Modular Function Field Research Articles

The equations for modular function fields of principal congruence subgroups of prime level

We determine the equation with integral coefficients of the modular, function fields with respect to the principal congruence subgroup ofSL2 (ℤ) of prime level.

Some stability properties of Teichm�ller modular function fields with pro-l weight structures

Some stability properties of Teichm�ller modular function fields with pro-l weight structures

Some singular moduli for 𝑄(√3)

In an earlier paper in this journal, the authors derived the equations which transform the Hilbert modular function field for Q ( 3 ) {\mathbf {Q}}(\sqrt 3 ) when the arguments are multiplied by ( 1 + 3 , 1 − 3 ) (1 + \sqrt 3 ,1 - \sqrt 3 ) . These equations define a complex V 2 {V_2} , but we concentrate on special diagonal curves on which the values of some of the singular moduli can be evaluated numerically by using the "PSOS" algorithm. In this way the ring class fields can be evaluated for the forms ξ 2 + 2 t A η 2 {\xi ^2} + {2^t}A{\eta ^2} , where A = 1 , 2 , 3 , 6 A = 1,2,3,6 and t > 0 t > 0 . These last results are based partly on conjectures supported here by numerical evidence.

Stickelberger elements and modular parametrizations of elliptic curves

In the present paper we shall give evidence to support the claim (Conjecture I below and (1.3)) that every elliptic curve A/o which can be parametrized by modular functions admits a canonical modular parametrization whose properties can be related to intrinsic properties of A. In particular, we will see how such a parametrizat ion can be used to prove some rather pleasant integrality properties of Stickelberger elements ad p-adic L-functions attached to A. In addition, if Conjecture I is true then we can give an intrinsic characterization of the isomorphism class of a special elliptic curve in the Q-isogeny class of A distinguished by modular considerations. For most of the paper we have opted for the concrete approach and defined modular parametrizations in terms of X I (N) (Definition 1.1). However, to justify our view of these parametrizations as being canonical, we begin here with a more intrinsic definition. Recall that Shimura ([19], Chap. 6; see w 1 of this paper) has defined a compatible system of canonical models of modular curves {Xs, SeS~}, where 5 p is a certain collection of open subgroups of the group GL(2, Az) over the finite adeles A I of Q. We define the adelic upper half-plane to be the pro-variety )~=lL_m Xs and give )~ the Q-structure induced by the s field of modular functions whose q-expansions at the 0-cusp have coefficients in Q. A modular parametrization of A is a Q-morphism ~: ) ( ~ A which sends

An explicit modular equation in two variables for 𝑄(√3)

A system of modular equations of norm 2 had been found for the Hilbert modular function field of Q ( 2 ) {\mathbf {Q}}(\sqrt 2 ) in an earlier issue of this journal. Here an analogous system is found for Q ( 3 ) {\mathbf {Q}}(\sqrt 3 ) but with the help of MACSYMA. There are special difficulties in the fact that two spaces of Hilbert modular functions exist for Q ( 3 ) {\mathbf {Q}}(\sqrt 3 ) that can be interchanged by the modular equations. The equations are also a remarkable example of hidden symmetries in the algebraic manifold V 2 {{\mathbf {V}}_2} which is defined in C 4 {{\mathbf {C}}^4} by the modular equation.

Gitter ohne Automorphismen der Determinante -1

Positive definite integral quadratic forms having no automorphism of determinant -1 are considered. We give sufficient conditions for a form to be of this kind, and we construct infinite families of indecomposable even unimodular forms which satisfy our conditions. By taking orthogonal sums, examples are obtained in all dimensions n≡0 (mod 8), n≧24. After E. Freitag this implies that the field of modular functions for Sp2n (ℤ), n as above, is not rational.

An explicit modular equation in two variables and Hilbert’s twelfth problem

The Hilbert modular function field over Q ( √ 2 ) {\mathbf {Q}}(\surd 2) has generators satisfying modular equations when the arguments are multiplied by factors of norm two. These equations are found by machine use of Fourier series and are further used to show computationally that Weber’s ring class field theory for rationals has an illustration of Hecke’s type for Q ( √ 2 ) {\mathbf {Q}}(\surd 2) . This has bearing on Hubert’s twelfth problem, the generation of algebraic fields by transcendental functions.

Quadratic relations for generators of units in the modular function field

Quadratic relations for generators of units in the modular function field

On the factors of the jacobian variety of a modular function field

On the factors of the jacobian variety of a modular function field

On rational points of the generic elliptic curve with level N structure over the field of modular functions of level N*

On rational points of the generic elliptic curve with level N structure over the field of modular functions of level N*

Fields of modular functions of genus $0$

Fields of modular functions of genus $0$

The group of automorphisms of the modular function field

Let F be the field of modular functions of all levels on the upper half plane, with, say, Fourier coefficients in the field of all roots of unity. One can define in an obvious way two types of automorphisms of F. First we note that F is a Galois extension of Q(j), where j is the modular function. We have therefore the Galois group U of automorphisms of F over Q(j). Second, if ~ is an element of G(~ (rational 2 x 2 matrices with positive determinant), then the map f~foe gives an automorphism of F. Shimura [4] has determined the structure of the group of automorphisms of F, and in particular has shown that it is equal to UGh. We shall give another proof of this fact, based on an entirely different principle than Shimura's arguments. Let a be an automorphism ofF. If crj =j, then (~sU and we are done. The whole point is to prove that if ~r moves j, then we can compose cr with some e so as to fix j, Let A be an abelian curve having invariant j, defined over Q(j), say by a Weierstrass equation y2=4x3-g 2 x-g3. As is standard (cf. [4]), we can identify F with the field obtained by adjoining to Q(j) all the x-coordinates of the points of finite order on A. For any positive integer N let A N be the group of points of period N on A. Let p be a prime number, and let

On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

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

ON REDUCTION MODULO A PRIME OF FIELDS OF MODULAR FUNCTIONS

We study the reduction modulo p of a subring of the field of modular functions K(p∞) modulo p. We obtain a generalization of a known congruence of Weber.

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

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