(3τ) Equations for the other seven theta functions may be obtained by shifting and reﬂecting these two. For some parameter t, the various modular quantities are related by

1 t3 m = 1 − , m = , 1 (t + 1)3 3 (t + 3)3 t + 3 2 t2 + 3t + 3 M = , N = , 3 t + 1 3 (t + 1)(t + 3) and the ϑ function identities can be recast as

3(t+2) 2 2 − u1u2 t t + 3t + 3 e (t+1)2 σ (M u|m ) = σ (u)σ (u)σ (u) − (σ (u)σ (u)σ (u) + σ (u)σ (u)σ (u)), 3 3 3 3 4 44 (t + 1)3(t + 3) 1 2 14 11 12 22 3(t+2) − u1u2 1 1 e (t+1)2 σ (M u|m ) = σ (u)σ (u)σ (u) + σ (u)σ (u)σ (u) + σ (u)σ (u)σ (u). 4 3 3 2 4 22 t + 1 3 11 14 (t + 1)2 1 12 44

Proof. The parametric representations are obtained in Section 8, so we need to verify the ﬁrst and second Θ function identities. By Proposition 7.4, we need to verify that both sides of the proposed identities agree up to the ﬁrst order at all of the characteristics. Actually, much less calculation is required. Both sides of each identity are invariant under the shift u → u + 4, so we only need to check the characteristics 3, 2 and 22. Furthermore Θ3(u) is an even function, so we do not need to check 22 in that case. Shifting by 3 gives

Shifting by 2 gives

2 Θ3(τ)Θ4(τ) Θ3(3τ) Θ2(3u|3τ) = Θ2(u)Θ1(u)Θ14(u) + Θ12(u)Θ22(u)Θ11(u) + Θ44(u)Θ4(u)Θ3(u), Θ2(3τ) Θ2(3τ) 2 Θ4(τ)Θ2(τ) Θ1(3u|3τ) = Θ12(u)Θ4(u)Θ14(u) + Θ22(u)Θ1(u)Θ3(u) + Θ2(u)Θ44(u)Θ11(u). Θ4(3τ)

Shifting by 22 gives

2 Θ4(τ)Θ2(τ) Θ12(3u|3τ) = Θ4(u)Θ1(u)Θ11(u) + Θ3(u)Θ12(u)Θ2(u) + Θ22(u)Θ14(u)Θ44(u). Θ4(3τ) It is easily verified that the series expansions of of both sides of these five identities agree at least up to the first order, so the identity is indeed valid.

46 A degree 4 analogue of the previous two theorems is given by

Θ2(4τ) 2 Θ4(4τ) 2 Θ3(4u|4τ) = 3 Θ2(u)Θ22(u)Θ3(u) + 3 Θ4(u)Θ44(u)Θ3(u) Θ3(τ)Θ2(τ) Θ3(τ)Θ4(τ) 2Θ2(4τ)Θ4(4τ) (Θ3(τ) + 4Θ3(4τ)) − 3 3 Θ2(u)Θ22(u)Θ4(u)Θ44(u) (7.3) Θ2(τ) Θ4(τ) 4Θ2(4τ)Θ4(4τ) − 2 2 Θ1(u)Θ11(u)Θ12(u)Θ14(u). Θ3(τ)Θ2(τ) Θ4(τ) This identity was found by trying various characteristics on the right hand side until a set of four coeﬃcients could be found so that expansions agree to the second order at the four characteristics; the stated identity is the most concise of the identities found. Unfortunately, it seems that there are no ﬁve and six term identities for Θ3(5u|5τ) and Θ3(6u|6τ), respectively.

8 Cubic Modular Equations

Recall that a modular equation of degree p is relation between the modular quantities m1, mp, Mp and Np. In addition to the three modular equations of degree 2, 5 and 11 stated in the introduction, Ramanujan (see pg. 120-132 in [3]) gives modular equations of degrees 3, 4, 7, 9, 14, and 20. The modular equations of degree 4 and 7 read as:

4 m1/3(1 − m )1/3 m1/3 (1 − m )1/3 M + 1 1 = 1 + 1 , (8.1) 4 1/3 1/3 1/3 M4 1/3 (1 − m4) m4 (1 − m4) m4

7 m1/3(1 − m )1/3 m1/3 (1 − m )1/3 m1/6(1 − m )1/6 M + 1 1 = 1 + 1 − 3 1 1 . (8.2) 7 1/3 1/3 1/3 1/6 M7 1/3 (1 − m7) 1/6 m4 (1 − m4) m7 m7 (1 − m7) In this section we will obtain many modular equations and recover the ones derived by Ra- manujan. In fact, both (8.1) and (8.2) posses smaller reﬁnements from which the larger whole is easily deduced. As usual, let Γ(1) denote the full modular group, and set

a b Γ (n) := ∈ Γ(1) : c ≡ 0 mod n . 0 c d

12 3 η(3τ)12 12 Under the action of Γ(1), the function f(τ) := 3 η(τ)12 has three other conjugates fi(τ) . These four functions are the cubic analogues of Weber’s invariants, and we set

η(3τ) η τ η τ+4 η τ−4 f(τ) = 31/4 , f (τ) = 3−1/4 3 , f (τ) = 3−1/4 3 , f (τ) = 3−1/4 3 . η(τ) 0 η(τ) 1 η(τ) 2 η(τ) The precise eﬀect the modular group has on these functions is given in the following table.

1 τ → − t τ → τ + 1

2πi f f0 e 12 f − 2πi f0 f1 e 12 f1 2πi 2πi f1 e 12 f2 e 12 f2 − 2πi − 2πi f2 e 12 f1 e 12 f0

47 These four function are connected with the modular function

3 Θ2(τ) m(τ) = 3 Θ3(τ) through the following proposition.

Proposition 8.1. With m = m(τ),

m1/12 f(τ) = , (1 − m)1/12 1/3 1 − m1/3 f0(τ) = , 31/6m1/36(1 − m)1/12 6 6 6 6 f(τ) = f0(τ) + f1(τ) + f2(τ) , √ 3f(τ)f0(τ)f1(τ)f2(τ) = 1.

Proof. If we solve the three equations in Corollary 6.6 for η(τ), η(3τ), and η(τ/3), we obtain

−1/8 1/8 3/8 −1/8 1/24 1/8p η(τ) = 3 Θ2(τ) Θ4(τ) = 3 m (1 − m) K3(m), −3/8 3/8 1/8 −3/8 1/8 1/24p η(3τ) = 3 Θ2(τ) Θ4(τ) = 3 m (1 − m) K3(m), −1/24 1/24 1/8 1/3 (8.3) η (τ/3) = 3 Θ2(τ) Θ4(τ) (Θ3(τ) − Θ2(τ)) 1/3 −1/24 1/72 1/24 1/3 p = 3 m (1 − m) 1 − m K3(m), hence the formulas for f(τ) and f0(τ) are evident. To prove the last two, note that

2 6 6 6 6 f0(τ) + f1(τ) + f2(τ) − f(τ) and f(τ)f0(τ)f1(τ)f2(τ) are both invarient under Γ(1) and must be constant if they do not have a pole at the cusp τ = i∞. With the expansions

f(τ) = 31/4q1/12 (1 + O(q)) , −1/4 −1/36 1/3 f0(τ) = 3 q 1 + O(q ) ,

2πi −1/4 −1/36 1/3 f1(τ) = e 9 3 q 1 + O(q ) ,

−2πi −1/4 −1/36 1/3 f2(τ) = e 9 3 q 1 + O(q ) , these expressions are found to be 0 and 3−1/2, respectively.

8.1 The Modular Equations

If p is any natural number, then m1 and mp are related by a polynomial equation called the cubic modular equation of degree p (the actual degree of this polynomial is higher than p). If p is prime, the roots of the this equation with respect to mp are the p + 1 quantities

τ τ + 1 τ + p − 1 m (pτ), m , m , ... , m , p p p

48 where 3 Θ2(τ) m(τ) = 3 Θ3(τ) is the main modular function. Examples of these modular equations when p is 2 or 3 can be obtained my eliminating the parameter t from the parameterizations in Theorems 7.3 and 7.5. For many purposes, though (including the calculation of Mp and Np), this parametric form is often the most useful. We will use this method of parametrization to derive further cubic modular equations. The main obstacle to overcome in this approach is that a rational parametrization exist only when p = 2, 3, 4 and 6, for the modular function m(τ) is actually a generator for the function field of X0(3) = Γ0(3)\H, and curve defined by m1 and mp is isomorphic to X0(3p). There are four cases where the genus of X0(3p) is zero, namely p = 2, 3, 4 and 6 (see [17]). In the other cases, we must work over the function field of some non- rational curve F (t, s), and below we give useful models of X0(3p) as some simple-looking curve F (t, s) = 0 where t and s are given explicitly as products of η functions that generate the function field A0(Γ0(3p)) of X0(3p). The generators t3p when X0(3p) is rational are taken directly from [17], as these are, in some sense, the most canonical. In this table, [a] stands for η(aτ).

p g t3p s3p model of X0(3p)

3 2 [2][6]5 2 0 2 3 [3][1]5 rational 2 [9]3 3 0 3 [1]3 rational 2 3 4 0 223 [3][2] [12] rational [1]3[4][6]2 [2][3][18]2 6 0 6 [1]2[6][9] rational

2 2 3 3 2 2 5 1 3 [3] [15] [3] [5] s −1 = 3t +5t+3 [5]2[1]2 [1]3[15]3 s t 2 2 2 2 7 1 7 [7][21] [3] [7] s +1 = t +3t+7 [3][1] [1]2[21]2 s t [2][3]2[4][24]2 [2][12][3]2[8]2 s2−1 t2+4t+3 8 1 3 [12][8]2[6][1]2 [4][6][1]2[24]2 s = t 3 4 9 1 3 [9] [3] s3 = 3t2 + 3t + 1 [1]3 [1]3[9] [2][9][36] [1][36] s2+1 t2+2t+3 12 1 3 [1][4][18] [4][9] s = t

2 2 (t2−t−1)(t2+2t−4)(t2−4t+2) 10 3 2 [6][10] 5 [5][10][15][30] (s−1) = [1][15] [1][2][3][6] s 5t2(t−1)2(t−2)2 2 6 6 2 (3t2+t+1)(9t4+15t3+14t2+5t+1) 11 3 [3][33] [3] [11] (s−1) = [1][11] [1]6[33]6 s t5 2 6 6 2 (t4+t3−t2+t+1)(t5−7t4+9t3+9t2−7t+1) 13 3 [3][13] 33 [3] [39] (s+1) = [1][39] [1]6[13]6 s 27t7 2 2 [2]2[3][8][12][48] [2][3][8]2[12][48] s2−4s+1 (t −2) 16 3 [1][4][6][16][24]2 2 [1][4][6]2[16][24] s = t(t+1)(t+2) √ [9][45] [3][5][45]2 2 √ 2 15 3 3 15s −t = 1−t 1 − 3 7t+4√ t +4 [1][5] [1]2[9][15] 3s 2 t √ √ √ √ [2][6][7][21] [2][14]2[21]3 ( t2−5t+1+ t2−t+1)( t4+t3+4t2+t+1+ t4+t3+t+1) 14 5 [1][3][14][42] [1][7]2[42]3 s = 4t

49 We also have the following table of identities between the generators given above and related η products.

p η product in terms of t3p and s3p p η product in terms of t3p and s3p

[2]9[3]2[12] [2]2[3]4 4 2 [1]6[4]3[6]3 t + 2 4 3 [1]4[6]2 t + 3 [3][4]3 [4][6]7 4 4 [1]3[12] t + 4 4 6 [1]2[2][3]2[12]3 t + 6

[2]3[3] [2][9]2 6 2 [1]3[6] t + 2 6 3 [1]2[18] t + 3 [2]3[3]8[18] 2 [6]6[9] 2 6 3 [1]6[6]4[9]2 t + 3t + 3 6 12 [1]3[3]2[18]2 t + 6t + 12

[2]3[3][4]3[24] [6]2[12]2 8 [1]3[6][8]3[12] t + 1 8 3 [1][3][8][24] t + 3

[3][27]3 9 9 [1][9]3 1 + t − s

[2]6[3][12] [2][3][12][18]5 12 [1]3[4]3[6]2 t + 1 12 3 [1][4][6]2[9]2[36]2 t + 3 [2]3[3][12][18]3 [2][3][18]2[36] 12 [1][4]2[6]2[9]2[36] s + 1 12 6 [1][4][6][9]2 3s − t [2]3[3][36] [2][12][18]2 12 2 [1]2[4][6][9] t − s 12 3 [4]2[6][36] 3 − ts [2]3[12] [2]2[3][12]4[18]3 12 [4]3[6] 1 − ts 12 3 [1][4]3[6]3[9]2[36] t + 3s + 2ts [3][4][18]3 [1]2[12][18]3 12 2 [1][6][9]2[36] 2 + t − s 12 [4]2[6][9]2[36] 1 − 2s − ts

[2][3]2[5]2[30] [2]2[3][5][30]2 10 [1]2[6][10][15]2 t − 1 10 2 [1]2[6][10][15]2 t − 2

[2]3[3]2[21] st+s+t 14 [1]3[6]2[42] s−t−1

[3]2[15]2 15s−t/s 15 3 [1][5][9][45] 6 + t−1

[2]3[3][16][24]4 [2][3][16][24]2 16 [1][6]4[8]3[48] t + 1 16 2 [1][6]2[8][48] t + 2

In stating a modular equation of degree p, the associated quantities

uc = fc(τ), vc = fc(pτ), x = m(τ)1/12m(pτ)1/12, y = (1 − m(τ))1/12 (1 − m(pτ))1/12 ,

t = t3p(τ), s = s3p(τ) will be used, and any relation among these that is algebraically independent of the equations deﬁning these quantities will be accepted as a modular equation of degree p. The purpose of giving this list of generators for A0(Γ0(3p)) is revealed in the follow proposition.

Proposition 8.2. For any natural number p and any divisor d of p, the three quantities md, Md, and Nd are all Γ0(3p)-modular, hence they are rational functions of t3p and s3p when X0(3p) has non-zero genus or rational functions of t3p when X0(3p) has genus zero.

50 Proof. Consider ﬁrst the case when p is prime. We have 3 12 m1 Θ2(τ) 4 η(3τ) = 3 = 3 12 , (8.4a) 1 − m1 Θ4(τ) η(τ) 3 12 mp Θ2(pτ) 4 η(3pτ) = 3 = 3 12 , (8.4b) 1 − mp Θ4(pτ) η(pτ) 3 3 3 9 mp Mp Θ2(pτ) η(τ) η(3pτ) = 3 = 9 3 . (8.4c) m1 p Θ2(τ) η(3τ) η(pτ) In order to prove that the three quotients of η functions on the right hand sides of these identities are all Γ0(3p)-modular, we need a transformation formula for the η function (see for example, pg. 330 of [2]). This formula is aτ + b a b √ η = v cτ + d η(τ) cτ + d c d a b where ∈ Γ(1), and c d

 2  d e2πi(bd(1−c )+c(a+d)−3c)/24 : c is odd  |c| a b  2 v = c e2πi(ac(1−d )+d(b−c)+3(d−1))/24 : d is odd and c ≥ 0 or d ≥ 0 . c d |d|  2  c 2πi(ac(1−d )+d(b−c)+3(d−1))/24  − |d| e : d is odd and c < 0 and d < 0 Using this transformation formula, we ﬁnd that a b 3 a 3bp 9 aτ+b 3 v v Θ2(p ) Θ (pτ)3 3cp d c d 3pcτ+d = 2 aτ+b 3 3 9 3 Θ2( ) Θ2(τ) a 3b a bp 3pcτ+d v v cp d 3c d 3 Θ2(pτ) 2πibd(p−1) = 3 e Θ2(τ) 3 Θ2(pτ) = 3 , Θ2(τ)

From which we deduce that the left hand side of (8.4c) is Γ0(3p)-modular. Similarly, we ﬁnd 3 that the left hand sides of (8.4a) and (8.4b) are also Γ0(3p)-modular. Hence, m1, mp, and Mp 2 are all rational functions of t3p and s3p. By (7.1), Mp is rational, hence Mp is rational. Finally, by (7.2), we see that Np is rational as well. When p is not prime, we can argue by induction on the prime factors pi of p. Since Γ0(3p) ⊂ Γ0(3pi), any function that is Γ0(3pi)-modular is also Γ0(3p)-modular.

The rational representation of elements of A0(Γ0(3p)) in terms of t and s may be computed by comparing the q-series expansions of the quantities involved. The valence formula (see Theorem 4.1.4 on pg. 98 of [21]) allows us to prove that a given representation is indeed correct by verifying that a ﬁnite number of terms in the q-series expansions agree. To describe this formula, let α ∈ Q be a cusp of X0(p) and M ∈ Γ(1) with α = M(∞). Then the width of the cusp is deﬁned by 1 k width(α) = min k > 0 : M · · M −1 ∈ Γ (p) , 0 1 0

51 and the invariant order of a modular function f ∈ A0(Γ0(3p)), ord(f; α), is the exponent on q in the lowest order term in the expansion of f(M(τ)) in the variable q = e2πiτ . If f is a non-constant function in A0(Γ0(p)) with possible poles only at the cusps of X0(p) , then the valence formula says that X width(α)ord(f; α) ≤ 0, α where the sum is over all Γ(p)-nonequivalent cusps. For example, let us prove a modular equation of degree 5 from which all of the modular equations in Theorem 8.7 may be deduced. We compose the following table about the inequivalent cusp of X0(15).

12 12 cusp α width(α) ord(f(τ) ; α) ord(f(5τ) ; α) ord(t15(τ); α) ord(s15(τ); α) 1/1 15 −1/3 −1/15 −1/15 −1/15 1/3 5 1 1/5 1/5 1/5 1/5 3 −1/3 −5/3 −1/3 1/3 1/15 1 1 5 1 −1 Thus, to prove the modular equation of degree 5,

2m 1 1 = 2u12 = 3t 3t4 + 10t3 + 15t2 + 10t + 3 + t2 3t2 + 5t + 3 s + , 1 − m1 s we only need to verify that, when expanded about the cusp ∞ (Γ(15)-equivalent to 1/15) in the variable q = e2πiτ , the coeﬃcients up to and including those of the

−15 · (−1/3) − 5 · (1) − 3(−5/3) = 5th order on both sides agree. Both sides are

27q + 324q2 + 2430q3 + 13716q4 + 64557q5 + O q6 , so the proof is complete. A u − v modular equation is a relation strictly between the uc and vc that is algebraically independent of the relations given in Proposition 8.1. There are many “false” u − v cubic modular equations such as 3 3 3 3 3 3 3 3 u v = u0v0 + u1v1 + u2v2 or 3 3 3 3 3 3 3 3 2 6 6 6 6 6 6 6 6 u v0 + u0v − iu1v1 + iu2v2 = 3u v0 + 3u0v − 3u1v1 − 3u2v2 that, rather surprisingly, hold for any p. Such modular equations are really two-variable identities and cannot be independent of the relations given in Proposition 8.1. An example of a true u − v modular equation is the modular equation of degree 17 given by

2 2 2 2 2 2 2 2 2 (uv + u0v0 + u2v1 + u1v2) = 3u v + 3u0v0 + 3u2v1 + 3u1v2 − 6, (8.5) which will be obtained in Section 8.2. The u − v modular equations for p ≤ 17 take on a particularly simple form, and for cubic modular equations of prime degree greater than 17, see the thesis of Hart [14], where a method to construct an equation of the form

uk vk 1 Ω ± , ulvl ± = 0 p vk uk vlul

52 is presented. However, for large p this quickly produces intractable modular equations, whereas the goal of this section is to produce modular equations in the simplest form. For this reason, we give rational parameterizations of the modular equations constructed by Hart when they exists, and resort to general methods to construct an equation of the form

Ωp (u, v, u0, v0, u1, v1, u2, v2) = 0 for certain large primes p.

8.1.1 Modular Equations of Degrees 2, 3, 4, and 6

When p = 2, 3, 4, 6, X0(3p) has genus 0, and the coordinates m1 and mp can be parametrized in terms of the generator t3p (or “Haupftmodul”) of A0(Γ0(3p)). Theorem 8.3. The following are modular equations of degree 2.

t(t + 9)2 27 (t + 8) m = = 1 − , (8.6a) 1 (t + 6)3 (t + 6) 3 t2(t + 9) 27(t + 8)2 m2 = = 1 − , (8.6b) (t + 12)3 (t + 12)3 t + 12 M = , (8.6c) 2 t + 6 N2 = 1, (8.6d)

1/3 1/3 1/3 1/3 m1 m2 + (1 − m1) (1 − m2) = 1, (8.7a) m2/3 (1 − m )2/3 M = 1 − 1 , (8.7b) 2 1/3 1/3 m2 (1 − m2)

2 2 u1v1 2 = ζu0v0 − ζ uv, (8.8a) u2v2 2 2 u2v2 2 = ζ u0v0 − ζuv, (8.8b) u1v1 2 2 2 2 u1u2v1v2 = u v + uvu0v0 + u0v0. (8.8c)

Proof. Set u = 0 in the ﬁrst equality of Theorem 7.3 to deduce that

1/3 1/3 1/3 1/3 m1 m2 + (1 − m1) (1 − m2) = 1.

Hence for some parameter a, we have

1/3 1/3 1/3 1/3 m1 m2 = a, (1 − m1) (1 − m2) = 1 − a.

3b+2 These equations have a rational solution for m1 and m2 when a = b(b+2) given by

3b + 2 (3b + 2)2 m = , m = 1 b3 2 (b + 2)3

53 Hence, by (7.1), s m1(1 − m1) dm2 b + 2 M2 = 2 = − . m2(1 − m2) dm1 b

Care must be taken in choosing the correct square root. By (7.2), we get N2 = 1. Thus we only need to obtain the relation between t and the parameter b. By (8.3), we have

η(2τ)η(6τ)5 t = 2332 η(3τ)η(τ)5 m2/3(1 − m )1/3 = 3M 3 2 2 2 1/3 2/3 m1 (1 − m1) 2 + 3b = −3 , 1 + b hence (8.6) and (8.7) are clear. To prove (8.8), it is wisest to introduce the parameter

η(τ)η(6τ)3 t1/3 ξ(τ) := 2 = . η(2τ)η(3τ)3 (t + 9)1/3

Then by substituting (8.6a) and (8.6b) into Proposition 8.1 and solving for the uc and vc we ﬁnd that 27ξ3 27ξ6 u12 = , v12 = , (1 − ξ3)2 (8 + ξ3) (1 − ξ3) (8 + ξ3)2 i 8 i 4 2i 4 2i 8 12 1 − ζ ξ 2 + ζ ξ 12 (1 − ζ ξ) (2 + ζ ξ) ui = , vi = . 27ζiξ (1 − ξ3)2 (8 + ξ3) 27ζ4iξ2 (1 − ξ3) (8 + ξ3)2

With these parameterizations, we readily ﬁnd that

u2v2 ζ(1 − ζξ) 2 + ζ2ξ 1 1 = = ζu v − ζ2uv, √ 1/4 1/4 0 0 u2v2 3ξ1/4 (1 − ξ3) (8 + ξ3) u2v2 ζ2 1 − ζ2ξ (2 + ζξ) 2 2 = = ζ2u v − ζuv, √ 1/4 1/4 0 0 u1v1 3ξ1/4 (1 − ξ3) (8 + ξ3) and (8.8c) is obtained by multiplying these two.

Theorem 8.4. The following are modular equations of degree 3.

t t2 + 3t + 3 1 m = = 1 − , (8.9a) 1 (t + 1)3 (t + 1)3 t3 9 t2 + 3t + 3 m = = 1 − , (8.9b) 3 (t + 3)3 (t + 3)3 t + 3 M = , (8.9c) 3 t + 1 2 t2 + 3t + 3 N = , (8.9d) 3 (t + 1)(t + 3)

54 1 − m1/3 (1 − m )1/3 = 3 , (8.10a) 1 1/3 1 + 2m3 3 1/3 = 1 + 2m3 , (8.10b) M3 1/3 1/3 N3 = 1 + (1 − m1) + m3 . (8.10c)

Proof. The theta function identity

Θ3(3τ) = Θ2(3τ) + Θ4(τ) was obtained in the proof of Corollary 6.6, and it is equivalent to the modular equations

3 1 − m1/3 = 3 , 1/3 M3 (1 − m1) 1 − (1 − m )1/3 M = 1 . 3 1/3 m3 (The second is obtained by substituting τ → −1/9τ in the ﬁrst.) After solving these two equations, m1 and m3 are parametrized in terms of M3, and the rest of the calculations are easily completed.

Theorem 8.5. The following are modular equations of degree 4.

t (t + 3) 4 (t + 6) 27 (t + 2) (t + 4) m = = 1 − , (8.11a) 1 (t2 + 6t + 6) 3 (t2 + 6t + 6) 3 t2 (t + 3) 2 (t + 6) 2 27 (t + 2) 2 (t + 4) 2 m = = 1 − , (8.11b) 2 (t2 + 6t + 12) 3 (t2 + 6t + 12) 3 t4 (t + 3) (t + 6) 27 (t + 2) (t + 4) 4 m = = 1 − , (8.11c) 4 (t2 + 12t + 24) 3 (t2 + 12t + 24) 3 t2 + 12t + 24 M = , (8.11d) 4 t2 + 6t + 6 3 (t + 2) (t + 6) t2 + 6t + 12 N = , (8.11e) 4 (t2 + 6t + 6) (t2 + 12t + 24)

M + 2 m1/3 (1 − m )1/3 M 4 = 1 + 1 , (8.12a) 4 1/3 1/3 4 − M4 m4 (1 − m4) M − 1 m1/3(1 − m )1/3 M 2 4 = 2 1 1 , (8.12b) 4 1/3 4 − M4 1/3 m4 (1 − m4) R S R + S = 4 + + , (8.12c) S R where m1/3 (1 − m )1/3 R = 2 , S = 2 . 1/3 1/3 1/3 1/3 m1 m4 (1 − m1) (1 − m4)

55 Proof. The parameterizations of the modular quantities may be obtained by using the identities

t6(τ) = t12(t12 + 6), (8.13a) 2 t12 t6(2τ) = , (8.13b) t12 + 2 which are derived from the the table of η product identities, and composing the modular equation of degree 2 with itself. For example,

2 4 t6(t6 + 9) t (t + 3) (t + 6) m1 = 3 = 2 3 , (t6 + 6) (t + 6t + 6) 2 2 2 2 t6(t6 + 9) t (t + 3) (t + 6) m2 = 3 = 2 3 , (t6 + 12) (t + 6t + 12) 2 4 t6(2τ) (t6(2τ) + 9) t (t + 3) (t + 6) m4 = 3 = 2 3 . (t6(2τ) + 12) (t + 12t + 24) We then obtain t2 + 6t + 6 t2 + 12t + 24 R = , t(t + 3) (t2 + 6t + 12) t2 + 6t + 6 t2 + 12t + 24 S = , 3(t + 4) (t2 + 6t + 12) and three equations of (8.12) become identities in t.

At this point we can prove Ramanujan’s modular equation (8.1) by noting that by (8.12), both sides reduce to the same rational function of M4. Theorem 8.6. The following are modular equations of degree 6. 27 (t + 2) 3 m = 1 − , (8.14a) 1 (t3 + 6t2 + 12t + 6) 3 27 (t + 2) 6 m = 1 − , (8.14b) 2 (t3 + 6t2 + 12t + 12) 3 t3 (t + 3) 6 m = , (8.14c) 3 (t3 + 6t2 + 18t + 18) 3 t6 (t + 3) 3 m = , (8.14d) 6 (t3 + 12t2 + 36t + 36) 3 t3 + 12t2 + 36t + 36 M = , (8.14e) 6 t3 + 6t2 + 12t + 6 5t6 + 66t5 + 378t4 + 1206t3 + 2268t2 + 2376t + 1080 N = , (8.14f) 6 (t3 + 6t2 + 12t + 6) (t3 + 12t2 + 36t + 36)

P 2 1 − 1 = S + , (8.15a) 3 S 1 P 2 + 1 = Q + , (8.15b) Q 3 1 P − = R + . (8.15c) P R

56 where

m m (1 − m ) (1 − m )1/12 P = 1 2 3 6 , (1 − m1) (1 − m2) m3m6 m (1 − m ) m (1 − m )1/4 Q = 1 2 3 6 , (1 − m1) m2 (1 − m3) m6 m (1 − m ) (1 − m ) m 1/4 R = 1 2 3 6 , (1 − m1) m2m3 (1 − m6) m m m m 1/12 S = 1 2 3 6 , (1 − m1) (1 − m2) (1 − m3) (1 − m6) Proof. The fundamental relations derived from the table of η product identities that are needed when composing the modular equations of degree 2 and 3 are

2 t6 = t18 t18 + 6t18 + 12 , 2 t18 (t18 + 3) 3t9 = . t18 + 2 It turns out that P , Q, R and S all have simple expressions in terms of t, i.e.,

P t2 + 6t + 12 = , R t(t + 2) (t + 2)(t + 3) Q = , t t2 + 3t + 3 PR = , t + 3 t(t + 3) S = . 3(t + 2) With these parameterizations, the three equations in (8.15) are readily veriﬁed.

8.1.2 Modular Equations of Degrees 5, 7, and 8

When p = 5, 7, 8, 9, or 12, X0(3p) has genus 1, and the coordinates m1 and mp can be parametrized rationally terms of the t3p and a square root of a quartic polynomial in t3p −1 (that is, s3p ± s3p ). All parameterizations in this section were obtained by comparing q-series coeﬃcients. Theorem 8.7. The following are modular equations of degree 5.

t6 − 1 t2 3t2 + 5t + 3 1 2m1 − 1 = + s + , (8.16a) (t2 + 3t + 1)3 (t2 + 3t + 1)3 s t6 − 1 t2 3t2 + 5t + 3 1 2m5 − 1 = − s + , (8.16b) (t2 + 3t + 1)3 (t2 + 3t + 1)3 s 2 − 2t2 t 1 M = + s + , (8.16c) 5 t2 + 3t + 1 t2 + 3t + 1 s 2t2 + 3t + 2 N = 2 , (8.16d) 5 t2 + 3t + 1

57 x4 + 3x2y2 + y4 = 1, (8.17a) u3 v3 3 3 − = − 4 = 3u2v2 + + 5, (8.17b) v3 u3 x2y2 u2v2 5 4 4 M5 − = 4y − 4x , (8.17c) M5 3 3 5 2 2 u v M5 + = 2x y 3 + 3 , (8.17d) M5 v u 4 4 N5 = 2 + 2x + 2y , (8.17e)

Proof. We will ﬁrst prove the following modular equations from which the rest may easily be derived.

u2v2 = t, (8.18a) u3 1 1 2 = 5 + 3 t + + s + , (8.18b) v3 t s v3 1 1 −2 = 5 + 3 t + − s + , (8.18c) u3 t s

Equation (8.18a) follows from the deﬁnition of u, v, and t. As before, we have the following table about the inequivalent cusp of X0(15). 12 12 cusp α width(α) ord(u ; α) ord(v ; α) ord(t15(τ); α) ord(s15(τ); α) 1/1 15 −1/3 −1/15 −1/15 −1/15 1/3 5 1 1/5 1/5 1/5 1/5 3 −1/3 −5/3 −1/3 1/3 1/15 1 1 5 1 −1 From this table, we have the following table of the invariant orders of the quantities in (8.18b). u3 1 1 cusp α width(α) ord(−2 v3 + 5 + 3 t + t + s + s ; α) 1/1 15 ≥ −1/15 1/3 5 ≥ −1/5 1/5 3 ≥ −1/3 1/15 1 n The valence formula says that

1 · (n) + 3 · (−1/3) + 5 · (−1/5) + 15 · (−1/15) ≤ 0, unless (8.18b) is true. We have veriﬁed though q-series expansions that n > 3 and so the proof of (8.18b) is complete. The proof of (8.18c) is exactly the same. After solving these three 12 12 parametrizations for u and v we obtain the parametrizations for m1 and m5. The formulas for M5 and N5 may be obtained with (7.1) and (7.2), respectively.

58 Theorem 8.8. The following are modular equations of degree 7. t t3 + t2 + 5t − 1 1 2m1 − 1 = s − , (8.19a) (t + 1)3 (t2 + t + 7) s t t3 − 35t2 − 49t − 343 1 2m7 − 1 = s − , (8.19b) (t + 7)3 (t2 + t + 7) s t + 7 M = , (8.19c) 7 t + 1 6t 1 N = s − , (8.19d) 7 t2 + 8t + 7 s

u2 v2 u2 v2 √ 1 − + − 7 = 3 3 u3v3 + , (8.20a) v2 u2 v2 u2 u3v3 3 u v 2 u v 2/3 = − + 7 − , (8.20b) x2y2 v u v u !1/3 √ m7/12(1 − m )5/12 m5/12(1 − m )7/12 M = 1 + 2 3 1 1 − 1 1 , (8.20c) 7 1/12 −1/12 −1/12 1/12 m7 (1 − m7) m7 (1 − m7) 2/3 7 2 2 u v M7 + = 8 − 12x y − , (8.20d) M7 v u 2 1/3 1/3 M7 5M − 4M7 + 35 m (1 − m ) 7 = 1 + 1 , (8.20e) 2 1/3 1/3 (7 − M ) (1 − m7) 7 m7 M − 1 m1/6 (1 − m )1/6 M 7 = 1 1 , (8.20f) 7 1/6 7 − M7 1/6 m7 (1 − m7) 1 7 7 m1/3m2/3 + (1 − m ) 2/3 (1 − m ) 1/3 − 1 (M − 1) M − = 1 7 7 1 , (8.20g) 24 M 7 7 M 2/3 1/3 1/3 2/3 7 7 −m1 m7 − (1 − m7) (1 − m1) u v u v 5/3 N = 2x2y2 + − , (8.20h) 7 v u v u Proof. We proceed as in the degree 5 case. Here, the fundamental relation needed is √ 1 1 1 3 3 u3v3 + = s + − 7 s − , (8.21) u3v3 s s which we prove by comparing q-series coeﬃcients. As before, from the table √ 3 3 3 3 1 1 1 cusp α width(α) ord(u v ; α) ord(s; α) ord(3 3 u v + u3v3 − s + s − 7 s − s ; α) 1/1 21 -2/21 -1/21 ≥ −2/21 1/3 7 2/7 1/7 ≥ −2/7 1/7 3 -2/3 1/3 ≥ −2/3 1/21 1 equation (8.21) is true if both sides agree up to the 6th order when expand in the variable q = exp 2πiτ. This is indeed the case, so the proof is complete. By combining (8.21) with the trivial equality u2 = s v2

59 we can solve for u12 and v12 as rational functions of t and s and ultimately deduce the parametrizations of m1 and m7, from which M7 and N7 are found by (7.1) and (7.2), respectively. The rest of the modular equations then reduce to identities in the parametrizating variable t. For example, both sides of (8.20d) are 36t 8 − . (t + 1)(t + 7)

Ramanujan’s modular equation (8.2) is now seen to easily follow by combining (8.20e) and (8.20f) of this theorem. Note that (8.20f) and (8.20g) are equivalent to the theta function identities 7Θ (7τ) − Θ (τ) Θ (τ)Θ (τ) 1/2 3 3 = 2 4 Θ3(τ) − Θ3(7τ) Θ2(7τ)Θ4(7τ) and 1 (Θ (τ) − Θ (7τ)) (7Θ (7τ) − Θ (τ)) 7Θ (7τ)2 − Θ (τ)2 24 3 3 3 3 3 3 2 2 2 2 = Θ2(τ)Θ2(7τ) Θ3(τ) − Θ2(τ) Θ2(7τ)Θ3(7τ) + Θ4(τ)Θ4(7τ) Θ3(τ) − Θ4(τ) Θ4(7τ)Θ3(7τ) of which a direct proofs would be of interest.

Theorem 8.9. The following are modular equations of degree 8.

t3 (t + 3) m1/3m1/3 = , (8.22a) 1 8 t4 + 12t3 + 36t2 + 36t + 9 9 (t + 1) (1 − m )1/3(1 − m )1/3 = , (8.22b) 1 8 t4 + 12t3 + 36t2 + 36t + 9 t2(t + 3)2 m1/3m1/3 = , (8.22c) 2 4 t4 + 6t3 + 18t2 + 18t + 9 9(t + 1)2 (1 − m )1/3(1 − m )1/3 = , (8.22d) 2 4 t4 + 6t3 + 18t2 + 18t + 9 8 −7t4 − 30t3 + 90t + 63 M8 − = 4 3 2 , (8.22e) M8 t + 12t + 36t + 36t + 9 7t4 + 48t3 + 126t2 + 144t + 63 N = , (8.22f) 8 t4 + 12t3 + 36t2 + 36t + 9

4 12 34 x4 + y4 − 1 2x4 + 2y4 + − = 18x4y4 − , (8.23a) 4 2 9x4y4 P 2 Q2 √ 1 − = 3 Q − , (8.23b) Q2 P 2 Q 8 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 − M8 = 4m1 m8 − 4(1 − m1) (1 − m8) + 3m2 m4 − 3(1 − m2) (1 − m4) , M8 (8.23c) 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 N8 = 2 + 2m1 m8 + 2(1 − m1) (1 − m8) + 3m2 m4 + 3(1 − m2) (1 − m4) , (8.23d)

60 where m m 1/12 P = 1 8 , (1 − m1) (1 − m8) m m 1/12 Q = 2 4 . (1 − m2) (1 − m4) Proof. These modular equations are obtained from the modular equations of degree 4. We ﬁrst need the relations

t12(τ) = s24t24 − 3, (8.24a)

(s24 − 4)t24 − 3 t12(2τ) = . (8.24b) 1 + t24 The ﬁrst one follows from √ f(τ)4 s t = 3 24 24 f(2τ)2 √ = t6 + 9

= t12 + 3, by (8.13a), and the second one follows via (8.13b) from p t12(2τ) = t6(2τ) + 9 − 3 s (t + 3) (t + 6) = 12 12 − 3 t12 + 2 s s t (s t + 3) = 24 24 24 24 − 3 s24t24 − 1 s t − t = 24 24 24 − 3, t24 + 1 where the defining relation 1 3 s24 − = t24 + + 4 s24 t24 is needed to obtain the final simplification. We then have by (8.11b), (8.11c), and finally (8.24a) that 2 1/3 1/3 t12(τ) (t12(τ) + 3) (t12(τ) + 6) m2 m4 = 2 2 (t12(τ) + 6t12(τ) + 12) (t12(τ) + 12t12(τ) + 24) t2(t + 3)2 = t4 + 6t3 + 18t2 + 18t + 9 Fortunately, s disappears when this expression is simplified. Similarly, by (8.11a), (8.11b), and (8.11c) 2 1/3 1/3 1/3 1/3 t12(τ)(t12(τ) + 3) (t12(τ) + 6) m1 m2 m4 m8 = 2 2 (t12(τ) + 6t12(τ) + 6) (t12(τ) + 6t12(τ) + 12) 2 t12(2τ) (t12(2τ) + 3) (t12(2τ) + 6) · 2 2 (t12(2τ) + 6t12(2τ) + 12) (t12(2τ) + 12t12(2τ) + 24) t3 (t + 3) t2(t + 3)2 = · . t4 + 12t3 + 36t2 + 36t + 9 t4 + 6t3 + 18t2 + 18t + 9

61 Hence, (8.22a) and (8.22c) are clear. Equations (8.22b) and (8.22d) are obtained from (8.22a) and (8.22c) by the substitution τ → −1/24τ, which sends t24 to 3/t24. Finally, to obtain the expressions for the multipliers, we solved (8.22a) and (8.22b) for m1 and m8 rationally in terms of t and s and applied (7.1) and (7.2). By good fortune, the two quantities 8 − M M8 8 and N8 where rational in t only. The rest of the stated modular equations are then identities in the variable t.

Ramanujan has derived a modular equation of degree 8 equivalent to (8.30a). This modular equation is (see pg. 132 of [3])

P 4 = RP (5P + 9T ) + 2R2, where

1/3 1/3 1/3 1/3 P = 1 − m1 m8 − (1 − m1) (1 − m8) 1/3 1/3 1/3 1/3 T = m1 m8 + (1 − m1) (1 − m8) 1/3 1/3 1/3 1/3 R = 9m1 m8 (1 − m1) (1 − m8) . Since, in the notation of Theorem 8.9, 9t(t + 1)(t + 3) P = , t4 + 12t3 + 36t2 + 36t + 9 t4 + 3t3 + 9t + 9 T = , t4 + 12t3 + 36t2 + 36t + 9 81t3(t + 1)(t + 3) R = , (t4 + 12t3 + 36t2 + 36t + 9)2 this modular equations is easily veriﬁed to be an identity in t.

8.1.3 Modular Equations of degrees 11, 13, and 14

The degrees 11, and 13 are the interesting cases were X0(3p) is hyperelliptic of genus 3. We also compose the modular equations of degrees 2 and 7 to deduce the modular equation of degree 14 given by Ramanujan (see pg. 129 of [3]). Here we state the parameterizations that come out cleanly in terms of t only. As usual, these parameterizations were obtained by comparing q-series expansions.

Theorem 8.10. The following are modular equations of degree 11.

√ u6 uv = 3t, = s, (8.25a) v6 9t4 + 1 x4 + y4 = , (8.25b) 9t4 + 27t3 + 18t2 + 9t + 1 3t2 x2y2 = , (8.25c) 9t4 + 27t3 + 18t2 + 9t + 1 11 3t2 − 1 15t2 + 9t + 5 − M11 = 2 4 3 2 , (8.25d) M11 9t + 27t + 18t + 9t + 1 45t4 + 54t3 + 45t2 + 18t + 5 N = 2 . (8.25e) 11 9t4 + 27t3 + 18t2 + 9t + 1

62 √ 2 4x2y2 + 3 3xy x2 + y2 + x2 + y2 = 1, (8.26a) √ 11 2 2 2 2 M11 − = 2 y − x 5x + 3 3xy + 5y , (8.26b) M11 4 2 2 4 N11 = 4 + 6(x + x y + y ), (8.26c) 3 Proof. We begin by noting that x2y2 has possible poles only at the cusps of Γ0(33), so we compute the invarient order of it along with t. cusp α width(α) ord(t; α) ord(3/x2y2; α) 1/1 33 -2/33 -1/33 1/3 11 -2/11 1/11 1/11 3 -2/3 -1/3 1/33 1 -2 1 Thus, in order to prove (8.25c), we need to show that the reciprocals of both sides argee to the th 2 2 6 order. Actually, since we haven’t shown that x y is Γ0(33)-modular, but we know that it’s sixth power surely is, we checked 3 6 9t4 + 27t3 + 18t2 + 9t + 16 = x2y2 t2 and found that both sides agree up to the 36th order in q = exp 2πiτ. Next, 1 x4 + y4 = x2y2 u2v2 + u2v2 9t4 + 1 = , 9t4 + 27t3 + 18t2 + 9t + 1 and p x2 + y2 = x4 + y4 + 2x2y2 3t2 + 1 = √ , 9t4 + 27t3 + 18t2 + 9t + 1 so Ramanujan’s modular equation (8.26a) is clear. The multipliers can be found as usual by solving (8.25a) for m1 and m11 rationally in terms of t and s and applying (7.1) and (7.2). Theorem 8.11. The following are modular equations of degree 13. u = t, u6v6 = s, (8.27a) v 27t7 x6y6 = , (8.27b) (t + 1)2(t4 + 4t3 − 7t2 + 4t + 1)3 13 7t4 − 8t3 + 5t2 − 8t + 7 M13 + = 2 4 3 2 , (8.27c) M13 t + 4t − 7t + 4t + 1 √ √ ! √ √ 3 u v 4 u v 2/3 = √ + √ − 13 √ + √ , (8.28a) x2y2 v u v u 13 2x2y2 M + = 7u4 − 8u3v + 5u2v2 − 8uv3 + 7v4 (u + v)2/3, (8.28b) 13 7/3 7/3 M13 3u v 2x2y2 N = 2u2 − 3uv + 2v2 (u − v)(u + v)5/3. (8.28c) 13 u7/3v7/3

63 Proof. The derivations of these modular equations bear many similarities to those of Theorem 8.10, so little detail will be given. In order to verify (8.27b) we ﬁnd that we have to verify that the reciprocals of both sides argee to the 21st order in q = exp 2πiτ. We then solve for 12 12 u and v rationally in terms of t and s to get parameterizations for m1 and m13, at which point (7.1) and (7.2) may be used to ﬁnd the multipliers.

Theorem 8.12. The following are modular equations of degree 14.

9t2 m1/3m1/3 + (1 − m )1/3(1 − m )1/3 = 1 − , (8.29a) 1 14 1 14 t3 + 5t2 − t + 1 9t m1/3m1/3 + (1 − m )1/3(1 − m )1/3 = 1 − , (8.29b) 2 7 2 7 t3 − t2 + 5t + 1 9t3 m1/3m1/3(1 − m )1/3(1 − m )1/3 = , (8.29c) 1 14 1 14 (t2 − t + 1) (t3 + 5t2 − t + 1) 2 9t5 m1/3m1/3(1 − m )1/3(1 − m )1/3 = , (8.29d) 2 7 2 7 (t2 − t + 1) (t3 − t2 + 5t + 1) 2 3 2 M14 t + 5t − t + 1 = 3 2 , (8.29e) M2M7 t − t + 5t + 1

1/3 1/3 1/3 1/3 1 M (1 − m2) (1 − m7) + m m + 14 = 2 7 2 , (8.30a) 1/3 1/3 M2M7 1/3 1/3 1 (1 − m1) (1 − m14) + m1 m14 + 2 M 2 (1 − m )1/6m1/6(1 − m )1/6m1/6 (1 − m )1/3(1 − m )1/3 + m1/3m1/3 − 1 14 = 2 2 7 7 × 2 7 2 7 . 1/6 1/6 1/3 1/3 M2M7 1/6 1/6 1/3 1/3 (1 − m1) m1 (1 − m14) m14 (1 − m1) (1 − m14) + m1 m14 − 1 (8.30b)

1/3 1/3 1/3 1/3 Proof. We ﬁrst show that m1 m14 and m2 m7 are Γ0(42)-modular. The modular equations 1/3 m7 of degree 7 imply that 1/3 is Γ0(21)-modular. Speciﬁcally, the solution of (8.20e) and (8.20f) m1 1/3 1/3 m7 m14 for 1/3 is rational in t21 and s21. Hence, 1/3 is Γ0(42)-modular as well. A direct consequence m1 m2 1/3 1/3 1/3 1/3 of the modular equation of degree two is the fact that m1 m2 and m7 m14 are Γ0(42)- modular. By

1/3 1/3 m7 m2 (m1)(m14) m1/3 m1/3 m1/3m1/3 = 1 14 , 1 14 1/3 1/3 1/3 1/3 m1 m2 m7 m14

1/3 1/3 m1 m14 (m2)(m7) m1/3 m1/3 m1/3m1/3 = 7 2 , 2 7 1/3 1/3 1/3 1/3 m1 m2 m7 m14

1/3 1/3 1/3 1/3 1/3 1/3 we see that m1 m14 and m2 m7 are Γ0(42)-modular. The quantities (1−m1) (1−m14) 1/3 1/3 and (1 − m2) (1 − m7) are also Γ0(42)-modular as the substitution t → −1/42τ has the eﬀect (t, s) → (t, s/t2) on the parameterizing variables. At this point, we immediately know that the left hand sides of (8.29a) through (8.29e) are rational in t alone since any element of

64 A0(Γ0(42)) can be written uniquely as

3 X i Ri(t)s i=0 and the invariance of this expression under the substitution τ → −1/42τ is equivalent to the vanishing of R1 through R3. Now, (8.29a), (8.29c), and (8.29e) are equivalent to the representations

!2 2 m1/6m1/6 (1 − m )1/6(1 − m )1/6 t2 − t + 1 t3 − 4t2 − t + 1 1 14 + 1 14 = , 1/6 1/6 1/6 1/6 3 (1 − m1) (1 − m14) 9t m1 m14 2 1 t2 − t + 1 t3 − t2 + 5t + 1 = , 1/3 1/3 1/3 1/3 9t3 m1 m14 (1 − m1) (1 − m14) Θ2(τ)Θ2(14τ) Θ4(τ)Θ4(14τ) 1 = 2 . Θ2(2τ)Θ2(7τ) Θ4(2τ)Θ4(7τ) t By computing the invariant orders of the quantities involved at the eight cusps 1/1, 1/2, 1/3, 1/6, 1/7, 1/14, 1/21, 1/42 of Γ0(42) we find that we need to verify the first two parameterizations to the 35th order in q = exp 2πiτ. They do indeed agree up to the 35th order, and the third parameterization follows directly from the definition of t42 and Corollary 6.6. The same argument may be used to deduce the pair (8.29b) and (8.29d), and we find that the parameterizations are identical except that t has been replaced by 1/t. We have been unable to find an explanation for this curious property.

Equation (8.30a) was given by Ramanujan (pg. 129 of [3]). If Ramanujan were aware of the parameterizations (8.29a) through (8.29e) he surely would have deduced the analogous modular equation (8.30a). Thus, his methods still remain elusive.

8.1.4 Modular Equations of Higher Degrees For the three primes 17, 23, and 29 in arithmetic progression, the modular equations constructed by Hart [14] and Chan and W.-C. Liaw [8] are actually rational. The modular equations constructed by Hart have the form

uk vk 1 Ω ± , ulvl ± = 0, p vk uk vlul whereas Chan and W.-C. Liaw present a method to derive a modular equation of the form

k k l l Ψp x + y , x y = 0.

It is our good fortune that both Ωp and Ψp are rational for these three primes. Thus, by working out the rational parameterizations, we can simplify the modular equations tremen- dously to a tractable form. The rational parameterizations of these modular equations along with simpliﬁed forms deduced from these parameterizations are given here, and the massive calculations required in calculating these parameterizations were performed with Mathematica.

65 Theorem 8.13. The modular equation of degree 17 can be parametrized as 2 2 (t − 3) (t + 1) x2 + y2 = , t3 + 4t2 − 6t + 3 3 x2y2 = , t3 + 4t2 − 6t + 3 u3 v3 − = (t − 1) t3 + 2t − 1 , v3 u3 1 2 3 uv + = (t − 3)2(t + 1), uv and the multipliers satisfy 3 3 17 2 2 3t − 1 u v M17 + = 2x y 2 3 + 3 , M17 t + 1 v u 17 4t − 3 4 4 M17 − = 4 y − x , M17 t − 3 4 4 N17 = −2 − 2 x + y − 1 t. Therefore, we have the following modular equations of degree 17. x4 − 9x2y2 + y4 − 12 x2 + y2 2 u3 v3 + 3 = 17 + − , 3x2y2 xy v3 u3 8x2y2 − 3 p4 − 4x4 + 3x2y2 − 4y4 6 = 11 + √ . x4 + 3x2y2 + y4 − 1 3xy Proof. Setting u3 v3 A = − , v3 u3 1 2 B = 3 uv + , uv we ﬁnd by comparing q-series expansions that −A3 +17A2B +136A2 −34AB2 +850AB −5120A+B4 −58B3 +1217B2 −10632B +32768 = 0. (8.31) As (A, B) = (4, 5) is a singular point on this curve, we introduce the parameter T by A = 4 + T (B − 5) When this substitution is performed in (8.31), the resulting equation has (beside the double t2+3 solution at 5) two other solutions for B. These two solution will be rational when T = t−4 . This gives the parametrizations exactly as stated. Also, setting 2 C = x2 + y2 , D = x2y2, we obtain C3 + 90C2D − 3C2 − 615CD2 + 117CD + 3C + 1024D3 − 384D2 + 36D − 1 = 0. This singular cubic may be parametrized as just done, and the equality of the two parameterizing variables follows from B = 3C/D.

66 Theorem 8.14. The modular equation of degree 23 may be parametrized as

2 2 2 t − t − 3 x2 + y2 = , t4 + 7t3 + 4t2 − 3t + 3 3 x2y2 = , t4 + 7t3 + 4t2 − 3t + 3 3 3 2 u v 2 − = (t + 1)2 t3 − t + 1 t3 + 2t2 + 2t − 1 , v3 u3 √ 1 3 uv + = t2 − t − 3. uv

Therefore, we have the following modular equation of degree 23.

√ 2 44x2y2 + 15 3xy x2 + y2 + 2 x2 + y2 = √ √ √ q √ √ 2 − 3 xy 3x2 + 4xy + 3y2 4 3x2 + 13xy + 4 3y2.

Proof. Setting

u3 v3 2 A = − , v3 u3 √ 1 B = 3 uv + , uv 2 C = x2 + y2 , D = x2y2, we ﬁnd that

−A2 + 23AB5 + 736AB4 + 8004AB3 + 39698AB2 + 92736AB + 83072A + B11 + 23B10 + 266B9 + 2001B8 +10585B7 + 40457B6 + 111196B5 + 210703B4 + 252096B3 + 161920B2 + 36864B − 4096 = 0, and

C4 − 614C3D − 4C3 + 4857C2D2 − 804C2D + 6C2 − 9824CD3 +3624CD2 − 681CD − 4C + 256D4 − 1408D3 + 1968D2 − 88D + 1 = 0, and

B2 = 3C/D.

The ﬁrst equation is a quadratic equation in A; when B has the indicated parametrization, the solution for A is rational. The next equation is a singular quartic and may be parametrized with the method used in the previous proof.

67 Theorem 8.15. The modular equation of degree 29 may be parametrized as

2 3 2 2 t t − 2t − t − 1 x2 − y2 = , t5 + 7t4 + 8t3 + 17t2 + 9t + 6 3 x2y2 = , t5 + 7t4 + 8t3 + 17t2 + 9t + 6 u3 v3 − = t2 + 1 t2 + t + 2 t3 + 2t2 + 4t + 1 , v3 u3 1 2 3 uv − = t2 t3 − 2t2 − t − 1 , uv 1 2 3 uv + = (t − 2)2 t3 + 2t2 + 3t + 3 . uv Proof. Setting u3 v3 A = − , v3 u3 1 2 B = 3 uv − , uv 2 C = x2 − y2 , D = x2y2, we ﬁnd that

−A5 + 116A4B + 3364A4 − 2871A3B2 + 43094A3B − 13519A3 + 29A2B4 +10614A2B3 − 54520A2B2 + 191226A2B + 27115A2 − 58AB5 − 9744AB4 −57420AB3 + 134792AB2 + 142390AB − 53800A + B7 + 42B6 + 2027B5 + 28696B4 +207239B3 + 430370B2 + 762925B + 53500 = 0, and

C5 + 3712C4D − 5C4 + 19198C3D2 + 6617C3D + 10C3 + 33524C2D3 +20085C2D2 + 6477C2D − 10C2 + 1769CD4 − 9781CD3 + 3324CD2 + 2648CD +5C + 116D5 − 716D4 + 1529D3 − 1054D2 + 229D − 1 = 0, and

B = 3C/D.

It is wisest to parametrize the C −D equation ﬁrst. This is a quintic equation with six singular points and a rational point (C,D) = (1, 0). Consider a general cubic

X i j aijC D = 0. (8.32) i+j≤3

There are 8 = 32 − 1 degrees of freedom in choosing this curve. If we impose the constraint that this cubic passes through the six singular points of the quintic and the rational point (1, 0), we have one more degree of freedom. Hence if we let t denote one of the coeﬃcients of the cubic, we can solve the cubic-quintic system for C and D rationally in terms of t.

68 8.2 u − v Modular Equations Besides the modular equations derived in the last few sections, there are u − v modular equations of degrees 5, 7 and 11 that assume very simple forms. These are: degree equation 2 2 2 2 5 u1v1u2v2 = 1 + u v + uvu0v0 + u0v0 2 uvu0v0 − 1 = (u1v2 − u2v1) ζ u1v2 − ζu2v1 2 2 2 2 2 7 ζu2v1 − ζ u1v2 = iu v0 + iuvu0v0 + iu0v √ √ 2 2 2 2 11 u1v1u2v2 = 2 + 3uv + 3u0v0 + u v + uvu0v0 + u0v0 2 uvu0v0 − 1 = (u2v1 − u1v2 − i) ζ u2v1 − ζu1v2 + i We ﬁrst brieﬂy indicate how these may be proven, as most of the detail lie in tedious but straightforward calculations, and the same techniques apply in deriving u−v modular equations of larger degrees. It must be noted that these modular equations were initially found by comparing the q-series expansions of the quantities uc and vc up to a very large order, (typically around the 100th order). For each prime p > 3, let

(∞) vc = fc(pτ) (= vc) and for 0 ≤ j < p, let τ + 36i v(j) = f (8.33) c c p

(j) 12 where i is the smallest positive integer with 36i ≡ j mod p. The (vc ) are permuted by Γ(1) according to

(j) 12 (j+1) 12 (j) 12 (−1/j) 12 (vc ) (τ + 1) = (vc+p ) (τ), and (vc ) (−1/τ) = (v−1/c ) (τ)

th (j) The exact 12 root of unity involved in permuting the vc is what requires tedious calculation. Now let G denote the subgroup of Γ(1) generated by τ → τ + 3 and τ → −1/τ. This is a congruence subgroup of level 3 with two cusps at ∞ and 1, each with width 3. Let

(c) 2 (c) 2 (c) (c) 2 (c) 2 (c) (c) F5 = 1 + u (v ) + uv u0v0 + u0(v0 ) − u1v1 u2v2 , 3  (∞) 2 (∞) 3 ζu v − ζ u v : c = ∞ (c) 2 (c) 2 (c) (c) 2 (c) 2  2 1 1 2 F7 = iu (v0 ) + iuv u0v0 + iu0(v ) − 3 , (c) 2 (c)  ζu1v2 − ζ u2v1 : c 6= ∞ √ √ (c) (c) (c) 2 (c) 2 (c) (c) 2 (c) 2 (c) (c) F11 = 2 + 3uv + 3u0v0 + u (v ) + uv u0v0 + u0(v0 ) − u1v1 u2v2 .

(c) We have veriﬁed that, for each p = 5, 7, or 11, the p + 1 quantities Fp are permuted by G, That is, (c) (c+3) (c) (−1/c) Fp (τ + 3) = Fp (τ), and Fp (−1/τ) = Fp (τ) (8.34) It follows that Y (c) Fp ∈ A0(G). c (c) Since the only possible poles of each Fp are at the cusps ∞ and 1, we will have shown that (c) (c) (c) each Fp vanishes if all Fp vanish at these two cusps. So we need to verify that when Fp (τ)

69 (c) 1 2πiτ and Fp (1 − τ ) are expanded in fractional powers of the variable q = e , only positive powers of q appear. We have checked this and have found that indeed this is the case. The (∞) (∞) fact that F5 and F11 vanish gives the modular equations of degrees 5 and 11 immediately. (∞) The modular equation of degree 7 is the vanishing factor of F7 , as q-series expansions show (∞) that the other two factors of F7 do not vanish identically. The modular equations just obtained are notable for their extreme simplicity and the fact that the total degree of the terms involved never exceeds four. However, the proof of these modular equations is anything but enlightening, and their existence seems to be a singularity stemming from the mysterious vanishing of q-series coeﬃcients. It would be nice to have a general method to easily proof relatively simple modular equations for any prime p. Unfor- tunately, we have only been able to ﬁnd such a modular equation for primes p with p ≡ ±1 mod 18. For p ≡ 1 mod 18, this modular equation involves the quantities

!i !i !i u i u u u Y (j) = + 0 + 1 + 2 , i v(j) (j) (j) (j) v0 v1 v2 while for p ≡ −1 mod 18, this modular equation involves the quantities

!i !i !i !i uv(j) u v(j) u v(j) u v(j) X(j) = √ + 0√0 + 2√1 + 1√2 . (8.35) i 3 3 3 3

(∞) (∞) Setting Xi := Xi and Yi := Yi , we immediately obtain the false modular equations

2 2 27 X1 − X2 = X−1 − X−2, 2 2 Y1 − Y2 = Y−1 − Y−2, that is, they constitute two-variable identities. Indeed, through the follow proposition, we have calculated that both of these identities remain true for two distinct primes p. It follows that they cannot be algebraically independent of the equations deﬁning the quantities uc and vc, hence they remain true for all p.

Proposition 8.16. Let p be a prime with p ≡ −1 mod 18 and let Xi be deﬁned by (8.35). Let P (X−2,X−1,X1,X2) be a polynomial with

p+1 p+1 P (X−2,X−1,X1,X2) = P (X−2, (−1) 18 X−1, (−1) 18 X1,X2)

(∞) (∞) (∞) (∞) (∞) (∞) (∞) (∞) and with P (X−2 ,X−1 ,X1 ,X2 ) having a zero the cusp τ = ∞. Then P (X−2 ,X−1 ,X1 ,X2 ) = 0 identically.

(c) Proof. Calculations show that the Xi satisfy the transformations: ( (c) (−1/c) 1 : c = 0, ∞ Xi (−1/τ) = Xi (τ) × i p+1 , (−1) 18 : c 6= 0, ∞ (8.36) (c) (c+1) p+1 i 18 Xi (τ + 1) = Xi (τ) × (−1) ,

(∞) (0) Xi (τ/p) = Xi (τ). (8.37)

70 The relations in (8.36) imply that

Y (c) (c) (c) (c) P (X−2,X−1,X1 ,X2 ) ∈ A0(Γ(1)). c

(c) (c) (c) (c) Thus we need to verify that each P (X−2,X−1,X1 ,X2 ) has a zero at the cusp ∞. By the (0) (0) (0) (0) hypothesis, this has a zero for c = ∞. By (8.37), we see that P (X−2 ,X−1 ,X1 ,X2 ) also has a zero at the cusp ∞. Finally, by the second equation of (8.36), we see that all other (c) (c) (c) (c) P (X−2,X−1,X1 ,X2 ) have a zero at the cusp ∞. This completes the proof. The modular equations thus obtained are, for example: degree equation(s) 2 17 X−1 = X1 X1 − 4 2 X1 = 3X2 − 2 2 2 2 2 19 Y1 − 3Y2 + 36 Y−1 + 58Y1Y−1 + 3Y1 (Y2 + 11) = 3Y2 − 3Y2 − 16 2 2 2 2 53 X−1 X1 + 15X2 + 10 = X1 57X1 X2 + 7X1 − 117X2 − 33X2 + 50 3 6 5 4 3 2 71 X−1 2X1 + 5X1 − X−1 − 5 = X1 − 16X1 + 57X1 − 32X1 − 18X1 − 17X1 + 6 2 107 X1 2 (1 − X−1) + (X1 + 2) 5X1 − 5X1 + 2 = 3X2 ((X1 + 2) (2X1 − 1) + 3X2) √ 8.3 Evaluation of Θ3( −n)

This section will be concerned with evaluating Θ3(τ), Θ2(τ), Θ4(τ) when τ is a quadratic irrationality. In view of the relation

3 3 3 Θ3(τ) = Θ2(τ) + Θ4(τ) , it suffices to evaluate only Θ (τ) and Θ4(τ) . Consider the classical elliptic modular function 3 Θ2(τ) λ(τ) defined by √ η(τ/2)η(2τ)2 λ(τ)1/8 := 2 . η(τ)3 This function satisfies K (1 − λ(τ)) i 2 = τ, K2(λ(τ)) and K2(λ(τ)) has been calculated for many quadratic irrational τ (see pg. 298 of [5]). The second equation of (6.4) now becomes

1 t2 + 6t + 12 Θ3(τ) = √ K2(λ(τ)) (8.38) 3 (t + 2)3/2(t + 6)1/2

Θ4(τ) where, as before, t = t12(τ/2). Hence if we can deduce the value of from a modular Θ2(τ) equation, we can deduce the value of t from

3 2 2 Θ4(τ) (t + 2) (t + 4) 3 = 27 2 2 2 (8.39) Θ2(τ) t (t + 3) (t + 6) and ultimately deduce the value of Θ3(τ). The following proposition is useful in actually carrying out these computations.

71 Proposition 8.17. Let √ η τ 4 η(2τ)4η(3τ)10 ξ(τ) = 3 2 . 10 3τ 4 4 η(τ) η 2 η(6τ) Then √ 2 Θ (τ) 2 3 ξ(τ)2 + 1 3 = , K2(λ(τ)) 16ξ(τ) where ξ(τ) is a solution to

Θ (τ)3/2 1 − 3ξ2 4 = = − cot 3 tan−1(ξ) . 3/2 3 Θ2(τ) ξ − 3ξ Proof. Set √ t + 6 3ξ = t + 2 in (8.38) and (8.39). The η product representation of ξ follows from the second table in Section 8.1.

We have found that the polynomial equation satisfied by ξ(τ) is often simpler than the equation satisfied by t12(τ/2). This may be attributed to the fact that ξ satisfies the transformation formula ξ(τ)ξ(−1/3τ) = 1. Theorem 8.18. The following table of values holds.

Θ4(τ) τ Θ3(τ) Θ2(τ)

√ 1/2 √ √ 1/2 q 1 3 7 9 √ ( 3−1) (4+ 3+3 5) Γ( )Γ( )Γ( )Γ( ) √ √ 1/3 √i 20 20 20 20 3 2 − 3 4 − 15 1+ 5 5 25/231/2 π3/2 2 √ 1/2 √ √ 1/2 q 1 3 7 9 √ ( 3+1) (4− 3+3 5) Γ( )Γ( )Γ( )Γ( ) √ √ 1/3 √5i 20 20 20 20 3 2 + 3 4 + 15 1+ 5 5 25/231/251/2 π3/2 2 r √ q √ √ 7/4 5+ 21 7+ 3 − 3 1 2 4 √ 5/6 √ √ 6 2 2 2 Γ( )Γ( )Γ( ) q q √i 7 7 7 31/2 5+ 21 5+ 21 − −3+ 21 7 25/231/87−1/4 π2 2 8 8 r √ q √ √ 7/4 5+ 21 7+ 3 + 3 1 2 4 √ 5/6 √ √ 6 2 2 2 Γ( )Γ( )Γ( ) q q √7i 7 7 7 31/2 5+ 21 5+ 21 + −3+ 21 7 25/231/871/4 π2 2 8 8

Proof of (iii) and (iv). Consider the function

η(7τ)4 t (τ) := 72 = t (τ)2s (τ), 7 η(τ)4 21 21 and set τ := √i throughout the proof. The second equality follows by the deﬁnitions of t 7 21 and s21 given in Section 8.1. The special value t7(τ) = 7 is trivial and is a consequence of the formula η(−1/τ) = pτ/i η(τ). 2 From t21s21 = 7 and the modular equation 1 7 s21 + = 3 + t21 + s21 t21

72 we deduce that s √ s √ 6 5 + 21 −3 + 21 s = + , 21  8 8  s √ s √ 3 5 + 21 −3 + 21 t = 71/2 − . 21  8 8 

Thus, s √ s √ 12 Θ2(τ)Θ4(7τ) 2 −3 + 21 5 + 21 = s21 =  +  . (8.40) Θ2(7τ)Θ4(τ) 8 8

From (8.20a), namely, ! √ Θ (τ)Θ (7τ)3/4 Θ (τ)Θ (7τ)−3/4 3 3 4 4 + 4 4 = Θ2(τ)Θ2(7τ) Θ2(τ)Θ2(7τ) ! ! Θ (τ)Θ (7τ)1/2 Θ (τ)Θ (7τ)−1/2 Θ (τ)Θ (7τ)1/2 Θ (τ)Θ (7τ)−1/2 2 4 − 2 4 2 4 + 2 4 − 7 , Θ2(7τ)Θ4(τ) Θ2(7τ)Θ4(τ) Θ2(7τ)Θ4(τ) Θ2(7τ)Θ4(τ) we deduce that √ !5/3 Θ (τ)Θ (7τ) 5 + 21 4 4 = 3 , (8.41) Θ2(τ)Θ2(7τ) 2

3/2 Θ4(τ) Θ4(7τ) Θ4(τ) and the stated values for and follow from (8.40) and (8.41). Now, both 3/2 Θ2(τ) Θ2(7τ) Θ2(τ) 3/2 Θ4(7τ) and − 3/2 are roots of Θ2(7τ)

x4 + 783x3 − 504x2 + 162x − 27 = 0

By setting 1 − 3ξ2 x := ξ3 − 3ξ and factoring we deduce that ξ(τ) and −ξ(7τ) are both roots of

3ξ4 + 6ξ3 − 3ξ − 1 = 0, √ or by factoring again this last equation over Q[ 21], that

ξ(7τ) − ξ(τ) = 1, √ 3 + 21 ξ(7τ)ξ(τ) = , 6 Thus, s r7 ξ(7τ) + ξ(τ) = 3 + 2 . 3

73 The stated evaluations of Θ(τ) and Θ(7τ) follow from

√ 2 √ 2 Θ (7τ) 2 Θ (τ) 2 3 ξ(7τ)2 + 1 3 ξ(τ)2 + 1 3 − 3 = − K (λ(7τ)) K (λ(τ)) 16ξ(7τ) 16ξ(τ) 2 2 √ 3 3 = , 8 √ 2 √ 2 Θ (7τ) 2 Θ (τ) 2 3 ξ(7τ)2 + 1 3 ξ(τ)2 + 1 3 + 3 = + K2(λ(7τ)) K2(λ(τ)) 16ξ(7τ) 16ξ(τ) √ s√ √ 1 5 + 21 7 + 3 = . 4 · 31/4 2 2 and the evaluations ([5], pg. 298)

71/4 Γ 1 Γ 2 Γ 4 K (λ(τ)) = 7 7 7 , 2 2 π2 7−1/4 Γ 1 Γ 2 Γ 4 K (λ(7τ)) = 7 7 7 . 2 2 π2

Proof of (i) and (ii). Now set τ := √i . By Theorem 8.7 and (8.3) the Haupftmodul of Γ (5) 5 0 has the evaluation η(5τ)6 t (τ) : = 53 5 η(τ)6 m1/4(1 − m )3/4 = 5 5 M 3 1/4 3/4 5 m1 (1 − m1) 2 1 = 9s15t15 + 3s15t15 − − 9 s15 = 53/2

When this is combined with the modular equation 1 3 s15 − = 3t15 + + 5, s15 t15 we deduce that √ 5 − 1 t = , 15 6 √ 3/2 √ 1/2 s15 = 2 + 3 4 + 15 .

Hence, the stated evaluations of Θ4(τ) and Θ4(5τ) follow from Θ2(τ) Θ2(5τ)

Θ4(τ) −1 −2/3 = t15 s15 , Θ2(τ)

Θ4(5τ) −1 2/3 = t15 s15 , Θ2(5τ)

74 which are consequences of the deﬁnitions of t15 and s15 and Corollary 6.6. We now have 3/2 3/2 −3/2 Θ4(τ) −3/2 Θ4(5τ) deduced that both 3 3/2 and 3 3/2 are roots of Θ2(τ) Θ2(5τ) x8 − 1732x6 − 2746x4 − 68x2 + 1 = 0. By setting 1 − 3ξ2 x = 3−3/2 ξ3 − 3ξ and factoring, we deduce that ξ(τ) and ξ(5τ) are roots of 81ξ8 − 108ξ6 − 522ξ4 + 372ξ2 + 1 = 0 √ Thus, by factoring again over Q[ 5], we obtain that 1 √ ξ(τ)2 + ξ(5τ)2 = 2 + 4 5 , 3 √ !2 1 1 + 5 ξ(τ)ξ(5τ) = , 3 2 and so √ Θ (τ) Θ (5τ) 3 ξ(5τ)2 + 1 ξ(τ)2 + 1 3 3 = , 1/2 1/2 K2(λ(τ)) K2(λ(5τ)) 16ξ(5τ) ξ(τ) (8.42) 1q √ = 2 + 5 5. 6 Equation (8.16c) states that 2 5Θ3(5τ) 2 − 2t15 t15 1 = 2 + 2 s15 + . Θ3(τ) t15 + 3t15 + 1 t15 + 3t15 + 1 s15 Hence, √ √ √ Θ (5τ)2 3 + 1 4 − 3 + 3 5 5 3 = √ √ √ . (8.43) 2 Θ3(τ) 3 − 1 4 + 3 + 3 5

The stated values of Θ3(τ) and Θ3(5τ) now follow from (8.42) and (8.43) and the values ([5], pg. 298)

q 1 3 7 9 √ Γ 20 Γ 20 Γ 20 Γ 20 K (λ(τ)) = 2−3/2(2 + 5)1/4 , 2 π3/2 q 1 3 7 9 √ Γ 20 Γ 20 Γ 20 Γ 20 K (λ(5τ)) = 5−1/22−3/2(2 + 5)1/4 . 2 π3/2

By utilizing the modular equations of degree 17, we have successfully evaluated the quotient Θ4 for τ = √i . The result, whose derivation follows the same lines as the proof of the previous Θ2 51 theorem, is

√ 1/3 √ 1/3 √ 1/3 !4 Θ i √ 2/3 5 − 17 + 23 − 17 − 4 + 17 4 √ = 17 − 4 . Θ2 51 3

For more evaluations of this type, see [7].

75 References

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