Modular Function Field Research Articles

Ramanujan continued fractions of order sixteen

We study the continued fractions [Formula: see text] and [Formula: see text] of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction [Formula: see text] with modularity and many interesting properties. Here we prove the modularities of [Formula: see text] and [Formula: see text] to find the relation with the generator of the field of modular functions on [Formula: see text]. Moreover we prove that the values [Formula: see text] and [Formula: see text] are algebraic integers for certain imaginary quadratic quantity [Formula: see text].

On two families of modular subgroups

In this work, we study various properties of two families of normal subgroups of the Hecke group Γ0(3) and their associated modular curves and function fields. As consequences, we obtain the stabilizer subgroups of the N-th roots of a uniformizer for X(Γ0(3)), and realize such N-th roots as uniformizers for the associated modular curves to their stabilizer groups.

Generators of Siegel modular function fields of higher genus and level

For positive integers g and N , let F N ( g ) be the field of meromorphic Siegel modular functions of genus g and level N whose Fourier coefficients belong to the N th cyclotomic field. We present explicit generators of F N ( g ) over F 1 ( g ) in terms of quotient of theta constants, when g ≥ 2 and N ≥ 3 .

Generation of Siegel modular function fields of even level

For positive integers g g and N N , let F N ( g ) \mathcal {F}_N^{(g)} be the field of meromorphic Siegel modular functions of genus g g and level N N whose Fourier coefficients belong to the N N th cyclotomic field. We construct explicit generators of F N ( g ) \mathcal {F}_N^{(g)} over F 1 ( g ) \mathcal {F}_1^{(g)} by making use of a quotient of theta constants, when g ≥ 2 g\geq 2 and N N is even.

A level 16 analogue of Ramanujan series for 1/π

The modular functionh(τ)=q∏n=1∞(1−q16n)2(1−q2n)(1−qn)2(1−q8n) is called a level 16 analogue of Ramanujan's series for1/π. We prove that h(τ) generates the field of modular functions on Γ0(16) and find its modular equation of level n for any positive integer n. Furthermore, we construct the ray class field K(h(τ)) modulo 4 over an imaginary quadratic field K for τ∈K∩H such that Z[4τ] is the integral closure of Z in K, where H is the complex upper half plane. For any τ∈K∩H, it turns out that the value 1/h(τ) is integral, and we can also explicitly evaluate the values of h(τ) if the discriminant of K is divisible by 4.

Generalized Lambda Functions and Modular Function Fields of Principal Congruence Subgroups

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.

Special values of generalized $\mathbf{\lambda}$ functions at imaginary quadratic points

We study a modular function $\Lambda_{k,\ell}$ which is one of generalized $\lambda$ functions. We show $\Lambda_{k,\ell}$ and the modular invariant function $j$ generate the modular function field with respect to the modular subgroup $\Gamma_1(N)$. Further we prove that $\Lambda_{k,\ell}$ is integral over $\mathbf Z[j]$. From these results, we obtain that the value of $\Lambda_{k,\ell}$ at an imaginary quadratic point is an algebraic integer and generates a ray class field over the Hilbert class field.

CONSTRUCTION OF CLASS FIELDS OVER IMAGINARY QUADRATIC FIELDS USING y-COORDINATES OF ELLIPTIC CURVES

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.

Completely normal elements in some finite abelian extensions

AbstractWe present some completely normal elements in the maximal real subfields of cyclotomic fields over the field of rational numbers, relying on the criterion for normal element developed in [Jung H.Y., Koo J.K., Shin D.H., Normal bases of ray class fields over imaginary quadratic fields, Math. Z., 2012, 271(1–2), 109–116]. And, we further find completely normal elements in certain abelian extensions of modular function fields in terms of Siegel functions.

Singular values of some modular functions

For an integer N greater than 5 and a triple ${\mathfrak{a}}=[a_{1},a_{2},a_{3}]$ of integers with the properties 0<a i ≤N/2 and a i ≠a j for i≠j, we consider a modular function $W_{\mathfrak{a}}(\tau)=\frac{\wp (a_{1}/N;L_{\tau})-\wp (a_{3}/N;L_{\tau})}{\wp (a_{2}/N;L_{\tau})-\wp(a_{3}/N;L_{\tau})}$ for the modular group Γ 1(N), where ℘(z;L τ ) is the Weierstrass ℘-function relative to the lattice L τ generated by 1 and a complex number τ with positive imaginary part. For a pair of such triples ${\mathfrak{A}}=[{\mathfrak{a}},{\mathfrak{b}}]$ and a pair of non-negative integers F=[m,n], we define a modular function $T_{{\mathfrak{A}},F}$ for the group Γ 0(N) as the trace of the product $W_{\mathfrak{a}}^{m}W_{\mathfrak{b}}^{n}$ to the modular function field of Γ 0(N). In this article, we study the integrality of singular values of the functions $W_{\mathfrak{a}}$ and $T_{{\mathfrak{A}},F}$ by using their modular equations. We prove that the functions $T_{{\mathfrak{A}},F}$ for suitably chosen ${\mathfrak{A}}$ and F generate the modular function field of Γ 0(N), and from Shimura reciprocity and Gee–Stevenhagen method we obtain that singular values $T_{{\mathfrak{A}},F}(\tau)$ for suitably chosen ${\mathfrak{A}}$ and F generate ring class fields. Further, we study the class polynomial of $T_{{\mathfrak{A}},F}$ for Schertz N-system.

Deformations and the rigidity method

In [D. Rohrlich, False division towers of elliptic curves, J. Algebra 229 (1) (2000) 249–279; D. Rohrlich, A deformation of the Tate module, J. Algebra 229 (1) (2000) 280–313], Rohrlich proved rigidity for PSL 2 ( Z p 〚 T 〛 ) for p > 5 , obtained this group as a Galois group over C ( t ) using modular function fields and derived from this interesting consequences for Galois representations attached to the Tate modules of elliptic curves. Furthermore in an unpublished preprint, he established that the corresponding Galois representation G C ( t ) : = Gal ( C ( t ) alg / C ( t ) ) → PSL 2 ( Z p 〚 T 〛 ) is universal. Here we will turn things around. We first provide a general framework for rigid deformations of (projective) representations of the absolute Galois group of a function field (in one variable) over a separably closed base. Under natural, rather general hypothesis, we will determine the corresponding universal deformation ring. If the residual representation is ‘geometrically rigid,’ which happens to be the case for many surjective representation to PSL 2 ( F p ) , p > 2 , which arise from Belyi triples, then certain universal deformations will be ‘geometrically rigid,’ too. This will give new proofs for most of the results of Rohrlich. Our method also applies to Thompson tuples. We then go on to give two further applications, which are based on the example computed by Rohrlich. Over F q ( t ) , where q is a power of a prime l, we construct infinite p-adic Galois extensions which have finite ramification and whose constant field is finite. Furthermore for p > 5 and p ≡ 1 ( mod 4 ) , we obtain a family of surjective Galois representations ρ ζ : Gal ( Q alg / Q ( ζ + ζ −1 ) ) ↠ SL 2 ( Z p [ ζ + ζ −1 ] ) , where the parameter ζ runs over all p-power roots of unity. Finally, we exhibit a general class of rigid universal deformation rings which are finite flat over Z p . In particular this shows that the above examples ρ ζ of Galois representations are not a singular event, but a general phenomenon.

Rational expression for J-invariant function in terms of generators of modular function fields

We give a method for expressing the modular j-invariant function J in a rational function of generators of the modular function field with respect to the modular group Γ0(N ). In the case the genus of the modular function field is positive, using this expression, we can determine isomorphism classes of elliptic curves corresponding to solutions of the defining equation, deduced from these generators, of the modular curve X0(N ). For every N from 6 to 50 and further for N = 52, we give computational results for the expression of J, the generators and the defining equation.

Construction of elliptic curves with cyclic groups over prime fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

The graded ring of Hermitian modular forms of degree 2 over [formula omitted

We describe the graded ring of symmetric Hermitian modular forms of even weights and degree 2 over Q ( −2 ) in terms of generators and relations. All the 8 generators of weight up to 12 are Maaß lifts and some of them can also be obtained from Borcherds products. Moreover, we construct generators for the module over this ring consisting of all Hermitian modular forms with respect to the commutator subgroup. As an application the field of Hermitian modular functions over Q ( −2 ) is determined. Finally, we construct 5 algebraically independent symmetric Hermitian modular forms in terms of theta series.

The graded ring of Hermitian modular forms of degree 2 over

We describe the graded ring of symmetric Hermitian modular forms of even weights and degree 2 over Q ( −2 ) in terms of generators and relations. All the 8 generators of weight up to 12 are Maaß lifts and some of them can also be obtained from Borcherds products. Moreover, we construct generators for the module over this ring consisting of all Hermitian modular forms with respect to the commutator subgroup. As an application the field of Hermitian modular functions over Q ( −2 ) is determined. Finally, we construct 5 algebraically independent symmetric Hermitian modular forms in terms of theta series.

Graded rings of Hermitian modular forms of degree 2

At first it is shown that any Siegel modular form of degree 2 and even weight can be lifted to a Hermitian modular form of degree 2 over any imaginary-quadratic number field K. In the cases \(\) and \(\) we describe the graded rings of Hermitian modular forms with respect to all abelian characters. Generators are constructed as Maaslifts or as Borcherds products. The description allows a characterization in terms of generators and relations. Moreover we show that the graded rings are generated by theta constants by analyzing the associated five-dimensional representation of PSp2(ℤ/3ℤ). As an application the fields of Hermitian modular functions are determined.

Generators and defining equation of the modular function field of the group Γ1(N)

Generators and defining equation of the modular function field of the group Γ₁(N)

Quotients of theta series as rational functions of j1,4

Let Q( n, 1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[ x] in Q( n, 1), the theta series 0 A(z): = ∑ XϵZ ne πizA¦X¦ (z ϵ h , the complex upper half plane) is a modular form of weight n 2 for the congruence group Γ 1(4) = {γ ϵ SL 2(Z)¦γ  1 ∗ 0 1 mod 4} . If n ≥ 24 and A[ X], B[ X] are two quadratic forms in Q( n, 1), then the quotient θ A(z) θ B(z) is a modular function for Γ 1(4). Since we can identify the field of modular functions for Γ 1(4) with the function field K( X 1(4)) over the modular curve x 1(4) = Γ 1(4) h∗ (the extended plane of h ) with genus 0, in this paper, we express it as a rational function j 1,4 which is a field generator over C of K( X 1(4)) and defined by j 1.4(z): = θ 2(2z) 4 (θ 3(2z) 4 . Here, θ 2 and θ 3 denote the classical Jacobi theta functions.

Generators and equations for modular function fields of principal congruence subgroups

C(X(N)) = C(s, t), FN (s, t) = 0, FN (X,Y ) ∈ Z[ζN ][X,Y ], where FN (X,Y ) is a polynomial of two variablesX and Y such that FN (s, Y ) = 0 is an irreducible equation of t over kN (s). Note that C(X(N)) can be identified with the field A(N) of all the modular functions with respect to Γ (N). Further, the function field kN (X(N)) of X(N) rational over kN is identified with the field FN of all the modular functions of A(N) with kN -rational Fourier coefficients at the cusp i∞. (See §6.2 of Shimura [6].) Thus such generators s and t may be taken from the field FN . The problem considered here is to give such two generators explicitly using Klein forms. Moreover, we would like to know the properties of the polynomial FN (X,Y ). For N prime, this problem was solved by Ishii [2] and by the author and Ishii [1]. In [2], Ishii defined a family of modular functions

Drinfeld modular functions

In 1973, Drinfeld introduced the notion of elliptic modules, which is now known as Drinfeld modules. After that the analogies between number fields and function fields have many interesting new aspects. Drinfeld modular function theory is one of these. Drinfeld modular functions were studied by several mathematicians and known to have many properties analogous to those of classical elliptic functions, such as the generators of modular function fields, Galois groups between them. ([5],[8],[9]). In the first part of this note, we establish some more properties of Drinfeld modular functions in analogy with those obtained by Shimura. In [12], Shimura proved his exact-sequence and his reciprocity law. Lang proved the exact-sequence another way in [11] using the isogeny theory. Shimura's proof of the reciprocity law is not easy. For example he used the parametrizations of the models of a modular function field over Q. In [11], Lang avoided Shimura's method using the decomposition gorups which is well-known in algebraic number theory. In this article, we will follow Shimura's method to prove the exact-sequence, and Lang's method to prove the reciprocity law in the function field case. In the second part, we go on to study two variable Drinfeld modular functions in analogy with two variable elliptic functions studied by Berndt. ([!]). In [2], he also gerneralized Shimura's exact-sequence and his reciprocity law corresponding to this extended modular function fields. We discuss the analogies of these in the Drinfeld setting. 1. Definitions and basic facts Let Λ=/yT], k=Fq(T k^ be the completion of k at 00= (£), and C the completion of the algebraic closure of k^. Then C has an absolute value extending that of k^. By an ^-lattice in C, we mean a projective Λ-submodule A of C which is discrete in the topology of C. A meromorphic function /on C is said to be even if/(μz)=/(z) for every μeF*. A meromorphic function /on C is