The Elliptic Modular Function and a Class of Analytic Functions First Considered by Hurwitz

The history of algebraic equations is very long. The necessity and the trial of solving algebraic equations existed already in the ancient civilizations. The Babylonians solved equations of degree 2 around 2000 B.C. as well as the Indians and the Chinese. In the 16th century, the Italians discovered the resolutions of the equations of degree 3 and 4 by radicals known as Cardano’s formula and Ferrari’s formula. However in 1826, Abel [1] (independently about the same epoch Galois [7]) proved the impossibility of solving general equations of degree ≥ 5 by radicals. This is one of the most remarkable event in the history of algebraic equations. Was there nothing to do in this branch of mathematics after the work of Abel and Galois? Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can solve the algebraic equation of degree 5 by using an elliptic modular function. Since \( \sqrt[n]{a} = \exp \left( {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} \right)\log a} \right) \) which is also written as exp((1/n) ∫ 1 a (1/x)dx), to allow only the extractions of radicals is to use only the exponential. Hence under this restriction, as we learn in the Galois theory, we can construct only compositions of cyclic extensions, namely solvable extentions. The idea of Hermite and Kronecker is as follows; if we use another transcendental function than the exponential, we can solve the algebraic equation of degree 5. In fact their result is analogous to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) . In the quintic equation they replace the exponential by an elliptic modular function and the integral ∫(1/x)dx by elliptic integrals. Kronecker [15] thought the resolution of the equation of degree 5 by an elliptic modular function would be a special case of a more general theorem which might exist. Kronecker’s idea was realized in few cases by Klein [11], [13]. Jordan [10] showed that we can solve any algebraic equation of higher degree by modular functions. Jordan’s idea is clarified by Thomae’s formula, 8 Chap, m (cf. Lindemann [16]). In this appendix, we show how we can deduce from Thomae’s formula the resolution of algebraic equations by a Siegel modular function which is explicitely expressed by theta constants (Theorem 2). Therefore Kronecker’s idea is completely realized. Our resolution of higher algebraic equations is also similar to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) In our resolution the exponential is replaced by tne Siegel modular function and the integral ∫(1/x)dx is replaced by hyperelliptic integrals. The existance of such resolution shows that the theta function is useful not only for non-linear differential equations but also for algebraic equations.

We know that the elliptic modular function j(z) gives the birational invariant of the elliptic curve with the analytic modulus z, and that the elliptic modular functions belonging to the congruence-subgroups are obtained from the values of elliptic functions at the points of finite order on elliptic curves. This is not only the origin of those functions, but one of the most essential points to which we may ascribe the significance of elliptic modular functions in number-theory. It is important to generalize these facts, namely, to investigate in a more general case the relation between automorphic functions with one or more variables and systems of algebraic varieties, especially, systems of abelian varieties. The object of this paper is to give some results on this subject. We shall deal with certain systems of polarized abelian varieties parametrized by holomorphic functions and show that there exist meromorphic functions whose values are considered as of the members of the systems (Theorem 1). The theory developed here will be chiefly concerned with systems of abelian varieties with non-trivial endomorphisms, since I have already given elsewhere a theory for Siegel's modular functions [11]. I am particularly interested, in the present paper, in the determination of the fields of definition for fields of automorphic functions. This is the first problem which confronts us when we proceed beyond a formal treatment in the arithmetic theory of automorphic functions. We obtain a criterion (Theorem 2) applicable even in the case of compact fundamental domain where one can not employ Fourier expansions. The last part of the paper is devoted to the theory of a certain type of automorphic functions of one variable known in the literature as functions belonging to indefinite ternary quadratic forms [8], [5]; they occur as moduli of abelian varieties of dimension 2 whose endomorphismrings are isomorphic to an order of an indefinite quaternion algebra. We get in this case the field of rational numbers as a field of definition for the function-field. We can also prove the congruence formulae for the modular correspondences, which give a generalization of what is obtained in [3], [10]. We shall treat these formulae in a subsequent paper together with the theory of the functions belonging to congruence-subgroups. Our method can be applied to other types of automorphic functions, for example, to Hilbert's modular functions, by considering a system of

The theory of the elliptic modular function plays an important role in many situations in number theory. The elliptic modular function is obtained as a one-to-one correspondence between the parameter space of the family of elliptic curves (given by the Weierstrass normal form) and its period domain (i.e., the complex upper half plane). The K3 surface is considered to be a two-dimensional counterpart of the elliptic curve. So, if we consider a family of algebraic K3 surfaces with some normal form, we can obtain its modular function. We call it a K3 modular function (see [18, 19], some mathematical physicists call it a mirror map for K3 surfaces).

Let $N$ be a positive integer greater than 1. We define a modular function of level $N$ which is a generalization of the elliptic modular lambda function. We show this function and the modular invariant function $j$ generate the modular function field with respect to the principal congruence subgroup of level $N$. Further we study its values at imaginary quadratic points.

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level 2 hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of “resolvent problems” formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension at $$p=2$$ for the symmetric groups $$S_n$$ is equal to the essential dimension at 2 of certain $$S_n$$ -coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic p, specifically Serre-Tate theory, as well as a family of remarkable mod 2 symplectic $$S_n$$ -representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the $$p=2$$ case. In the second half of this paper we introduce the notion of $$\mathcal {E}$$ -versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are $$\mathcal {E}$$ -versal. We use these $$\mathcal {E}$$ -versality result to deduce the equivalence of Hilbert’s 13th Problem (and related conjectures) with problems about congruence covers.

We present an asymptotically fast algorithm for the numerical evaluation of modular functions such as the elliptic modular function j j . Our algorithm makes use of the natural connection between the arithmetic-geometric mean (AGM) of complex numbers and modular functions. Through a detailed complexity analysis, we prove that for a given τ \tau , evaluating N N significative bits of j ( τ ) j(\tau ) can be done in time O ( M ( N ) log ⁡ N ) O(\mathcal {M}(N)\log N) , where M ( N ) \mathcal {M}(N) is the time complexity for the multiplication of two N N -bit integers. However, this is only true for a fixed τ \tau and the time complexity of this first algorithm greatly increases as I m ( τ ) \mathrm {Im}(\tau ) does. We then describe a second algorithm that achieves the same time complexity independently of the value of τ \tau in the classical fundamental domain F \mathcal {F} . We also show how our method can be used to evaluate other modular forms, such as the Dedekind η \eta function, with the same time complexity.

Kaneko and Yoshida introduced the kappa function κ by the property J(κ(z)) = λ(z), where J and λ are the elliptic modular functions. This paper constructs several variants of the kappa function using other pairs of modular functions, and gives their explicit Fourier expansions.

We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

By a change of variables we obtain new <TEX>$y$</TEX>-coordinates of elliptic curves. Utilizing these <TEX>$y$</TEX>-coordinates as meromorphic modular functions, together with the elliptic modular function, we generate the fields of meromorphic modular functions. Furthermore, by means of the special values of the <TEX>$y$</TEX>-coordinates, we construct the ray class fields over imaginary quadratic fields as well as normal bases of these ray class fields.

The theory of automorphic functions of one complex variable is now nearly a century old, and has antecedents -the theories of elliptic and elliptic modular functions -which can be t raced through ABEL'S far-reaching paper of 1827 [1] to the addition theorems for certain elliptic integrals discovered by FAO~A_~O in 1714 [18]. I t can be said to have at ta ined a state of matur i ty and to be, today, well understood in general al though a number of impor tan t specific problems still remain unsolved. Despite its origin as a branch of Analysis, and its reliance on essentially analyt ical techniques for the establishment of its major results, the theory maintains to this day a curiously algebraic outlook: its main results concern the algebraic structure of certain graded rings of functions, and the algebraic correspondences that persist amongst

Elliptic Functions and Modular Forms

This book provides an introduction to the theory of elliptic modular functions and forms, a subject of increasing interest because of its connexions with the theory of elliptic curves. Modular forms are generalisations of functions like theta functions. They can be expressed as Fourier series, and the Fourier coefficients frequently possess multiplicative properties which lead to a correspondence between modular forms and Dirichlet series having Euler products. The Fourier coefficients also arise in certain representational problems in the theory of numbers, for example in the study of the number of ways in which a positive integer may be expressed as a sum of a given number of squares. The treatment of the theory presented here is fuller than is customary in a textbook on automorphic or modular forms, since it is not confined solely to modular forms of integral weight (dimension). It will be of interest to professional mathematicians as well as senior undergraduate and graduate students in pure mathematics.

The elliptic norm curve En in space Sn I admits a group G2n2 of collineations and there is a single infinity of such curves which admit the same group. A particular Et of the family is distinguished by the coordinates of a point on a modular curve, the ratios of these coordinates being elliptic modular functions defined by the modular group congruent to identity (mod n). In the group G 2, 2 there are certain involutory collineations with two fixed spaces. If Et is projected from one fixed space upon the other, a family of rational curves Cm mapping the family of Et's is obtained. The quadratic irrationality separating involutory pairs on Et involves the co6rdinates of a point on the modular curve and the parameter t on a member of the family Cm. Miss B. I. Millert has discussed the elliptic norm curves for which n = 3, 4, 5. In these cases the genus of the modular group is zero and a point of the modular curve can be denoted by a value of the binary parameter r. The irrationality separating involutory pairs on En was used by her to define an elliptic parameter

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker-Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker-Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.