Elliptic Modular Functions Research Articles

The Galois unsolvability of the sextic equation of anisotropic elasticity

By an approximate numerical application of Galois theory it is proved that the sextic equation of anisotropic elasticity for cubic symmetry is in general unsolvable in radicals, elementary transcendental functions, or elliptic modular functions and that its group is the full symmetric group. This implies the same unsolvability for tetragonal, orthorhombic, monoclinic, and triclinic symmetry. A separate investigation proves the same unsolvability for trigonal symmetry. Special cases of cubic symmetry which might have solvable equations are examined. Directions restricted to {111} or {112} planes give unsolvable equations, in contrast to {100} and {110} planes. Three additional classes of elastic constants which give solvable equations are found but only two limiting cases are physically possible. An extensive survey suggests that any further special elastic constants are rather unlikely.

An extremal problem for quasiconformal mappings in an annulus

The following extremal problem is solved. We consider a family of continuously differentiable univalent quasiconformal mappings w = f ( z ) w = f(z) of the annulus r > | z | > 1 r > |z| > 1 onto the unit disk minus some continuum containing the origin. For a point b on a fixed circle, maximize | f ( b ) | |f(b)| within the family. The problem is solved by using a variational method due to Schiffer. The extremal function and the maximum are found in terms of the Weierstrass ℘ \wp -function and the elliptic modular function.

Book Review: Elliptic modular functions

Book Review: Elliptic modular functions

Automorphic forms and automorphy preserving differential operators on products of halfplanes

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

An Invariant Multiple Differential Attached to the Field of Elliptic Modular Functions of Characteristic p

An Invariant Multiple Differential Attached to the Field of Elliptic Modular Functions of Characteristic p

Uniformization in a soluble partial wave model

A non-relativistic partial wave model based on elastic unitarity and a simple left-hand cut is studied. Although the usual N/ D method leads to singular integral equations, the model can be solved exactly in terms of hypergeometric functions by use of a differential equation technique. This differential equation approach also reveals the existence of a manifold of other functions which solve the basic discontinuity relations ithout, however, obeying dispersion relations. Since the model is marginally singular, the exact solutions give examples of amplitudes of non-integer CDD class. Furthermore, the amplitude can be globally uniformized by use of the elliptic modular function. Because of this circumstance, a comparison can be made of approximation schemes which differ by their treatment of the underlying Riemann surface. The conclusion is that the exact solution of the model is best approximated in terms of the uniformizing variable, a procedure which maintains the correct Riemann surface at each stage of approximation.

Riemann surfaces and partial wave models

Within the context of simple partial wave models for elastic scattering the problem of uniformizing the partial wave amplitude and classifying its Riemann surface is studied. Starting with the analytic continuation of the amplitude an analysis of the Riemann surface is made through its group of covering transformations relative to a simpler base surface. A model based on the Yukawa potential is studied in this manner and the Riemann surface of interest is found to be the universal covering surface of the thrice punctured sphere. The uniformization of the amplitude can be done explicitly in this case by use of the elliptic modular function. In terms of the uniformizing variable, the original discontinuity relations for the amplitude then reduce to functional equations involving elements of the modular group.

On the algebraic theory of elliptic modular functions

On the algebraic theory of elliptic modular functions

A Note on Weierstrass Points

In (4) G. Lewittes proved some theorems connecting automorphisms of a compact Riemann surface with the Weierstrass points of the surface, and in (5) he applied these results to elliptic modular functions. We refer the reader to these papers for definitions and details. It is our purpose in this note to point out that these results are of a purely algebraic nature, valid in arbitrary algebraic function fields of one variable over algebraically closed ground fields (with an obvious restriction on the characteristic). We shall also make use of the calculation carried out in (5) to obtain a rather easy extension of a theorem proved in (6, p. 312).

On the genus of fields of elliptic modular functions

On the genus of fields of elliptic modular functions

On the endomorphism algebra of Jacobian varieties attached to the fields of elliptic modular functions

On the endomorphism algebra of Jacobian varieties attached to the fields of elliptic modular functions

On the theory of automorphic functions and the problem of moduli

The subject of this presentation is the problem of moduli for certain special types of algebraic-geometrical objects and the connection of this problem with the theory of automorphic functions for certain arithmetically defined discontinuous groups acting on bounded domains in complex Euclidean space. To begin with, the earliest such situation of non trivial content was the theory of elliptic modular functions, connected with the theory of moduli of elliptic curves. This situation was subsequently generalized to the theory of Siegel's modular functions connected with the moduli of normally polarized Abelian varieties. The theory of moduli of algebraic curves or of compact Riemann surfaces of genus n, which has received independent development from the analytical point of view by Teichmuller [9], Ahlfors [l], and Bers [4], may also be extracted to a large degree from the theory of moduli of normally polarized Abelian varieties of dimension n, from among which, by certain algebraic-geometrical criteria due to Matsusaka [6], may be distinguished those which are the canonically polarized Jacobian varieties of curves of genus n. The essential point is that the Jacobian varieties form a Zariski-open subset of an algebraic subset of the normally polarized Abelian varieties [2]. (Of course here we should be careful to point out the distinction between results of a more algebraic nature on the theory of moduli, and the results discussed here which are of a transcendental nature, linking such a theory of moduli with the theory of automorphic functions of some discontinuous group.) The fact that we speak only of the moduli of polarized Abelian varieties comes of course from the well-known phenomenon that if, in the space of nX2n period matrices of complex, w-dimensional tori, we identify those points which correspond to complex analytically isomorphic tori, what we get is a space which is not even Hausdorff, not to speak of being a complex space or algebraic variety. A polarization for a complex torus T is a class consisting of all divisors D

Ramification in Elliptic Modular Function Fields

Ramification in Elliptic Modular Function Fields

On the Jacobian varieties of the fields of elliptic modular functions

On the Jacobian varieties of the fields of elliptic modular functions

A Generalization of a Theorem of Hecke

Introduction. Suppose K denotes the field of elliptic modular functions of prime level p, and j the Weierstrass absolute invariant. Then K is a Galois extension of C (j) with LF (2, p) for Galois group and every element of this group gives rise to an automorphism of the vector space of differentials of the first kind in K. In this way we get a representation of LF (2, p) with corresponding character X say. In two papers (cf. [3], [4]), Hecke determined this character completely, i. e. he determined how X decomposes into simple characters. When p =3 mod 4, LF (2, p) has exactly two simple complex characters which happen to be complex conjugate, and Hecke discovered that the difference of the multiplicities of these characters il X yields the Dirichlet expression for h(- p), the class-number of Q(/ - p). However, Hecke derived this remarkable fact by computation and the underlying reason for it is not yet known. With this problem in mind we shall investigate whether Hecke's result is restricted to prime levels p satisfying p = 3 mod 4, or not. It turns out that there exists a rather natural generalization for those levels n t

Periodicity modulo 𝑚 and divisibility properties of the partition function

has infinitely many solutions in non-negative integers n. This conjecture seems difficult and I have only scattered results. In ?2 of this paper it will be shown that the conjecture is true for m= 5 and 13 by means of congruences derived from the elliptic modular functions, and similar theorems will be proved; for example that p(5n+4)/5 and p(7n+5)/7 fill all residue classes modulo 5 and 7 respectively, infinitely often. In ?1 it will be shown in an elementary way that the conjecture is also true for m=2. Thus p(n) is odd infinitely often and even infinitely often. In ?1 we will also consider the question of the periodicity of a sequence modulo m in some generality.

On the Theory of Automorphic Functions

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

Kroneckerian Model of Fields of Elliptic Modular Functions

Kroneckerian Model of Fields of Elliptic Modular Functions

The coefficients of certain infinite products

are classical, (a) being the so-called pentagonal number of Euler, and (b) being an identity of Jacobi's, well known in the theory of elliptic modular functions. It is remarkable that for no other values of r are the coefficients of HJ(1-Xn)t known explicitly. It will be the purpose of this paper to give various recursion formulas and identities for these coefficients, which are consequences of identities between functions on certain modular subgroups. In particular, the coefficients will be investigated thoroughly for r=2, 4, 6. A typical theorem is that every integer occurs as the modulus of the coefficients for r =2.

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

Introduction. In spite of its wide applicability in various branches of the theory of functions, the elliptic modular function is often used with a certain hesitation. This is mainly due to the fact that its application presupposes familiarity with a comparatively intricate formalism, in particular when the determination of numerical constants is involved. In fact, the endeavor to avoid the elliptic modular function has given rise to an extensive mathematical literature aiming at proving certain theorems in an way, the word being used here as a synonym for without making use of the elliptic modular function. As an impressive example, Picard's theorem on integral functions might be quoted. The difficulties which beset the numerical treatment of the elliptic modular function go essentially back to the fact that, on the one hand, the formalism of this function can only be developed with the help of the Jacobian elliptic functions while, on the other hand, what is needed in the applications are the conformal mapping properties of the modular function, and the connection between these two different aspects of the modular function has to be established through the medium of the theory of Schwarz' differential parameter or by a very detailed study of the periodic properties of the Jacobian elliptic functions. The object of the first part of this paper is to show how those properties of the elliptic modular function which are required for the applications may be derived in a simple way by the exclusive use of elementary principles of the theory of conformal representation. It will be shown that once the modular surface' is defined, the functional equation