c 2014 by Daniel Schultz. All rights reserved.

Transcription

1 c 24 by Daniel Schultz. All rights reserved.

2 CUBIC THETA FUNCTIONS AND IDENTITIES FOR APPELL S F FUNCTION BY DANIEL SCHULTZ DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 24 Urbana, Illinois Doctoral Committee: Professor Scott Ahlgren, Chair Professor Alexandru Zaharescu, Co-Chair Professor Bruce C. Berndt, Director of Research Professor Kenneth B. Stolarsky

3 Abstract This thesis is centered around three topics: the theory of the cubic theta functions as functions of two analytic variables, cubic modular equations, and a class of two-variable cubic modular equations. Chapter is dedicated to the first two topics, while Chapter 4 covers the last. First, the theory of cubic theta functions can be developed analogously to, but distinct from, the classical theory of elliptic functions. We will derive analogues of the Jacobian elliptic functions, and provide addition theorems, integral inversion formulas, differential equations, and modular transformations for these functions. Second, we revisit the cubic modular equations first derived by Ramanujan and study them in a systematic manner. The results obtained greatly extend previous work on cubic modular equations. Finally, in Chapter 4, we study modular equations for the Picard modular functions. These modular equations provide a two-variable generalization of the cubic modular equations studied in Chapter. ii

4 Acknowledgments I would like to thank my advisor Professor Bruce C. Berndt for encouraging me to pursue the topic of this thesis, as well as proof reading several drafts. My other committee members, Professors Scott Ahlgren, Kenneth Stolarsky, and Alexandru Zaharescu, also deserve thanks for their guidance over the years. Also, I would also like to give a special thanks to Tito Piezas III and Armin Straub for verifying the solvability of an equation that arises in Section 4.4 for some special values of s and t. iii

5 Table of Contents Chapter List of Symbols Chapter 2 Introduction Chapter Cubic Theta Functions Introduction The Theta Function and its Corresponding Riemann Surface X Definitions Differentials of the First, Second and Third Kinds on X Riemann Theta Function Cubic Theta, Sigma, Zeta and Weierstrass Functions Basic Properties Periodicity Properties of the Theta Functions Abel s Theorem and Properties of the Integrals of the First Kind Zeros of the Theta Functions Inversion Formulas Addition Formulas Cubic Elliptic Functions Modular Transformations Transformations of the Theta Functions Cubic Modular Equations The Modular Equations u v Modular Equations Evaluation of Θ n Chapter 4 Modular Equations for Appell s F Function Introduction Picard Modular Forms Partial Differential Equations and the Algebracity of the Multiplier Analysis of the Modular Equation and a Two-Parameter Solvable Nonic Concluding Remarks References iv

6 Chapter List of Symbols 2F a, b c x Gauss 2 F hypergeometric function, also 2 F a, b; c; x Θ a u τ Cubic theta function with characteristic ɛ a. Θ a τ Cubic theta function with zero argument and characteristic ɛ a. ϑ a u m Denormalized cubic theta function with characteristic ɛ a. ϑ a Denormalized cubic theta function with zero argument and characteristic ɛ a. fa, b Ramanujan s theta function a nn+/2 b nn /2. mτ the main modular function Θ 2τ Θ τ m p M p N p modulus mpτ associated with the p th order transformation multiplier associated with integrals of the first kind multiplier associated with integrals of the second kind K m generalized complete elliptic integral of the first kind, 2 F, 2 ; ; m E m generalized complete elliptic integral of the second kind, 2 F, ; ; m F a; b ; b 2 c Θ i τ, τ 2 x, y Appell s F hypergeometric function Two-variable cubic theta function

7 Chapter 2 Introduction The cubic theta function m,n qm2 +mn+n 2 e 2πimu e 2πimu 2, which is the main object of study in Chapter, is an entire function which is quasi-periodic with respect to the lattice Z 2 + 2τ τ τ 2τ Z 2, where τ H = {τ Imτ > }. We will develop the theory of this function analogously to the classical Jacobian theta functions by connecting it to an algebraic curve which has this lattice as its Jacobian. This curve turns out to be a hyperelliptic curve with the singular model y = x ξ x ξ 2 x ξ This kind of approach to the cubic theta functions has not been previously worked out in the literature which deals with these functions, and so many of the interesting properties of this function had gone undiscovered. For example, by considering characteristics supported at the four branch points ξ i and, it becomes clear that we need to consider nine associated cubic theta functions in order to arrive at a coherent theory. These nine functions are the cubic analogues of the four Jacobian elliptic theta functions. The other main objects of study in Chapter are the modular transformations of the cubic theta functions, also known as cubic modular equations. The moduli space of the curves 2.. depends on the single complex number m = ξ ξ 2 ξ ξ. 2

8 The map τ mτ which is obtained by recovering the curve 2.. in the form y = x x 2 mx from its Jacobian turns out to be a univalent function on H/Γ where Γ is a index 4 subgroup of the elliptic modular group. This abuse of notation for a left action of Γ on H is standard and it will be used throughout. The cubic modular equation of degree p describes how the modulus m and other related quantities transform under the transformation τ pτ. The function mτ is actually implicit in Ramanujan s work, but he used an equivalent formulation of the cubic modular equations by means of the relation τ = 2 F, 2 ; ; m 2F, 2 ; ; m. Ramanujan stated some cubic modular equations, but his work in this area was not as complete as his work on classical modular equations. For example, Ramanujan gives only one cubic modular equation of degree 7, while we will give a dozen such modular equations, all in the spirit of Ramanujan s work. In Chapter 4, modular equations associated with the Picard curve y = x ξ x ξ 2 x ξ x ξ are studied and a new equation is derived. The Jacobian of this curve has the form Jτ, τ 2 = C /Z + Ωτ, τ 2 Z where Ω is a certain matrix and τ, τ 2 B = {τ, τ 2 2 Reτ > τ 2 2 }. A modular equation for the isogenous Jacobian varieties Jτ, τ 2 and Jτ, τ 2 has been derived in the literature. We will consider the more general isogenous Jacobian varieties Jτ, τ 2 and Jαᾱτ, ατ 2 for any α Q and derive the explicit modular equation for α = 2. A key component in the derivation of this new modular equation is the system of partial differential equations satisfied by the period integrals of 2..2.

9 The moduli space of the curves 2..2 depends on the two complex numbers m = ξ ξ 2 ξ ξ 4, m 2 = ξ ξ ξ ξ 4, hence such modular equations constitute identities between functions of two complex variables. In fact, the map τ, τ 2 m, m 2 is univalent on B/Γ, where Γ is a certain index 24 subgroup of the Picard modular group. The modular equations which we will investigate for these Picard curves contain the cubic modular equations as a special case. When any two of the ξ i coincide, the Picard curve becomes the singular hyperelliptic curve 2.., hence any modular equation derived for the Picard curve 2..2 should contain a cubic modular equation through a specialization of the moduli m and m 2. 4

10 Chapter Cubic Theta Functions. Introduction For any positive integer n, the rising factorial is defined by a n := Γa + n Γa = aa + a + n, and the Gaussian hypergeometric function is defined for z < by 2F a, b; c; z := n= a n b n c n n z n. In his paper [2] on modular equations and approximations to π, Ramanujan gives two series for π that belong to the theory of q 2 where q 2 k 2 = exp 2π 2F, 2 ; ; k2 2F, 2... ; ; k2 Such an expression is an analogue of the classical relation between the nome q and modulus k 2 of the elliptic curve y 2 = x x k 2 x given by q = exp π 2 F 2, 2 ; ; k2 2F 2, 2 ; ; k2..2 in which the hypergeometric function K 2 z := 2 F 2, 2 ; ; z.. 5

11 has been replaced by K z := 2 F, 2 ; ; z...4 Ramanujan gives several results in the theory of q 2, 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. in [4] gives the solution for φ when u = φ cos sin m sin φ dφ..5 m sin 2 φ by dφ du = K m n= x n + q n,..6 + q n where x = exp 2iu K m, and q = q 2m is given by... Here it seems more natural to use m to denote the modulus instead of using Ramanujan s k 2, and Ramanujan s theory of q 2 is more appropriately called the theory of cubic elliptic functions or the theory of signature. A proof of this inversion formula is given in [4] or [5], and a simpler derivation is given in [4]. Ramanujan defined a cubic modular equation of degree p as a relation between two moduli m and m p induced by the relation K m p K m p = p K m,..7 K m and gave the following modular equations of degrees 2, 5, and respectively: x 4 + y 4 =,..8 x 4 + x 2 y 2 + y 4 =,..9 4x 2 y 2 + xy x 2 + y 2 + x 2 + y 2 2 =,.. where x = m /2 m /2 p and y = m /2 m p /2 in all cases. Ramanujan s modular equations may be found in Chapter of [4]. An example of a modular equation of degree 2 that 6

12 will be obtained here is 44x 2 y xy x 2 + y x 2 + y 2 2 = 2 xy x 2 + 4xy + y 2 4 x 2 + xy + 4 y 2... 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 which arise from the hypergeometric function K m in place of the hypergeometric function K 2 m. In fact, Ramanujan also developed results for the hypergeometric functions K 4 z := 2 F 4, 4 ; ; z..2 and K 6 z := 2 F 6, 5 6 ; ; z,.. but the case of K z is the one that gives the most interesting analogue to the classical theory of elliptic functions. This is the reason that is case is developed here. Two central objects in the theory of elliptic functions see, for example, [27] for a treatment of elliptic functions are the elliptic integral u = x πk 2 m dt 2 t t mt..4 and the group of four theta functions Θ u; q = Θ /2 u; q, /2 Θ 2 u; q = Θ /2 u; q, Θ u; q = Θ u; q, 7

13 Θ 4 u; q = Θ /2 u; q, where Θ b a u; q = n= q n+b2 e 2n+bπiu+a...5 The periods of the integral in..4 without the factor πk 2 m, are πk 2 m and iπk 2 m, hence the inversion of this integral gives rise to a doubly periodic function. More specifically, when u and x are related by..4 and q is given by..2 and k 2 = m, we have the fundamental inversion formulas which relate ratios of theta functions to x: Θ ; q Θ u; q Θ 2 ; q Θ 4 u; q = x, Θ 4 ; q Θ 2 u; q Θ 2 ; q Θ 4 u; q = x, Θ 4 ; q Θ u; q Θ ; q Θ 4 u; q = mx...6 These functions are the Jacobian elliptic functions snu m, cnu m, and dnu m respectively. There is another important set of inversion formulas which relate m and K 2 m to the theta constants: Θ ; q 2 =, Θ 2 ; q 2 = m /2 K 2 m, Θ ; q 2 = K 2 m,..7 Θ 4 ; q 2 = m /2 K 2 m. With the last set of inversion formulas, the relation K 2 m p K 2 m p = p K 2 m,..8 K 2 m which defines a modular equation of degree p in the classical theory, becomes where q is defined 8

14 by..2 m = Θ 2; q 4 Θ ; q 4, m p = Θ 2; q p 4 Θ ; q p 4, so that a modular equation is equivalent to an identity involving theta constants at q and q p. This modular equation is actually only part of the general p th order transformation for elliptic integrals, which asks for the relationship between x and x p induced by the equation pk 2 m p K 2 m x dt xp 2 t t m t = dt 2 t t m p t...9 The quantity M p = K 2m K 2 m p is known as the multiplier of degree p in the classical theory, and the case p = 2 of this relationship is known as the Gauss or Landen transformation. In terms of the parameter we have t = 2 Θ 2; q 2 2 Θ 4 ; q 2, tt + 2 m = t + 2, t 2 m 2 = t + 2 2, 2 = t + 2 M 2 t +, x2 = 2 x x, M 2 m x..2 and this is a complete statement of the transformation of degree 2. For q <, the q-product symbol is defined as x; q := xq k k= and Ramanujan s theta function fa, b is defined by fa, b = a; ab b; ab ab; ab = a nn+/2 b nn /2...2 n= 9

15 The equality of the two representations is the Jacobi triple product identity see Entry 9 in Ch. 6 of []. The theta constants responsible for inverting K m analogously to..7 were first identified by the Borweins in [8] and Hirschhorn, Garvan, and Borwein in [2], where they introduced the notations set ζ = e 2πi/ throughout aq = q m2 +mn+n 2, bq = cq = They proved the representations m,n= m,n= m,n= ζ m n q m2 +mn+n 2, q m+ 2 +m+ n+ +n+ 2. bq = q; q q ; q, cq = q / q ; q q; q, and that these functions satisfy the identities hence these functions should be called cubic theta functions. aq = bq + cq,..22 cq K aq = aq,..2 In [4], Shen defines three analogues of the Jacobian elliptic functions based on the integral..6, by sn u = sin φ, cn u = cos φ, dn u = dφ du. However, such functions do not provide satisfactory analogues of the Jacobian elliptic functions, as,

16 for example, sn and cn are related by a sum of squares identity, and not a sum of cubes identity. In [2] and [], the four functions a q, z = q m2 +mn+n 2 z n, m,n= aq, z = bq, z = cq, z = m,n= m,n= m,n= q m2 +mn+n 2 z m n, ζ m n q m2 +mn+n 2 z n, q m+ 2 +m+ n+ +n+ 2 z m n, were introduced by adding an analytic variable to aq, bq, and cq, and it was shown that these functions satisfy the identities a q, z = bq, z + cq 2 cq, z, aq, z = cq, z + bq 2 bq, z, which provide one-variable generalizations of the sum of cubes identity..22. In order to produce satisfactory cubic analogues of the Jacobian theta functions in as much generality as possible, 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 the cubic theta functions of two complex variables u = u, u 2, four of which are set z a i = e2πiau i Θ u; q := Θ 2 u; q := Θ u; q := Θ 4 u; q := m,n= m,n= m,n= m,n= ζ +m n q m+ 2 +m+ n+ +n+ 2 z m+ z n+ 2, q m+ 2 +m+ n+ +n+ 2 z m+ z n+ 2, q m2 +mn+n 2 z m z n 2, ζ m n q m2 +mn+n 2 z m z n 2.

17 These functions parameterize a Fermat curve, Θ u; q + Θ u; q = Θ 2 u; q + Θ 4 u; q,..24 and this parameterization is analogous to the equations snu 2 + cnu 2 =, k 2 snu 2 + dnu 2 =. The reason that the identity..24 was missed in previous work on these functions is that the theta constant associated with Θ namely, Θ ; q vanishes identically as a function of q, thus it was not identified as an integral part of the theory. In analogy with Jacobi s derivative formula for the differential coefficient of the vanishing classical theta constant Θ ; q, dθ du ; q = πθ 2; qθ ; qθ 4 ; q, we have the identity dθ du ; q = 2πi Θ 2; qθ 4 ; q for cubic theta functions. The other five cubic theta functions are denoted by Θ, Θ 2, Θ 4, Θ 22, Θ 44 and are defined in Section... We also have a triplication formula, bqcq 2 bq Θ 4 u; q = Θ u; qθ 2 u; qθ 44 u; q+ Θ 2 u; qθ 4 u; qθ 22 u; q + Θ u; qθ u; qθ 4 u; q, which is analogous to the classical theta function identity Θ 4 ; q 2 Θ 4 2u; q 2 = Θ u; qθ 4 u; q, which is a key component in deriving the Landen transformation. 2

18 Just as the classical theta functions are related to the elliptic curve y 2 = x x mx, the nine cubic theta functions are related to the genus 2 hyperelliptic curve y = x x 2 mx...25 The standard Rosenhain normal form for this curve is discussed in Section.2. Formulas of the type..2 or..7 are known as Thomae-type formulas for the curves..25 and y 2 = x x mx respectively, and these identities were established in [5]. In fact, we cite Example 6. of [5] for proofs of the following fundamental inversion formulas: where q = exp 2π K m K m. aq = K m, bq = m / K m, cq = m / K m, Section.2 discusses the basics of the hyperelliptic curve..25, while Sections. and.4 collect the necessary facts about theta functions and Riemann surfaces. Inversion formulas for the cubic theta functions that are direct analogues of..6 are given in Section.5, and some examples of the addition theorems satisfied by the cubic theta functions are presented in Section.6. In Section 6, Ramanujan s inversion formula in..6 is deduced in an entirely different fashion than in [4] or [4] along with an elegant series-product identity which follows from an addition theorem for the cubic theta functions. Finally Sections.8 and.9 are devoted to modular transformations of the cubic theta functions, and many cubic modular equations are derived and formulas for multipliers are given. These multipliers appear, for example, in Ramanujan s series for /π, i.e. π = n= 2n!n! n! 5 m τm p τ n 2 + 6nm τ m p τ p N p τ, 27

19 where τ = p, and m, m p and N p are the functions defined in Section.8..2 The Theta Function and its Corresponding Riemann Surface X Here we will attempt to give a full treatment of the properties of the function q m2 +mn+n 2 z m z2 n,.2. 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 y = x x 2 mx..2.2 That this is the corresponding algebraic curve will be clear when its period matrix is computed in..2. Much of the theory of theta functions on Riemann surfaces will be utilized in studying the theta function.2.. 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 [2], [], and [7]. The curve.2.2 can be viewed as a three sheeted covering of the Riemann sphere with four branch points at x =, x =, x = /m, and x =, which we also refer to as b, b 2, b and b 4. We can choose a branch of the function y so that its only discontinuities as a function of x are on the intervals [, ] and [/m, ]. For concreteness, this branch will be denoted y and its value will be fixed to y := 2/ x / x 2/ mx /, where the principal value of a b is taken. The y values on the other two sheets y and y 2 differ from y by a factor of a cube root of unity so that y i = ζ i y. Gluing the three sheets y,y,y 2 together gives a genus 2 compact Riemann surface, and the standard homotopy basis of loops A, A 2, B, B 2 is shown in Figure.. The loop A resp. A 2 lies entirely in the first 4

20 Figure.: Homotopy basis of the Riemann Surface X êm sheet resp. second sheet and loops around the branch cut [, ]. The loop B resp. B 2 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 x, y, y is a holomorphic function of x, so we have the coordinates x, y = x + t, y + y m 2 t + Ot 2. x mx x At the branch point,, we have coordinates x, y = t, t + Ot 2, at,, x, y = t, m / t 2 + Ot, at /m,, x, y = t / m, / t + Ot m m 2, and at the infinite point on X, x, y = t, m / t 4 + Ot. As X has genus 2 and every curve of genus 2 is hyperelliptic, there is a function of order 2 on 5

21 X. Indeed, the function ẋ = y x can be verified by the charts constructed above to have zeros {, /m} and poles {, }. This function ẋ 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, AutX. For general m, AutX has order 2, generated by the automorphisms σx, y = x, ζy, mx m y γx, y =, m x m x 2, δx, y = mx, x, y mx.2. where, as before, ζ = exp 2πi/. The automorphism σ cycles the three sheets, while γ is the hyperelliptic involution. X. We need the following proposition describing the basic structure of meromorphic functions on Proposition.2.. The field of meromorphic functions on X, MX, is generated by x and y, and each meromorphic function with poles only at infinity can be written uniquely as y2 p x + p xy + p 2 x x, where each p i x is a polynomial in x. Proof. The first statement, which is true of any algebraic curve, follows from Proposition.2 in Chapter 6 of [29], where it is shown that the index of a function in MX is equal to its order. Since the function x MX has order and Cx has index in Cx, y by the defining equation for X, we see that Mx = Cx, y. We now know that each meromorphic function can be written uniquely as y2 r x + r xy + r 2 x x, where each r i x is a rational function in x. Calculations using the local coordinates given above 6

22 show that each r i x can have poles only at infinity, i.e. each function is a polynomial. Proposition.2.2. Let f be a function of order 2 with a pole at p = x, y and a zero at q = x, y. Then the other pole and the other zero of f are given by γp and γq, respectively, and f is constant multiple of ẋ x. ẋ x 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 most easily by bringing the equation defining X into the form w 2 = z z λ z λ 2 z λ z by a suitable bi-rational transformation in which z = ay+b x cy+d x The Rosenhain invariants λ i are given by for some constants a, b, c and d. λ = ξ ξ + ξ ξ +, λ 2 = ξ 2 ξ 2 ξ 2 + 2, λ = ξ 2 ξ ξ 2 + 2, where the parameter ξ is related to the modulus m by m = ξ2 ξ 2 2 ξ 2 +. The most notable property of this defining equation for X is that the Rosenhain invariants satisfy λ = λ λ 2, which has implications for the cubic elliptic functions associated with X. The hyperelliptic involution γ now corresponds to the transformation γz, w = z, w. At the infinite point in these variables, the local coordinates are z, w = t 2, t 5 + O t 4. 7

23 Let z, w, z 2, w 2 be the poles of f. Then z z z z 2 fz, w is a function of order at most four and has poles only at infinity. Thus, there are polynomials p z and p z with z z z z 2 fz, w = p z + p zw. If p z is non-zero, then the right-hand side has a pole at infinity of order at least 5, which is a contradiction. Thus fz, w = fz, w, that is, f is fixed by the hyperelliptic involution.. Definitions.. Differentials of the First, Second and Third Kinds on X The differentials of the first kind are -forms on X which are holomorphic everywhere. It is well known that the space of such forms has the same dimension as the genus of the curve. For the genus 2 curve X, a basis is given by du = du du 2 = dx y xdx y 2. The generalized incomplete integrals of the first kind are the integrals of these two differentials, which we put together into a vector-valued function x x du = x x du x x du 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 ω = 2πi K m ζ ζ 2 and ω 2 = 2πi K m ζ ζ 2. 8

24 It follows from Euler s integral representation 2F a, b; c; z = Γc t b t c b ΓbΓc b tz a dt.. of the hypergeometric function that K m = 2π = 2π dx x / x 2/ mx / dx x 2/ x / mx 2/, and these two integrals are directly related to the periods of x du in the following way. The i, jth component of ω or ω 2 is the integral of du i around the loop A j or B j. Since the loops A j and B j form a homotopy basis of X, x du will change only by integral multiples of the four columns of ω and ω 2 as the path of integration from to x changes. The ratio of these two period matrices is ω ω 2 = i K m K m 2 2 = 2τ τ τ 2τ,..2 which depends only on the single complex number τ = i K m K m matrix for the normalized basis of differentials of the first kind given by. This matrix is the B-period dûu = ω du. We now need a differential that is holomorphic everywhere except at an arbitrary point p = x, y where it has a residue of and at where it has a residue of. Such a differential is called a differential of the third kind for the points p and. Following the general method in Section 45 of [2], we find that the differential x x dx := + y y + xy2 dx x y 2 x x has the required properties. The differential x x, x 2 dx = x x dx x x 2 dx is then a differential of the third kind for the points x, y and x 2, y 2, for it has residues and at these two points 9

25 respectively. Finally, we seek a vector of differentials dvx such that z x z dxdz + dux dvz is a symmetric bi-meromorphic differential in x and z. With the choice dv = dv dv 2 = mx xdx y 2 +m 2mxdx y, 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 x x dv = x x dv x x dv 2. These integrals have associated period matrices η = 2πi E m ζ2 ζ and η 2 = 2πi K m E m ζ2 ζ, where E m : = 2 F, ; ; m = 2π mx / dx x 2/ x / is the generalized complete elliptic integral of the second kind. The integral representation follows from... Also, η ω = E m K m. 2

26 The relations between the period matrices ω, ω 2, η, and η 2 derived from the bilinear relations see Section 4 of [2] may be stated as. ω ω 2 is symmetric with positive definite imaginary part, 2. η ω is symmetric,. ω η T 2 ω 2η T = 2πiI = η 2ω T η ω T 2. The first two are easily confirmed by the calculations given above while the last one is equivalent to the important identity E m K m + E m K m = + 2πK mk m,.. which we refer to as the generalized Legendre identity. This is a special case of an identity due to Elliott [4]. We also list the system of differential equations satisfied by K m and E m for convenience: m m d dm K m = E m mk m, m d dm E m = E m K m Riemann Theta Function The Riemann theta function associated to X is Θ u τ := Θ u, u 2 T τ = = a,b= a,b= e iπa,b ω ω 2 a,b T e 2πiu,u 2 a,b T q a2 +ab+b 2 z a z b 2 where z i = e 2πiu i, q = e 2πiτ. It will be necessary to have notation for the shifts 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 Riemann theta 2

27 function with an arbitrary characteristic n n 2 m m 2 R 2 2 is defined as Θ n n 2 m m 2 u τ = a,b= q a+n 2 +a+n b+n 2 +b+n 2 2 e 2πim z a+n e 2πim 2 z 2 b+n2 = q n2 n n 2 n 2 2 e 2πin m +u +n 2 m 2 +u 2 Θ u + m + 2n τ + n 2 τ m 2 + n τ + 2n 2 τ τ...5 The nine characteristics are derived from the integrals of du between any two of the four branch points,, /m and. Using the definitions of the loops A j and B j, one can compute that /m du ω + ω 2 / / du ω / + ω 2, 2/, du ω / + ω 2 / 2/ /, where is taken modulo the period lattice ω Z 2 + ω 2 Z 2, and the characteristic symbols for these three vectors in C 2 are ɛ 4 := ɛ 2 := ɛ := / 2/ / / / / / 2/,,, respectively. Since the four columns of the period matrices ω and ω 2 are linearly independent over R, every vector in C 2 has a unique corresponding characteristic symbol with entries in R. 22

28 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 du between the branch points modulo the period lattice. Since ɛ = ɛ 2 + ɛ 4 and ɛ 4 ɛ 2, we see that this is a group of order nine generated by ɛ 2 and ɛ 4. We use the following notation for the elements of this group: ɛ = ɛ 2 + ɛ 4, ɛ 4 = ɛ 2 + ɛ 4, ɛ 44 = ɛ 2 + 2ɛ 4, ɛ 2 = ɛ 2 + ɛ 4, ɛ = ɛ 2 + ɛ 4, ɛ 4 = ɛ 2 + 2ɛ 4, ɛ 22 = 2ɛ 2 + ɛ 4, ɛ 2 = 2ɛ 2 + ɛ 4, ɛ = 2ɛ 2 + 2ɛ 4. While this not the most natural notation for the elements of this group, it is direct analogue of the notation for the four Jacobian theta functions, and it is also the most relevant for the purposes of the cubic theta functions. The cubic theta functions with simple characteristics ɛ, ɛ 2, ɛ, and ɛ 4 have zeros which are easy to describe in terms of the four branch points, while the theta functions with the remaining five compound characteristics have slightly more complicated zero sets... Cubic Theta, Sigma, Zeta and Weierstrass Functions Define the ϑ and σ functions for an arbitrary characteristic ɛ by ϑ [ɛ] u m := Θ [ɛ] ω u τm, σ [ɛ] u m := e 2 ut η ω u ϑ [ɛ] u m, where Θ [ɛ] ω u τm is defined by..5 and τm = i K m. K m We now define nine Θ functions Θ a u, nine ϑ functions ϑ a u, nine σ functions σ a u, two ζ functions ζ i u i 2, and three functions ij u i j 2 by Θ a u τ := Θ [ɛ a ] u τ, 2

29 ϑ a u m := ϑ [ɛ a ] u m, σ a u m := c a σ [ɛ a ] u m, ζ i u m := u i log σ 4 u m, 2 ij u m := log σ 4 u m, u i u j where c a is chosen so that the first non zero differential coefficient of σ a u is. For brevity, ζu will denote the gradient of log σ 4 u vector of ζ i functions, and u will denote the Hessian of log σ 4 u matrix of ij functions. Thus we have the series representations for the four Θ functions mentioned in the introduction and the remaining five. Θ u τ := Θ 2 u τ := Θ u τ := Θ 4 u τ := Θ u τ := Θ 2 u τ := Θ 4 u τ := Θ 22 u τ := Θ 44 u τ := m,n= m,n= m,n= m,n= m,n= m,n= m,n= m,n= m,n= ζ +m n q m+ 2 +m+ n+ +n+ 2 z m+ z n+ 2, q m+ 2 +m+ n+ +n+ 2 z m+ z n+ 2, q m2 +mn+n 2 z m z n 2, ζ m n q m2 +mn+n 2 z m z n 2, ζ m+n q m m+ 2 n+ 2 +n+ 2 2 z m+ 2 z n+ 2 2, ζ +m n q m m+ 2 n+ 2 +n+ 2 2 z m+ 2 z n+ 2 2, ζ m+n q m+ 2 +m+ n+ +n+ 2 z m+ z n+ 2, q m m+ 2 n+ 2 +n+ 2 2 z m+ 2 z n+ 2 2, ζ m+n q m2 +mn+n 2 z m z n 2, 24

30 where q a = e 2πiaτ and z a i = e 2πiau i. Whenever the arguments u, τ, or m are omitted from the functions, they assume the default values of, τ, and m, respectively. For example, Θ a u = Θ a u τ, Θ a 2u = Θ a 2u τ, Θ a 2τ = Θ a 2τ, Θ a = Θ a τ, while the Borweins functions aq, bq, and cq have the more systematic notation ae 2πiτ = Θ τ, ce 2πiτ = Θ 2 τ, be 2πiτ = Θ 4 τ. The theta constants Θ 22 τ, Θ 44 τ reduce to Θ 2 τ and Θ 4 τ, respectively. The other four theta constants Θ τ, Θ τ, Θ 2 τ, and Θ 4 τ were not introduced by the Borweins because they turn out to vanish identically. For example, Θ τ = q m2 +mn+n 2 +m+n+ ζ m n. m,n= If this sum is broken into two sums depending on the parity of n via the substitutions m = x y n = 2y and m = x y n = 2y we obtain Θ τ = q x2 +x+y 2 +y+ ζ x + q x2 +y 2 2y+ ζ x+. x,y= x,y= By the Jacobi triple product identity..2, this may be simplified as Θ τ = q 2 ; q 2 q 6 ; q 6 ζ 2 ; q 2 ζq 2 ; q 2 q 2 ; q 6 q 4 ; q 6 25

31 + ζ 2 q 2 ζ 2 /q; q 2 q ζ; q 2 /q; q 6 q 7 ; q 6 = q 2 ; q 2 q 6 ; q 6 + ζ 2 + ζ =. This important null value for the theta constant Θ τ is also Corollary in Ping Xu s thesis [8], where it was recognized as a consequence of a third order circular summation formula..4 Basic Properties In this section, the quasi-periodicity properties of the theta functions are stated as well as general theorems which hold on any compact Riemann surface..4. Periodicity Properties of the Theta Functions Proposition.4.. Let a and a 2 denote column vectors in R 2, and ɛ and ɛ 2 denote row vectors in R 2. Then with C = e 2πiɛ 2 a ɛ a 2, we have Θ ɛ 2 ɛ u + a + ω ω 2 a 2 = Ce πi2at 2 u+a T 2 ω ω 2 a 2 2πia T a 2 ϑ ɛ 2 ɛ u + ω a + ω 2 a 2 = Ce πi2at 2 ω u+a T 2 ω ω 2 a 2 2πia T a 2 σ ɛ 2 ɛ e 2πiɛ 2 a Θ u + ω a + ω 2 a 2 = Ce 2 η a +η 2 a 2 T 2u+ω a +ω 2 a 2 πia T a 2 ɛ 2 + a T 2 ɛ + a T e 2πiɛ 2 a ϑ ɛ 2 + a T 2 ɛ + a T e 2πiɛ 2 a σ u, ɛ 2 + a T 2 ɛ + a T u, u. 26

32 If a and a 2 are integer vectors, then the expressions in brackets are equal to Θ [ ɛ 2, ɛ T ] u, ϑ [ ɛ 2, ɛ T ] u, and σ [ ɛ 2, ɛ T ] u, respectively. Furthermore, in this case, ζ u + ω a + ω 2 a 2 = ζu + η a + η 2 a 2, u + ω a + ω 2 a 2 = 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 ω and ω 2, while the ζ function increases by the corresponding column of the matrices η and η 2 and the function is genuinely periodic with respect to the lattice ω Z 2 + ω 2 Z 2. If a holomorphic function fu satisfies f u + a + ω ω 2 a 2 = e 2πiɛ 2 a ɛ a 2 e πi2at 2 u+a T 2 ω ω 2 a 2 2πia T a 2 r fu for integer vectors a and a 2, then it is called a theta function of order r with characteristic ɛ 2 ɛ. For example, the functions ϑ u and ϑ u are theta functions of order and, respectively, with characteristics ɛ and ɛ, respectively. The space of all theta functions with order r and a given characteristic has dimension r 2 see pg. 24 of [] for a proof of this basic but very useful fact. Proposition.4.2. If fu = ϑ [ɛ ] uϑ [ɛ ] u ϑ [ ɛ n] u ϑ [δ ] uϑ [δ ] u ϑ [ δ n] u and ɛ + ɛ + + ɛ n δ + δ + + δ n, then f u + ω a + ω 2 a 2 = fu for integer vectors a and a 2. 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, 27

33 as the next three examples illustrate. Example.4.. Shifting u by ɛ b in ϑ a u gives ϑ c u, where ɛ c is the characteristic such that ɛ c ɛ a + ɛ 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.4., we find that 2/ 2/ Θ u + ɛ = Θ u + / + 2τ 2/ 4/ 2/ τ = e πi e πi e 2πi u +u 2 +τ e 4πi = e 2πi u +u 2 +τ Θ = e 2πi u +u 2 +τ Θ u. Θ u 2 τ 2τ u / / Similar calculations show that, regardless of the starting characteristic ɛ a, shifting by the characteristics ɛ, ɛ 4, or ɛ 44 gives no exponential factor, shifting by ɛ 2, ɛ, or ɛ 4 gives the factor e 2πi u +u 2 +τ, and shifting by ɛ 22, ɛ 2, or ɛ gives the factor e 4iπ u +u 2 τ. Example.4.4. Reflecting u in ϑ a u gives ϑ b u, where ɛ b is the characteristic such that ɛ b ɛ a. By the definition of the Riemann theta function with characteristics, the fact that Θu is an even function, and Proposition.4., we have Θ 4 u = Θ u + / 2/ = Θ u + / 2/ = Θ u + 2/ 4/ = Θ 2/ 4/ 28 u

34 = Θ 44 u. Similar calculations show that Θ u = Θ u, Θ 2 u = Θ 22 u, Θ 4 u = Θ 44 u, Θ u = Θ u, Θ 2 u = Θ 4 u. Example.4.5. The theta function ratio Θ u Θ 4 u has the factors ζ, ζ, ζ2, ζ associated with the four periods. That is, with Proposition.4. we find that Θ u +, T = ζ Θ u Θ 4 Θ 4 u, Θ u +, T = ζ Θ u Θ 4 Θ 4 u, Θ u + 2τ, τ T = ζ 2 Θ u Θ 4 Θ 4 u, Θ u + τ, 2τ T = ζ Θ u Θ 4 Θ 4 u..4.2 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 [29] or Chapter 2 of []. Proposition.4.6. The points {p,..., p n } and {q,..., q n } are the poles and zeros of a meromorphic function on X if any only if q qn du + + du. p p n When the automorphisms σ, γ and δ act on the integrals of the first kind to give γx du, etc., the result is, of course, still an integral of the first kind that can be written in terms of the original 29

35 integrals of the first kind x du. The precise effect they have is given in the following proposition whose proof is by direct calculation. Proposition.4.7. For m,, recall that the curve y = x x 2 mx has the automorphisms σx, y = x, ζy, mx m y γx, y =, m x m x 2, δx, y = mx, x. y mx The effects of these automorphisms on the integrals of the first kind are σx γx δx du = ζ2 ζ du = du = x x du, x /m du. du,.4. Zeros of the Theta Functions We will now explicitly describe the zeros of the theta functions when the theta functions are composed with the integrals of the first kind. Proposition.4.8. For any vector e C 2, the function Θ x dûu + e has two zeros z and z 2 as a function of x which satisfy z z2 dûu + dûu e for some fixed vector C 2 provided this function does not vanish identically. Also, satisfies ɛ. Proof. The first assertion is Theorem. in []. The only thing left to do is to compute the vector

36 of Riemann constants by considering the case e ɛ 4. The function Θ 4 x dûu does not vanish identically as a function of x since it is the non-vanishing theta constant Θ 4 when x =. This function does evaluate to the two vanishing theta constants Θ and Θ 2 when x = /m and x = respectively. Hence, the two zeros are z = /m and z 2 =, and we must have /m dûu + dûu ɛ 4, or 2ɛ ɛ. Proposition.4.9. For i =, 2,, 4, the zeros of ϑ i x du + x 2 du with respect to x are the i th branch point b i and γx 2. Proof. For general x 2, the function does not vanish identically, so its two zeros z and z 2 are determined from Since and x2 z z2 ɛ i + du + du + du ɛ. γx ɛ i = du = /m b i x du, du + /m du, this equation is satisfied with the choices z = b i and z 2 = γx 2..5 Inversion Formulas In this section we will obtain algebraic expressions for ratios of the theta functions ϑ a u, the zeta functions ζu, and the Weierstrass functions u when their argument is set to u = x,y du + x2,y 2 du. These algebraic expressions will be rational functions of x and x 2 and the corresponding

37 y coordinates on the curve y and y 2. Also, for brevity, we have ẋ i = y i x i. 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 ϑ a u, thus completing the definitions of the sigma functions σ a u. The formulas are also useful in that they give parametric representations of the functions, from which many identities may be found. Theorem.5.. With u = x du + x 2 du, the ϑ functions with simple characteristics satisfy ζ ϑ ϑ u ϑ 2 ϑ 4 u = x/ x / 2 = σ u σ 4 u, ϑ 4 ϑ 2 u ϑ 2 ϑ 4 u = x / x 2 / = σ 2u σ 4 u, ϑ 4 ϑ u ϑ ϑ 4 u = mx / mx 2 / = σ u σ 4 u, ζ ϑ uϑ u ϑ 2 uϑ 4 u = ẋ ẋ 2 = σ uσ u σ 2 uσ 4 u ; the ϑ functions with compound characteristics satisfy ϑ2 ϑ 4 ϑ 2 ϑ 4 ζ ϑ ϑ 4 ϑ 2 2 ϑ 2 ϑ 2 2 the functions satisfy ϑ 2 uϑ u ϑ 4 u 2 = x 2 x ẋ 2 ẋ = σ 2uσ u σ 4 u 2, ϑ uϑ 2 u ϑ 4 u 2 = x ẋ 2 x 2 ẋ ẋ 2 ẋ ϑ uϑ 2 uϑ 4 u ϑ 4 u = x y 2 y x 2 ẋ 2 ẋ = σ uσ 2 u σ 4 u 2, = σ uσ 2 uσ 4 u σ 4 u, ϑ 2 uϑ uϑ 22 u ϑ 2 2 ϑ ϑ 4 u = mx y 2 mx 2 y ẋ 2 ẋ ϑ 4 ϑ uϑ 44 u ϑ ϑ 4 u 2 = mx ẋ 2 mx 2 ẋ ẋ 2 ẋ = σ 2uσ uσ 22 u σ 4 u, = σ uσ 44 u σ 4 u 2 ; u = σ 4 u 2 mσ 2uσ u σ uσ 44 u σ uσ 44 u mσ 22 uσ 4 u ; 2

38 and if the same path of integration is used in the x i dv, the ζ functions satisfy ζu = x x2 dv + dv + m σ 4 u 2 σ 2 uσ u. Corollary.5.2. We have the following table of differential coefficients of the nine ϑ functions. a ϑ a ϑ a u ϑ a u 2 2 ϑ a u u 2 ϑ a u u 2 2 ϑ a u 2 u 2 K m mk E 2 m / ϑ ϑ 2 ϑ E 22 m / ϑ ϑ 2 ϑ E 4 m / ϑ 44 m / ϑ ϑ 4 ϑ K E ϑ 4 ϑ K E ζ 2 ϑ 2ϑ 4 ϑ ζ 2 ϑ 2ϑ 4 ϑ ζ 2 ϑ 2ϑ 4 ϑ ζ 2 ϑ 2ϑ 4 ϑ 2 ζ ϑ 2ϑ 4 ϑ ζ ϑ 2ϑ 4 ϑ 4 ζ ϑ 2ϑ 4 ϑ ζ ϑ 2ϑ 4 ϑ Proof of Corollary.5.2. The evaluations of the non-vanishing theta constants are a special case of the Thomae-type formulas derived in [5] see Example 6. there. They are also equivalent to Theorem 2. on page 99 of [4]. The fact that the remaining four theta constants vanish follows from Proposition.4.9; the function ϑ x du vanishes identically, and setting x to any of the four branch points gives the results. The first order differential coefficients of ϑ are zero since it is an even function. The rest of the entries will be established after the inversion formulas have been established. Proof of Theorem.5.. The function fu = ϑ u ϑ 4 u is periodic on ω Z 2 +ω 2 Z 2 by Proposition.4.2, hence the function F x, x 2 = f x du+ x 2 du is single valued and hence a meromorphic function of x and x 2. By Proposition.4.9, it has a triple zero at and triple pole at with respect to either variable. Therefore, F x, x 2 = cx x 2, where c is independent of x and x 2. By setting x = x 2 = and using the fact that du = ɛ 4, we deduce that c = f2ɛ 4 = ϑ 2ɛ 4 ϑ 4 2ɛ 4 = ϑ 2. ϑ

39 Substituting this value for c, taking the cube root, and rearranging gives the formula for ϑ u ϑ 4 u. The cube roots are indeed arbitrary as the calculations in Example.4.5 show that as x 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 function ϑ u. The function fu = ϑ 2uϑ u ϑ 4 u 2 is periodic on ω Z 2 + ω 2 Z 2 by Proposition.4.2, and hence the function F x, x 2 = f x du + x 2 du is a meromorphic function of x and x 2. To calculate this function, we need to know the zeros of ϑ with respect to x, say. By Proposition.4.8, the zeros z and z 2 of ϑ u with respect to x satisfy x2 z z2 2ɛ + du + du + du ɛ. Since ɛ and du, this is equivalent to x2 z z2 du + du + du. That is, there is a meromorphic function on X with zeros {x, z, z 2 } and a triple pole at. By Proposition.2., such a function must be of the form A + Bx. Since x 2 is a zero of this function, it must be a constant multiple of x x 2. Now, with respect to x the zeros and poles of F x, x 2 are {, z, z 2 } and {,, γx 2 }. However, the function x x 2 ẋ ẋ 2 has exactly the same set of zeros and poles since the zeros and poles of ẋ ẋ 2 are {x 2, γx 2 } and {, } respectively, and the zeros and poles of x x 2 are {x 2, z, z 2 } and {,, } respectively. Therefore, F x, x 2 = c x x 2 ẋ ẋ 2, where c is independent of x. Since both sides are symmetric in x and x 2, c is independent of both x and x 2. Unfortunately F x, x 2 is zero or infinite when x and x 2 are at any of the branch 4

40 points,, /m and, so a simple substitution will not determine c. By combining the formulas ϑ 4 ϑ 2 u ϑ 2 ϑ 4 u = x / x 2 / and we deduce that ϑ 2 ϑ u ϑ 4 ϑ 4 u = ϑ 2 uϑ u ϑ 4 u 2 = c x x 2 ẋ ẋ 2, c x / x 2 / x x 2 ẋ ẋ 2. As x 2 tends to and x tends to /m, the right-hand side tends to /m2/ c m / = cζ ϑ ϑ 4 ϑ 2 2 and the left hand side tends to ϑ 2 ϑ 4 ϑ ɛ +ɛ 4 ϑ 4 ɛ +ɛ 4 =. Thus c = ζ ϑ ϑ 4 ϑ 2 2 formula ζ ϑ ϑ 4 ϑ 2 2 ϑ 2 uϑ u ϑ 4 u 2 = x 2 x ẋ 2 ẋ. and the result is the stated 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 2.2 of [7]. This formula in the notation used here is dux T u duz = z x z dxdz + dux dvz 2 du x du 2 x x z du x du x + x z du x du 2 x du x 2 du 2 x 2 x 2 z du x 2 du 2 x 2 z x du z du z x x du x du 2 x x 2 x du x 2 du 2 x 2. Recall that du, dv, and x z dx are the differentials of first, second and third kinds, respectively. Since dux = t x m t x + dt x, 5

41 where t x is the local coordinate at x =, and dux = m / + +2m m 4/ t x + m 2/ t x + dt x, where t x is the local coordinate at x =, will be isolated by expanding both sides at x = and z = and extracting the coefficients of t xt zdt x dt z. Similarly, 2 and 22 will be extracted with the choices x =, z = and x =, z =, respectively. Carrying out these plans and expanding the right-hand side gives m 2/ ut xt zdt x dt z + = m/ x x x 2 x 2 x 2 y x y 2 y + y 2 t xt zdt x dt z +, m / 2 ut xt zdt x dt z + = x 2 y mx 2 x y 2 mx m / t x 2 y x y 2 y + y 2 xt zdt x dt z +, m x y 2 x 2 y x 2 x y 22 ut xt 2 2 x 2 2 x 2 y 2 zdt x dt z + = x x 2 x 2 y + x y 2 + y y 2 2 t xt zdt x dt z +. With some easy manipulations, these expressions can be brought into the forms given in the stated formula for u using formulas for the functions σ 2uσ u σ 4 u 2 from the formulas stated in the first and second parts of the theorem., σ uσ 44 u, and σ 22uσ 4 u σ 4 u 2 σ 4 u 2 obtained At this point it is possible to finish deriving all the entries of the table. At x = the integrals of the first kind have the expansion x du = 2 t2 x + m+2 5 t5 x + t x + +2m 2 t4 x +. Expanding, for example, the identity ϑ ϑ u ϑ 4 ϑ 4 u = mx / mx 2 / at x = x 2 =, substituting the known values ϑ u = ϑ u 2 =, and equating coefficients of 6

42 t x t x 2, t x t x 2, and t 2 x t x 2 gives, respectively, ϑ 4 u = 2 ϑ 4, ϑ 4 u = ϑ = ϑ 4 u 2 2 ϑ ϑ u 2 u 2 ϑ 2 ϑ u 2 u 2 ϑ 2 2 ϑ 4 u 2 u 2 ϑ 4, 2 ϑ 4 u 2 u 2 ϑ 4 u ϑ 4 2 ϑ 4. From these equations the values ϑ 4 u = ϑ 4 u 2 = easily follow. In this way, all of the 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, ij u = 2 log σ 4 u = η ω u i u j ij + ϑ4 u i u ϑ 4 u j u ϑ 4 u 2 2 ϑ 4 u i u j u. ϑ 4 u Combining these with the algebraic evaluations of the functions gives ϑ 4 u u ϑ 4 u u ϑ 4 u 2 E ϑ4 m K m + u u ϑ 4 u 2 u ϑ 4 u 2 ϑ 4 u 2 u ϑ 4 u 2 u ϑ 4 u 2 2 ϑ 4 u u u = m x x x 2 x 2, ϑ 4 u x 2 y x y 2 y + y 2 2 ϑ 4 u u 2 u ϑ 4 u = x 2 y mx 2 x y 2 mx, x 2 y x y 2 y + y 2 x 2 x y2 2 x 2 2 x 2 y 2 x x 2 x 2 y + x y 2 + y y 2 2, 2 ϑ 4 u 2 u 2 u m x y 2 x 2 y = ϑ 4 u and expanding these identities at x = x 2 =, using the known values ϑ 4 u = ϑ 4 u 2 =, and equating coefficients of t x t x 2 gives, respectively, = 2 ϑ 4 u u ϑ 4 = E m K m = 2 ϑ 4 u 2 u 2 ϑ 4, 2ϑ4 u u 2, ϑ 4, from which the remainder of the differential coefficients of ϑ 4 follow. Once all of the zeroth and first order differential coefficients are known and the second order ones are known for ϑ 4, the second 7

43 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 ζu. The function x x2 F x, x 2 = ζu dv + dv is actually a single valued function of x and x 2, hence a vector of meromorphic functions of x and x 2. 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 x goes around the period loop A, u increases by the first column of ω and thus ζu increases by the first column of η by Proposition.4. while x dv also increases by the first column of η by the definition of the period matrices. Thus the difference is unaltered when x goes around A 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 x F x, x 2 = ζ u F 2 x, x 2 = ζ 2 u x x2 dv + dv 2 + x2 dv, dv 2 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 dv at x = gives dv = m/ t 2 x 2m 2/ t x + m m 2/ t x + + m m / + dt x, so that x dv = m/ t x + Ot x. m 2/ t 2 x Now, by the chain rule and the algebraic expressions just derived for the functions ij, the partial 8

44 derivative of ζ u with respect to x is ζ u = du u + du 2 2 u x dx dx = m x x x 2 x 2 y x 2 y x y 2 y + y 2 x x2 y mx 2 x y 2 mx + x 2 y x y 2 y + y 2 y 2 = m/ t2 x + m 9m 2/ t5 x +, where t x is the local coordinate at x =, and thus integrating in x gives ζ u = m / t x +. Upon comparing the principal parts of ζ u and x v at x =, we see that F x, x 2 does not have a pole at x =. Since the only other possibility of a pole is at x = γx 2, and this would give at most a simple pole, F x, x 2 is a function of order at most one in x. Since there are no functions of order one on X, F x, x 2 must be constant in x. Since it is symmetric in x and x 2, it must be an absolute constant. Therefore, F x, x 2 = F, =. Since ζ 2 u has poles at most {, γx 2 } and x dv has a double pole at, we see that F 2 x, x 2 has poles at most {,, γx 2 } as a function of x. The expansion of ζ 2 u at x = may be derived exactly as the expansion for ζ u was, and it is ζ 2 u = m / ẋ 2 t x +. By combining this expansion and the expansion of x v 2 at x =, we deduce that F 2 x, x 2 = m 2/ t 2 x + m / ẋ 2 t x +, where t x is the local variable at x =. Coincidentally, m x x 2 = m 2/ t 2 x ẋ ẋ + m / ẋ 2 t x + 2 9

45 has exactly the same principal part. As noted earlier, x x 2 has poles {,, γx 2 } as a function x x 2 of x. Therefore, F 2 x, x 2 = m x x 2 + c, x x 2 where c is an absolute constant. Upon letting x and x 2 tend to, we deduce that c =. 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. Corollary.5.. We have ϑ u + ϑ u = ϑ 2 u + ϑ 4 u. Proof. Elimination of x and x 2 from the first three identities in Theorem.5. along with the values of the theta constants given in Corollary.5.2 gives this identity. Corollary.5.4. When u = x du + x 2 du, the points x, y and x 2, y 2 may be determined uniquely up to a permutation of the pair from the following formulas, except when u ɛ in which case the infinite solution set is parametrized by x = x, x 2 = γx for any point x: σ u σ 4 u = x x 2, σ 2 u σ 4 u = x x 2, σ u σ 4 u = mx mx 2, σ uσ u σ 2 uσ 4 u = y x y 2 x 2, m σ 2uσ 22 u σ 2 uσ 4 u + mσ 4uσ 44 u σ 2 uσ 4 u = y x + y 2 x 2. Proof. The stated formulas follow by combining the various inversion formulas in Theorem.5.. Any two of the first three give x and x 2 up to a permutation. The corresponding y coordinates y and y 2 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 x x 2, x x 2, and mx mx 2, a u such that x and x 2 are not uniquely determined must be a common zero of 4

46 σ 4 u and at least two of σ u, σ 2 u, and σ u. The only characteristic with this property is ɛ. To show that this is indeed the only such u, suppose that u = x du + x 2 du z du + z 2 du. Then z x du+ z 2 x 2 du, and thus there is a meromorphic function with poles {x, x 2 } and zeros {z, z 2 }. By Proposition.2.2, we must have x 2 = γx and z 2 = γz, and so u = x du + γx du = ɛ is the only such point in C 2 /ω Z 2 + ω 2 Z 2. If O u n denotes a remainder in which all terms have total degree at least n, it is also clear from the expansions σ σ 4 ɛ + u = m / u + u Ou u 2 + u 2 + Ou, σ 2 m/ u 2 + u 2 ɛ + u = + Ou σ 4 m / u 2 + u 2 + Ou, σ ɛ + u = m / u + u Ou σ 4 u 2 + u 2 + Ou, that the expressions for x x 2 and mx mx 2 do not have unique limits as u tends to ɛ but that x x 2 tends to m m. This is in agreement with what was obtained above since x γx = x mx m x = m m. In Weierstrass s approach to elliptic functions, the x and y coordinates on an elliptic curve are given by u and 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 u. A similar list of generators can be given for the field of hyperelliptic functions on ω Z 2 + ω 2 Z 2, but the situation is not as simple, as each point u C 2 /ω Z 2 + ω 2 Z 2 is associated with two points on X. If fu is periodic on ω Z 2 + ω 2 Z 2, then f x du + x 2 du is a symmetric bi-meromorphic function of x and x 2 on X. While x and x 2 cannot individually be solved uniquely as meromorphic functions of u, any symmetric function of x and x 2 is single valued, and hence a meromorphic function of u. In this way, the field of hyperelliptic functions on Z 2 ω + Z 2 ω 2 corresponds to the field of symmetric bi-meromorphic functions on X. Combining this with Proposition.2., we then have the following theorem. Theorem.5.5. The field of invariants of C [x, y, x 2, y 2 ] under the action of g : x, y, x 2, y 2 4

47 x 2, y 2, x, y is generated by the five elements x + x 2, x x 2, y + y 2, y y 2, y /x + y 2 /x 2, and thus the field of hyperelliptic functions on ω Z 2 + ω 2 Z 2 is generated by σ 2 uσ u σ 4 u 2, σ uσ 2 u σ 4 u 2, σ u σ 4 u, σ 2 u σ 4 u, σ uσ 2 u 2 σ u σ 4 u 4. Proof. A theorem due to Noether see Theorem 5 of Chapter 7 in [2] 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 x + x 2, y + y 2, x 2 + x 2 2, x x 2, y 2 + y 2 2, y y 2, x y + x 2 y 2, x 2 y + x y 2, which can be reduced to x + x 2, y + y 2, x x 2, y y 2, x 2 y + x y 2. Every element in the field of invariants can be written as N D where N and D are ring elements with no common factors. If this ratio is fixed by g, we should have N gn = D gd, and since N and D have N no common factors, gn = D gd = c where c is a constant. Since g has order 2, c must be or. If c =, both N and D are in the ring of invariants, and we are done. If c =, write N D = x x 2 N x x 2 D, and the numerator and denominator are in the ring of invariants. So we see that the same five elements generate the field of invariants as well. It now suffices to establish that each of the five elements x + x 2, x x 2, y y 2, y + y 2, and y /x + y 2 /x 2 can be written rationally in the five displayed sigma function quotients R,..., R 5. From the first four equations stated in Theorem.5.4, x + x 2, x x 2, and y y 2 are rational in R, 42

48 R 4, and R 5. For y /x + y 2 /x 2 we have y x + y 2 x 2 = mx 2 y x mx mx 2 y x mx y 2 y x 2 x + y 2 x 2 y 2 x 2 mx x 2 y y 2 x x 2 x x 2 + mx 2y 2 x x mx y2 2 x 2 x 2 y y x mx 2 x 2 = x x 2 mx 2 { } { mx x 2 y y 2 x x 2 = y x x 2 x x 2 mx 2 x mx { x x 2 mx mx 2 y y 2 mx 2 y 2 x 2 } } { y x y 2 x 2 y x mx and the first and third terms in brackets are rational in R, R 4, and R 5, while the second and fourth are R mr 2 and mr 2 respectively from the inversion formulas. Similarly, y mx 2 y x mx 2 x 2 y + y 2 y + y 2 = y mx 2 y x mx 2 x 2 { } { + m m x + x 2 y y 2 x x 2 = y x x 2 mx 2 y x mx 2 x 2 { } { y mx mx 2 x x 2 x y 2 x x 2 y y 2 mx 2 from which y + y 2 is rational in the R i as before. } x 2 y x mx y 2 x 2 y 2 x 2 } },,.6 Addition Formulas The classical addition formula for the elliptic integral, x dt x 2 t t mt + dt x 2 t t mt = dt 2 t t mt.6. where x = x x2 mx2 + x 2 x mx mx x 2, is most easily derived via the inversion formulas..6 by deriving an addition formula for the various products Θ a u + vθ b u v as in Section.4 of [27] and then taking quotients. The 4

49 cubic analogue we are interested in is x x2 x x4 x5 du + du + du = du + du,.6.2 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 inversion formulas in Theorem.5., we will have the ability to solve for x 4 and x 5 if we have addition formulas for the ratios σau+v σ 4 u+v. Although there are 8 expressions of the form ϑ a u + vϑ b u v, it suffices to consider only the expression ϑ u + vϑ u v since the shifts u u ɛ a ɛ b, and v v ɛ a + ɛ b convert this expression to the general form. Theorem.6.. For u, v C 2, the ϑ functions satisfy the addition formula ϑ u + vϑ u v = ϑ 2uϑ 22 uϑ 2 vϑ 22 v ϑ 2 ϑ + ϑ uϑ uϑ 2 vϑ 4 v ϑ 2 ϑ 4 = ϑ u 2 ϑ v 2 ϑ 2 /ϑ ϑ u 2 ϑ 4 vϑ 44 v ϑ 2 /ϑ 4 + ϑ 4vϑ 44 vϑ 4 uϑ 44 u ϑ 2 ϑ 4/ϑ 2 + ϑ 4uϑ 44 uϑ 4 vϑ 44 v ϑ ϑ 4 + ϑ 2uϑ 4 uϑ vϑ v ϑ 2 ϑ 4 ϑ 4uϑ 44 uϑ v 2 ϑ 2 /ϑ 4 + ϑ uϑ uϑ 2 vϑ 4 v ϑ 2 ϑ 4 + ϑ 2uϑ 4 uϑ vϑ v ϑ 2 ϑ 4. Proof. Fix v, and consider the first identity as a function of u. Every term is a second order theta function with zero characteristic ɛ. But the space of all such functions is a vector space of dimension 2 2. In particular, if the four functions ϑ 2 uϑ 22 u, ϑ 4 uϑ 44 u, ϑ uϑ u, ϑ 2 uϑ 4 u are linearly independent, the left-hand side is such a linear combination. Now at each of the four points ɛ 2, ɛ 4, ɛ, and ɛ 2 exactly three of the functions vanish and the other becomes a product of 44

50 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 ϑ u + vϑ u v = Aϑ 2 uϑ 22 u + Bϑ 4 uϑ 44 u + Cϑ uϑ u + Dϑ 2 uϑ 4 u. By setting u in turn to ɛ 2, ɛ 4, ɛ, and ɛ 2, 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 ϑ u 2, ϑ 4 uϑ 44 u, ϑ uϑ u, ϑ 2 uϑ 4 u in exactly the same way. The addition formulas just obtained may be exploited by expanding both sides as power series in u and u 2. The series expansions of the σ functions are much cleaner than the expansions of the ϑ functions since the introduction of the factor c a e 2 ut η ω u eliminates the appearance of the transcendental functions K m and E m from the series coefficients. The expansions up to the third order are listed here for convenience. σ u = + mu u 2 + O u 4, σ 2 u = mu / + mu 2/ + O u 4, σ 22 u = + mu / mu 2/ + O u 4, σ 4 u = + u u 2 mu / mu 2/ + O u 4, σ 44 u = + u u 2 + mu / + mu 2/ + O u 4,.6. σ u = u u 2 2/2 + u 2 u 2 /2 + O u 4, σ u = u + u 2 2/2 + u 2 u 2 /2 + O u 4, σ 2 u = u 2 u 2 /2 + u u 2 2/2 + O u 4, σ 4 u = u 2 + u 2 /2 + u u 2 2/2 + O u 4. Armed with these series expansions and addition formulas, we can easily evaluate all derivatives of 45

51 ratios of σ functions. These formulas are the cubic analogues of the differential equations sn u = cnudnu, cn u = snudnu, dn u = msnucnu. Corollary.6.2. With the notation fu = σ u σ 4 u = σ2 uσ 44 u σ 2u σ 4 u = σ u σ 4 u = σ u σ 4 u = σ 2u σ 4 u = σ 4u σ 4 u = σ 22u σ 4 u = σ 44u σ 4 u = σ 4 u 2, σ uσ 4 u σ 4 u 2 fu,u 2 u, fu,u 2 u 2,, m σ 2uσ 22 u σ 4 u 2 m σ 4uσ 44 u σ 4 u 2, σ uσ u σ 4 u 2 m σ 2uσ 2 u σ 4 u 2, m σ uσ 22 u, m σ uσ 4 u σ 4 u 2 σ 4 u 2 + σ uσ 22 u σ 4 u 2, σ 2 uσ 44 u σ 4 u 2, σ uσ u σ 22 uσ 44 u σ 4 u 2, σ 4 u 2, σ uσ 44 u σ 2 uσ u σ 4 u 2, σ 4 u 2 + m σ uσ 2 u σ 4 u 2 σ uσ 2 u σ 4 u 2, m σ uσ 44 u σ 4 u 2 m σ uσ u σ 4 u 2, m σ 2uσ 4 u σ 4 u 2., m σ uσ 2 u σ 4 u 2 Proof. Use the addition formula to get a formula for σau+vσ b u v, and expand both sides up to σ b u 2 the first order in v and v 2. The left-hand side is σ a u σ b u + u σa u σa u v + u2 v 2 + Ov 2. σ b u σ b u,, The expansion of the right hand side may be found using.6., and equating the coefficients of v and v 2 in each case gives the identities. Recall the addition formula for elliptic integral of the second kind. That is, when.6. holds, a similar equation holds for the integrals of the second kind but with an error term. This equation 46

52 see pg. 65 of [27] is: x mt x2 2 t t dt + mt 2 t t dt = m x x x2 x + mt 2 t t dt. The error term we wish to evaluate is F in the equation x x2 x x4 x5 dv + dv + dv = F x, x 2, x, x 4, x 5 + dv + dv when.6.2 holds. By the inversion formulas, this is equivalent to an addition formula for the ζ function. Theorem.6.. For u, v C 2, the ζ function satisfies the addition formula ζu+v ζu ζv = m σ 4 uσ 4 vσ 4 u + v σ uσ vσ 2 u + v + σ 2 uσ 2 vσ u + v σ 4 uσ 4 vσ 22 u + v + σ uσ vσ 2 u + v. Proof. We will only show how the formula for ζ may be proved as the proof of the formula for ζ 2 is essentially the same. We would like to show that ζ u + v ζ u ζ v = m σ uσ vσ 2 u + v σ 4 uσ 4 vσ 4 u + v + mσ 2uσ 2 vσ u + v σ 4 uσ 4 vσ 4 u + v. By multiplying both sides by σ 4 uσ 4 vσ 4 u + v and using the definition of ζ, this is seen to be equivalent to a formula for the order 2 theta function with characteristic ɛ 44 + v given by σ 4uσ 4 vσ 4 u + v + σ 4 uσ 4vσ 4 u + v σ 4 uσ 4 vσ 4u + v.6.4 where f ij u denotes fu,u 2 u i u j. This expression may be represented as a linear combination of four linearly independent theta functions of u of the same type as Aσ uσ vσ 2 u + v + Bσ 2 uσ 2 vσ u + v +Cσ uσ vσ 44 u + v + Dσ 4 uσ 4 vσ 4 u + v..6.5 Since.6.4 is also symmetric in u and v, A, B, C, and D are independent of v as well. By a 47

53 direct calculation, the series expansion of.6.4 is found to be σ 4 v 2 vu + σ 4 v v u 2 + Ou while the series expansion of.6.5 is Cσ vσ 44 v + Dσ 4 vσ 4 v + Cσ vσ 44v + Dσ 4 vσ 4v + Aσ 2 vσ v u Cσ vσ 2 44v + Dσ 4 vσ 2 4v + Bσ vσ 2 v u 2 + Ou 2. Thus, C = D = and A and B are then given by A = σ 4v 2 v σ 2 vσ v = m, B = σ 4v v σ vσ 2 v = m, where the final simplification in both cases may be obtained by substituting the algebraic expressions in Theorem.5. for the relevant functions..7 Cubic Elliptic Functions The main aim of this section is a derivation of Ramanujan s inversion formula for..5 as a special case of Theorem.5. and obtaining an elegant cubic analogue of the addition theorem for elliptic integrals of the second kind see pg. 65 of [27], namely, Zu + v = Zu + Zv k 2 snusnvsnu + v. where Zu is Jacobi s zeta function. This addition theorem is given as a series-product identity via q; q 2 q 2 ; q 2 4 n= n q n q 2n xn y n x n y n = q xy f x, q2 x f y, q2 y f f qy, q y f f qx, q x xy, q2 xy qxy, q xy

54 Recall that fa, b is defined by..2 as a; ab b; ab ab; ab. We will also make extensive use of the Dedekind η function defined by ητ = e 2πiτ/24 e 2kπiτ = q /24 q; q, where q = e 2πiτ. The addition theorem.7. has the following cubic analogue. k= Theorem.7.. For q < x, y, xy < q, q; q q ; q n= n q n + q 2n q n xn y n x n y n = q xy f x, q x f y, q y f qxy, q2 xy f q 2 x, q x f q 2 y, q y f q 2 xy, q xy f x, q x f y, q y f q 2 xy, q xy + q. f qx, q2 x f qy, q2 y f qxy, q2 xy In order to deduce this identity as a special case of Theorem.6., 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.. The periods of a related basis and the corresponding differentials of the second kind are given in the table below. A A 2 B B 2 du 2 du 2 πik m 4 πik m 2π K m du 2 + du 2π K m 4 πik m 2 πik m dv 2 dv 2 πie m 4 πie m 2π K m E m dv 2 + dv 2 πe m 4 πi K m E m 2 πi K m E m Thus the differentials du 2 du and du 2 + du, 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:. The equation defining X can written in the Rosenhain normal form w 2 = z z λ z λ 2 z λ z in which λ = λ λ 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 are quite complicated, but the interested reader can find the equations in [6]. 49

55 . The normalized period matrix 2τ τ τ 2τ can be transformed by a transformation of order 2 to a diagonal matrix. This results in expressions for the cubic theta functions as sums of products of genus theta functions. The explicit factorization can be given as: a,b= q a2 +ab+b 2 z a z b 2 = = = f A,B= qz, qz f b even q A2 +B 2 z A B z 2B 2 + q a2 +ab+b 2 zz a 2 b + q a2 +ab+b 2 zz a 2 b q z 2 2 z, q z z 2 2 A,B= + qz 2 f b odd q A B+ 2 2 z A B z2 2B+ q 2 z, z f q 6 z2 2, z z z Integrals of both the differentials du 2 du and du 2 + du can be inverted easily with specializations of the cubic theta functions ϑ a u. Since the formulas for du 2 du seem to come out nicer and there would be no new ideas introduced by considering du 2 + du as well, only the inversion of the integral of du 2 du will be treated in detail here. Furthermore, the integral in..5, when properly interpreted, is directly related to du 2 du. Define the one variable cubic theta functions for a characteristic ɛ a in terms of the two variable functions by θ a u = Θ a u, ϑ a u = ϑ a ζu ζ 2 u, σ a u = d a σ a ζu ζ 2 u ζu = u log σ 4u,, u = 2 u 2 log σ 4u, where d a is chosen so that the first non-zero differential coefficient of σ a u is one. Since ϑ a u was 5

56 defined as Θ a ω u and ω ζu, ζ 2 u T = 2πiK m u, T, we see that ϑ u = a,b= q a2 +ab+b 2 e au K m, E m σ u = K m e K m u2 ϑ u, θ u = q a2 +ab+b 2 e 2aπiu, a,b= ϑ a u = θ u a. 2πiK m Also define the differentials of the first and second kind as x d u = du 2 du = y 2 dx, y mx x d v = dv 2 dv = y 2 + m 2mx dx. y These have periods ω = 2πi K m, ω 2 = 2π K m, η = 2πi E m, η 2 = 2π K m E m, respectively. As we shall see shortly, integrals of these differentials are inverted by the functions defined above. For the functions ϑ a, σ a, ζ, and, the period lattice is ω Z + ω 2 Z. The θ a u have period lattice Z + τz. The functions ϑ a u, θa u all have two zeros in the fundamental parallelogram they are theta functions of order two. The generalized Legendre identity becomes ω η 2 ω 2 η = 2πi, and the quasi-periodicity of ζu is expressed by ζu + a ω + bω 2 = ζu + 2a η + 2b η 2 for integers a and b, while θ u satisfies θ u + = θ u, 5

57 θ u + τ = e 2πi2u+τ θ u. Theorem.7.2. For the elliptic sigma functions σ a u we have the following inversion formula. Let m /2 denote one of the fixed points of the automorphism δ the point on the second sheet with x coordinate m /2. Recall the notation ẋ = y x, and define the new variable φ in Ramanujan s integral..5 up to additive multiples of π by e 2iφ = ẋ mx 2. Then with we have u = x m /2 d u 2i φ cos sin m sin φ dφ, m sin 2 φ x / m / σ u σ 4 u = + ẋ, x / m 2/ σ u σ 4 u = + ẋ, x / m / σ 2u σ 4 u = + ẋ x + ẋ, x / m 2/ σ 22u σ 4 u = + ẋ + mx ẋ, σ u σ 4 u = + ẋ + ẋ, σ 4 u = σ u, σ 2 u = σ u, σ 44 u = σ 4 u, and therefore σ u + σ 2 u = e 2iφ/ = σ u σ 4 u σ u, σ 22 u σ u = e 2iφ/ = σ u σ 4 u σ u, σ u σ 4 u = 2 cos 2 sin m sin φ = 2i dφ du. 52

58 Also, ζu = 2 m 2 ẋ 2mx ẋ u = m σ 2u σ u σ 4 u 2 = 2 + mx2 x m σ u σ 22 u σ 4 u 2 + ẋ ẋ 2. x + 2 d v, m /2 + 2 σ u σ 4 u, Proof. In Theorem.5., set x = δσx, x 2 = γσ 2 x. Then from Proposition.4.7, u = = δσx du + = ζ ζ ζ 2 ζ 2 γσ 2 x ζ2 ζ = ζ ζ ζ 2 ζ 2 ζu ζ 2 u x x + ɛ. du x du + ζ ζ 2 du + ζ2 ζ x /m du du + ζ ζ 2 du + ζ2 ɛ 2 + ζ ζ ζ 2 /m ɛ du This last equality uses the fact that m /2 du ω 2, an identity that follows easily from the fact that m /2 is a fixed point of δ see Proposition.4.7. Performing these substitutions in Theorem.5., shifting the characteristics by ɛ and simplifying gives the first group of equations. The next three follow from the relationship between φ and the point x. The equation for u follows from the equation for u given in Theorem.5. and the chain rule. The equation for ζu follows from the fact that by the periodicity properties of ζu, x F x = ζu 2 d v m /2 is a meromorphic function of x on X. With the help of the equation for u we can find df dx, 5

59 integrate this, and use the known values at x = m /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.7.2 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.7.2. This integral maps X surjectively and generically two-to-one onto the fundamental parallelogram. For example the two zeros of θ u are determined from the four solutions of + ẋ + ẋ =, thus the x-coordinates of the zeros are all ± m. Care must be taken in choosing the correct y coordinate, and the result of simplifying the integrals obtained for the zeros of the four functions θ 4 u, θ u, θ u, and θ u is given in the following table. a zeros of θ a u 4 τ,2τ,2τ,τ 2 + τ c, 2 + 2τ + c where, at least for < m <, c = c m = + m 4πiK m = i log 2 2π im 24π im2 576π + Om. t / t 2/ mt / t 2/ t / dt mt 2/.7. 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. Proposition.7.. With x = e 2πiu, and q = e 2πiτ = exp 2π K m K m and c as in.7., θ 4 u = q; q q ; q f qx, q2 f q 2 x, q, x x 54

60 θ u = x / q / q; q q ; q f qx, q2 x θ u = x / q / q; q q ; q f qx, q2 x 2πiK m ζ u = 4πiE m u + K m n= n f q x,, x f q 2 x, q, x q n + q 2n q n xn for q < x < q. 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 c. Since θ 4 u = Θ u + / 2/, we have, from the factorization of Θu into genus sums by.7.2, q θ 4 u = f x, q x f = c f qx, q2 x ζqx, q q + ζ 2 6 qf ζx f q 2 x, q x. x, x f ζq 2 x, ζx Upon setting x = ζ 2, the second term on the right vanishes, and we are left with c = f q, qf ζq, ζ 2 q f ζ 2 q, ζq 2 f ζ 2 q 2, ζq = q; q q ; q after some manipulation of the q-products. The proofs of the remaining three product representations are similar. The formula for ζu follows from the definitions ζu = u log σ 4 ζu ζ 2 u = u 2 η ω ζu ζu log θ u 4 ζ 2 u ζ 2 u 2πiK m = [ ] E m u K m u2 log θ u 4, 2πiK m 55

61 along with the product formula for θ 4. Corollary.7.4. The theta constants are given by Θ 2 τ = ητ ητ, Θ 4 τ = ητ ητ, Θ τ = η τ/ ητ + ητ ητ. Proof. The second equality follows from letting u = in Proposition.7.. The first then follows from the second by replacing τ with /τ see Theorem.8.2 for the effect of this transformation on the theta functions. The last equality follows from the identity Θ τ = Θ 2 τ + Θ 4 τ, which we may prove as follows. If we let, for i =,, 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. Also, A 2 + AB + B 2 = a + i 2 + a + i b + i + b + i 2 so that Θ 4 τ = ζ A B q A2 +AB+B 2 A,B= 2 = i= a,b= ζ i q b+ i 2 +b+ i a+ i +a+ i 2 = Θ τ + ζθ 2 τ + ζ 2 Θ 22 τ = Θ τ Θ 2 τ. We are now in a position to easily deduce Theorem.7.. The addition formula for ζu may 56

62 be obtained from the addition formula for ζu by using the relation ζu = ζ ζu ζ 2 u ζ ζ 2. By setting u = ζu, ζ 2 u T and v = ζv, ζ 2 v T in Theorem.6., we obtain ζu + v ζu ζv = m σ u σ v σ u + v + σ 2 u + v σ 4 u σ 4 v σ 4 u + v m σ u σ v σ22 u + v σ u + v σ 4 u σ 4 v σ 4 u + v = m σ u σ v σ u + v σ 4 u σ 4 v σ u + v m σ u σ v σ u + v σ 4 u σ 4 v σ u + v,.7.4 where the last equality follows from the identities σ u+ σ 2 u σ 4 u given in Theorem.7.2. By setting x = exp u K m, y = exp = σ u σ u v K m and σ 22u σ u σ 4 u = σ u σ u 2π K and q = exp m K m in.7.4, and using the series representation of ζu and the product representations given in Proposition.7., we produce Theorem Modular Transformations In this section we will consider how the theta functions Θ a u τ are affected by the transformations τ /τ and τ pτ. Since τ = i K m, K m the transformation m m corresponds to τ /τ. Such a transformation corresponds to switching the period matrices ω and ω 2, and is a direct analogue of the imaginary transformation of Jacobi where snu is transformed into sniu, but here, as we will see, the role of the fourth root of unity is played by the matrix whose fourth power also is the identity see Theorem.8.2. For small p 2 or, the full effect of the transformation τ pτ on the theta functions, modulus, 57

63 and period matrices can be given explicitly, see Theorem.8. and Theorem.8.5. 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 m p denote the modulus associated with pτ, let M p = p K m p K m be the multiplier associated with the periods of the differentials of the first kind, and let N p = p 2E m p K m p K m 2E m K m K m p be the multiplier associated with the periods of the differentials of the second kind. For rational p, m and m p are algebraically related, while M p and N p are algebraic functions of m and m p. We refer loosely to these four quantities as the modular quantities associated with the transformation τ pτ. The effect of this transformation on the modulus m and the four period matrices is completely determined once the modular quantities are known. Proposition.8.. The various modular quantities are given in terms of Θ functions by: m p τ / = Θ 2pτ Θ pτ, N p τ = 2πi m pτ / = Θ 4pτ Θ pτ, d dτ log Θ 2pτΘ 4 pτ Θ 2 τθ 4 τ Θ τθ pτ M p τ = p Θ pτ Θ τ,, and M p and N p can be calculated by: Mp 2 = p m m dm p,.8. m p m p dm N p = 2m p M p 2m p M p + 6m p m p dm p dm p..8.2 Proof. The formulas for m p and M p in terms of Θ functions follow directly from Corollary.5.2, so only the formula for N p needs to be proved. First we note that by the generalized Legendre 58

64 identity.. and..4, dτ = d i K m p dm p dm p p K m p = p 2πim p m p K m p 2. Therefore, by a direct calculation, = dm p dτ 2πi d dτ log Θ 2pτΘ 4 pτ Θ 2 τθ 4 τ 2πi Θ τθ pτ d dm p {log m / p m p / K m p 2 } dm dτ K m K m p 2πi = p 2E m p K m p K m = N p. d dm {log m / m / K m 2 } K m K m p 2E m K m K m p Next, dm p dm = dm p dτ dm dτ = 2pπim p m p K m p 2 2πim m K m 2, and so the second formula for M p is clear. The formula for N p follows by differentiating the identity K m M p = pk m p with respect to m p and rearranging..8. Transformations of the Theta Functions Theorem.8.2. If f aɛ2 +bɛ 4 denotes the function f with characteristic aɛ 2 + bɛ 4 for a, b Z/Z, then, under the complementary transformation, the theta functions satisfy Θ aɛ2 +bɛ 4 u u 2 τ ϑ aɛ2 +bɛ 4 u, u 2 T m = ζ ab τ e 2πiτu2 u u 2 +u 2 2 Θaɛ4 bɛ 2 = ζ ab K m K m e 2π u u 2 K mk m ϑ aɛ4 bɛ 2 τu 2τu 2 τu τu 2 τ, u 2, u T m, 59

65 σ aɛ2 +bɛ 4 u, u 2 T m = aba b ζ ab e u u 2 σ aɛ4 bɛ 2 u2, u T m. Proof. These can be obtained by applying the general modular transformation for theta functions with arbitrary period matrices given in Ch. 5 of [], but here we will prove that Θ u u 2 τ = Ce 2πi τ u 2 u u 2 +u 2 2 Θ τ u 2u 2 u u 2 τ for some constant C simply by noting that both sides are theta functions on Z 2 + ω ω 2Z 2 of order with zero characteristic. That is, they satisfy the relations f u +, T = fu, f u +, T = fu, f u + 2τ, τ T = e 2πiu+τ fu, f u + τ, 2τ T = e 2πiu2+τ fu. The value of C may be deduced by setting u = to obtain Θ τ = CΘ /τ. Thus, by Corollary.5.2, C = Θ τ Θ /τ = K m / K m =. τ Replacing τ with /τ gives the first identity stated. The transformations for the remaining characteristics follow by shifting this identity; expect for an exponential factor, the complementary transformation just permutes the nine theta functions. In translating the identity to ϑ a u and σ a u, we only need the evaluations given in Corollary.5.2 and the generalized Legendre identity... Thus the transformation τ /pτ induces the transformations m m p, m p m, 6

66 M p p/m p N p N p on the modular quantities. To produce cubic analogues of the general p th order tranformation of the classical incomplete elliptic integral of the first kind see..9, we need formulas relating the Θ a pu pτ to polynomials in the Θ a u τ. Armed with these formulas and the inversion formulas in Theorem.5., we can provide the explicit relationship between the pair x and x 2 and the pair x and x 4 induced by M p x x2 x x4 du + du = du + du, and this constitutes a cubic analogue of the classical p th order transformation for elliptic integrals in..9. Theorem.8.. The theta functions satisfy the degree 2 modular identities Θ 2u 2τ = Θ 22τ Θ 2 τθ τ Θ 2uΘ 22 u + Θ 42τ Θ 4 τθ τ Θ 4uΘ 44 u = Θ 42τ Θ 4 τ 2 Θ u 2 2 Θ 22τΘ 4 2τ Θ 2 τθ 4 τ 2 Θ 2uΘ 22 u = Θ 22τ Θ 2 τ 2 Θ u Θ 22τΘ 4 2τ Θ 2 τ 2 Θ 4 τ Θ 4uΘ 44 u. Formulas for Θ[aɛ 2 + bɛ 4 ]2u 2τ may be obtained with the shift u u + aɛ 2 bɛ 4. For some parameter t, the various modular quantities are related by m = tt + 92 t + 6, m 2 = t2 t + 9 t + 2, M 2 = t + 2 t + 6, N 2 =, and the theta function identities may be recast for the sigma functions as e 9t+8 t+6 2 u u 2 σ M 2 u m 2 = = = t t + 9t + 2 σ u 2 + 9t + 8 t + 9t + 2 σ 4uσ 44 u t + 62 t + 2 σ u 2 tt + 9 t + 2 σ 2uσ 22 u tt + 9 t + 6t + 2 σ 9t + 8 2uσ 22 u + t + 6t + 2 σ 4uσ 44 u. 6

67 Proof. The substitution u u + aɛ 2 bɛ 4 turns Θ 2u 2τ into Θ 2 u + aɛ 2 2bɛ 4 2τ = Θ 2 u + aɛ 2 + bɛ 4 2τ, which is Θ[aɛ 2 + bɛ 4 ]2u 2τ up to an exponential factor. Thus it suffices to give formulas for Θ 2u 2τ only. On the period lattice Z 2 + ω ω 2Z 2, the theta function Θ 2u 2τ has order 2 and zero characteristic, and the five functions Θ u τ 2, Θ 2 u τθ 22 u τ, Θ 4 u τθ 44 u τ, Θ u τθ u τ, Θ 2 u τθ 4 u τ have the same property. Furthermore, as used in the proof of the first addition formula, the four functions Θ 2 uθ 22 u, Θ 4 uθ 44 u, Θ uθ u, Θ 2 uθ 4 u are linearly independent, and Θ 2u 2τ can be written as linear combinations of these. The coefficients may be isolated by evaluating both sides at u = ɛ 2, ɛ 4, ɛ, ɛ 2. This gives the first formula for Θ 2u 2τ exactly as it is stated. The next two follow similarly by using the sets of functions Θ u 2, Θ 4 uθ 44 u, Θ uθ u, Θ 2 uθ 4 u and Θ u 2, Θ 2 uθ 22 u, Θ uθ u, Θ 2 uθ 4 u. However, in both cases the parameterizations of the modular quantities are needed to simplify the coefficients of Θ 2 uθ 22 u and Θ 4 uθ 44 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.9 where modular equations 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... 62

68 The sum of cubes identity ϑ u + ϑ u = ϑ 2 u + ϑ 4 u from Corollary.5. is interesting for methods one might employ to prove it. We can prove it:. algebraically by setting u = x du + x 2 du, 2. by actually multiplying out and combining the series,. by verifying that both sides agree when written in terms of some basis of the theta functions of order, 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 {ϑ a uϑ b uϑ c u} a,b,c where ɛ a +ɛ b +ɛ c is some fixed characteristic has dimension 6 instead of the required dimension 9 = 2. Hence, we cannot be sure that a given theta function of order is a linear combination of functions of the form ϑ a uϑ b uϑ c u. The method employed here to obtain and prove cubic theta function identities of order or higher is that of the fourth. A precise statement of to what order the series expansions must be calculated is given in Proposition.8.4. This proposition is a genus 2 analogue of the fact that an elliptic theta function of order r has exactly r zeros counted according to multiplicity inside its fundamental parallelogram unless it vanishes identically. Proposition.8.4. Let fu be a cubic theta function of order r, and suppose that at each of the 9 characteristics fu has a zero of order at least n = max 2r+ 4, 2r that is, all differential coefficients of fu of order less than n vanish. In this case, fu vanishes identically. 6

69 Proof. Recall that a cubic theta function of order r is a holomorphic function fu satisfying f u + a + ω ω 2 a 2 = e 2πiɛ 2 a ɛ a 2 e πir2at 2 u+a T 2 ω ω 2 a 2 fu for integer vectors a and a 2. Fix e C 2, and suppose that gx = fe + x dûu does not vanish identically. The number of zeros of gx on X is given by d log gx 2πi X where X is obtained by dissecting X along its homotopy basis A, A 2, B, B 2. The path obtained is shown in Figure.2, and d log gx is a single-valued on this dissection of X. The integral may be evaluated as d log gx = 2πi X 2πi = 2πi = 2πi = 2r. 2 i= 2 i= A i + A i + B i + A i 2 2πirdu ˆ i A i i= A i + d log gx B i d log gx The second line follows since d log gx is the same on B and B 2 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 A i, d log gx + B i = 2πir ˆ du i + d log gx where x + B i is the point on A i after traversing B i. Finally, by definition of the differentials ˆ du i see Section.., A i ˆ dui =. Thus, we see that gx has 2r zeros counted according to multiplicity unless it vanishes identically. The hypothesis n > 2r/4 of the theorem implies that, for each ɛ a, the function f x ɛ a + dûu 64

70 Figure.2: Dissection of the Riemann Surface X B B A 2 X B 2 B 2 2 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 ɛ a. It follows that, for generic x, the function of x 2 given by f x x2 dûu + dûu has a zero of order at least one at any of the four branch points, and it has a zero of order at least 2r at x 2 = γx, since x dûu + γx dûu ɛ. Therefore, the total number of zeros with respect to x 2 is greater than 2r. Thus, for each x, this expression vanishes identically as a function of x 2, and fu must then vanish identically as a function of u. Theorem.8.5. The theta functions satisfy the degree modular identities Θ τθ 4 τ 2 Θ u τ = Θ τ Θ 2 τ Θ 2 τ Θ uθ 4 uθ 44 u + Θ uθ 2 uθ 4 u + Θ uθ 2 uθ 22 u, Θ 4 τθ 2 τ 2 Θ 4 u τ = Θ uθ 2 uθ 44 u + Θ 2 uθ 4 uθ 22 u + Θ uθ uθ 4 u. Θ 4 τ 65

71 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 m = t +, m = t +, M = t + t +, N = 2 t 2 + t + t + t +, t and the ϑ function identities can be recast as e t+2 t+ 2 u u 2 σ M u m = σ uσ 4 uσ 44 u t2 t 2 + t + t + t + σ uσ 2 uσ 4 u + σ uσ 2 uσ 22 u, e t+2 t+ 2 u u 2 σ 4 M u m = σ 2 uσ 4 uσ 22 u + t + 2 2t + 2 σ uσ uσ 4 u t 2 2t + 2 σ uσ 2 uσ 44 u. Proof. The parametric representations are obtained in Section.9, so we need to verify the first and second Θ function identities. By Proposition.8.4, we need to verify that both sides of the proposed identities agree up to the second order at all of the characteristics. Actually, much less calculation is required. Both sides of each identity are invariant under the shift u u + ɛ 4, so we only need to check the characteristics ɛ, ɛ 2 and ɛ 22. Furthermore Θ u is an even function, so we do not need to check ɛ 22 in that case. Shifting by ɛ gives Θ τθ 4 τ 2 Θ u τ = Θ τ Θ 2 τ Θ 2 τ Θ uθ 4 uθ 44 u + Θ uθ 2 uθ 4 u + Θ uθ 2 uθ 22 u, Θ 4 τθ 2 τ 2 Θ 4 u τ = Θ uθ 2 uθ 44 u + Θ 2 uθ 4 uθ 22 u + Θ uθ uθ 4 u. Θ 4 τ Shifting by ɛ 2 gives Θ τθ 4 τ 2 Θ 2 u τ = Θ τ Θ 2 τ Θ 2 τ Θ 2uΘ uθ 4 u + Θ 2 uθ 22 uθ u + Θ 44 uθ 4 uθ u, Θ 4 τθ 2 τ 2 Θ u τ = Θ 2 uθ 4 uθ 4 u + Θ 22 uθ uθ u + Θ 2 uθ 44 uθ u. Θ 4 τ 66

72 Shifting by ɛ 22 gives Θ 4 τθ 2 τ 2 Θ 2 u τ = Θ 4 uθ uθ u + Θ uθ 2 uθ 2 u + Θ 22 uθ 4 uθ 44 u. Θ 4 τ It is easily verified that the series expansions of of both sides of these five proposed identities agree at least up to the second order, so the identity is indeed valid. For example, let us complete the calculations for the first case. We may verify the series coefficients after recasting in terms of the σ functions. Using.6. and the parameterizations of m, m and M, we find that both e t+2 t+ 2 u u 2 σ M u m and σ uσ 4 uσ 44 u t2 t 2 + t + t + t + σ uσ 2 uσ 4 u + σ uσ 2 uσ 22 u have the expansion + 2t2 + 4t + t + 2 t + u u 2 + Ou 4, which agree at least to the fourth order. A degree 4 analogue of the previous two theorems is given by Θ 4u 4τ = Θ 24τ Θ τθ 2 τ Θ 2uΘ 22 uθ u 2 Θ 4 4τ + Θ τθ 4 τ Θ 4uΘ 44 uθ u 2 2Θ 24τΘ 4 4τ Θ τ + 4Θ 4τ Θ 2 τ Θ 4 τ Θ 2 uθ 22 uθ 4 uθ 44 u 4Θ 24τΘ 4 4τ Θ τθ 2 τ 2 Θ 4 τ 2 Θ uθ uθ 2 uθ 4 u..8. This identity was found by trying various characteristics on the right hand side until a set of four coefficients could be found that give an identity; the stated identity is the most concise of the identities found. Unfortunately, it seems that there are no five and six term identities for Θ 5u 5τ and Θ 6u 6τ, respectively. 67

73 .9 Cubic Modular Equations Recall that a modular equation of degree p is a relation between the modular quantities m, m p, M p and N p. In addition to the three modular equations of degree 2, 5 and stated in the introduction, Ramanujan see pg. 2 2 in [4] gives modular equations of degrees, 4, 7, 9, 4, and 2. The modular equations of degrees 4 and 7 read as: M M 4 m / m / m / 4 m 4 / = m / m / 4 + m /,.9. m 4 / M M 7 m / m / m / 4 m 4 / = m / m / 7 + m / m 7 / m/6 m /6 m / m 7 /6 In this section we will obtain many modular equations and recover the ones derived by Ramanujan. In fact, both.9. and.9.2 posses smaller refinements from which the larger whole is easily deduced. As usual, let Γ denote the full modular group, and set Γ n := a c b d Γ : c mod n. Also, A Γ n denotes the field of Γ n-invariant meromorphic functions, where a matrix acts on τ as a c b d : τ aτ + b cτ + d. The action of Γ on Θ u τ can be deduced from the transformation formula given in Ch. 5 of [] since the action of Γ on τ corresponds to the action Ω a a Ω + 2b b b 2b 2c c c 2c Ω + d d 68

74 on the normalized period matrix Ω = 2τ τ τ 2τ. The result is the transformation formula u Θ aτ + b cτ + d cτ + d = χdcτ + de 2πic u 2 u u 2 +u2 2 cτ+d Θ u τ,.9. which bears a resemblance to the complementary transformation given in Theorem.8.2. Here, χd is the non-principal modulus three Dirichlet character, as determined by the generators τ τ + and τ τ/τ + of Γ /±. Under the action of Γ, the function fτ 2 := ητ 2 ητ 2 has three other conjugates f i τ 2. These four functions are the cubic analogues of Weber s invariants, and we set /4 ητ fτ = ητ, f τ = /4 η τ ητ, τ+4 f τ = /4 η ητ, f 2 τ = /4 η τ ητ The precise effect the modular group has on these functions is given in the following table. τ /τ τ τ + f f e 2πi 2 f f f e 2πi 2 f f f 2 e 2πi 2 f 2 e 2πi 2 f 2 e 2πi 2 f e 2πi 2 f These four functions are connected with the main modular function mτ = Θ 2τ Θ τ through the following proposition. Proposition.9.. With m = mτ, m /2 fτ = m /2, m / / f τ = /6 m /6 m /2, fτ 6 = f τ 6 + f τ 6 + f 2 τ 6, 69

75 fτf τf τf 2 τ =. Proof. If we solve the three equations in Corollary.7.4 for ητ, ητ, and ητ/, we obtain ητ = /8 Θ 2 τ /8 Θ 4 τ /8 = /8 m /24 m /8 K m, ητ = /8 Θ 2 τ /8 Θ 4 τ /8 = /8 m /8 m /24 K m, η τ/ = /24 Θ 2 τ /24 Θ 4 τ /8 Θ τ Θ 2 τ /.9.5 = /24 m /72 m /24 m / / K m, hence the formulas for fτ and f τ are evident. To prove the last two, note that f τ 6 + f τ 6 + f 2 τ 6 fτ 6 2 and fτf τf τf 2 τ are both invariant under Γ and must be constant if they do not have a pole at the cusp τ = i. With the expansions fτ = /4 q /2 + Oq, f τ = /4 q /6 + Oq /, f τ = e 2πi 9 /4 q /6 + Oq /, f 2 τ = e 2πi 9 /4 q /6 + Oq /, these expressions are found to be and /2, respectively..9. The Modular Equations If p is any natural number, then m and m p are related by an equation called the cubic modular equation of degree p. The term cubic is used to distinguish these equations from the modular equations for elliptic curves in Legendre s form. The cubic modular equation can be stated as an identity of the form Φ p m, m p = 7

76 as we will show in Lemma.9.2. The actual degree of Φ p in either m or m p is higher than p, since, for example, if p is prime other than, the roots of this equation with respect to m p are the p + quantities where m pτ, m τ, m p τ + p,..., m m τ = mτ = Θ 2τ Θ τ τ + p p, is the main modular function. If the parameter τ is eliminated from the definitions m p = mpτ and m = mτ, the defining relation for a cubic modular equation can be stated as K m p K m p = p K m, K m which is the form that Ramanujan used for defining a cubic modular equation. Examples of these modular equations when p is 2 or can be obtained my eliminating the parameter t from the parameterizations in Theorems.8. and.8.5. For many purposes, though including the calculation of M p and N p, 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 of m and m p exists only when p = 2,, 4 and 6, for the modular function mτ is actually a generator for the function field of X = H/Γ, and the curve defined by m and m p is isomorphic to X p. There are four cases where the genus of X p is zero, namely p = 2,, 4 and 6 see [28]. 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 X p as some simple-looking curve F t, s = where t and s are given explicitly as products of η functions that generate the function field A Γ p of X p. The generators t p when X p is rational are taken directly from [28], as these are, in some sense, the most canonical. Lemma.9.2. The function field of X p is given by A Γ p = Cm τ, m p τ Proof. Let us first prove that the function mτ = m τ is univalent on H/Γ. The Schwarz map τm = i K m K m 7

77 Figure.: Image of the m-plane under the Schwarz triangle map τm. has the following local behavior at the singular points m =, m =, and m =. m 2πiτ = log 27 2πi τ = log m 27 τ = Om, + O m, + cm / + O m 2/, where c is a non-zero constant. Thus the points m =, m =, and m = are taken to the points τ = i, τ =, and τ = /2 + /6, where the local angles are, and π/, respectively. The resulting map is shown in Figure.. From these local expansions at the points m = and m = we have the monodromy of the function τ as m makes one trip around the points m = and m = : at m = : τ τ +, at m = : τ τ τ +. Since these two transformations generate Γ /±, the image of the Schwarz map must be a 72