The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X₀(pn) associated to congruence subgroups Γ₀(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X₀(pn)→X₀(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.     

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ₀(n)⊂GL₂(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b₁. We give elements in the Hecke annihilator of b₁. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.     

The Lehmer inequality and the Mordell-Weil theorem for Drinfeld modules

In this paper we prove several Lehmer type inequalities for Drinfeld modules which will enable us to prove certain Mordell-Weil type structure theorems for Drinfeld modules.     

On the Brocard–Ramanujan Diophantine Equation n! + 1 = m2

In 1876, H. Brocard posed the problem of finding all integral solutions to n! + 1 = m². In 1913, unaware of Brocard's query, S. Ramanujan gave the problem in the form, The number 1 + n! is a perfect square for the values 4, 5, 7 of n. Find other values. We report on calculations up to n = 10⁹ and briefly discuss a related problem.     

Serre's Conjecture for Imaginary Quadratic Fields

We study an analog over an imaginary quadratic field K of Serre's conjecture for modular forms. Given a continuous irreducible representation :Gal(Q/K) GL₂(F_l) we ask if is modular. We give three examples of representations obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.     

Flat Connections and Quantum Groups

We review the Kohno–Drinfeld theorem and a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection _C on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture for the cases when g=sl _n or when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection _C and Cherednik's generalisation of the Knizhnik–Zamolodchikov connection to finite reflection groups.     

Elliptic Curves Suitable for Pairing Based Cryptography

For pairing based cryptography we need elliptic curves defined over finite fields whose group order is divisible by some prime with where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields with arbitrary embedding degree k are given. Unfortunately, p is of size .We give a method to generate ordinary elliptic curves over prime fields with p significantly less than which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with where or depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50     

The Kernel of the Eisenstein Ideal

Let N be a prime number, and let J₀(N) be the Jacobian of the modular curve X₀(N). Let T denote the endomorphism ring of J₀(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J₀(N)[I] of this ideal (the set of points in J₀(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J₀(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.     

Boundedness of Mordell-Weil ranks of certain elliptic curves and Lang's conjecture

In this paper, we show that a special case of Lang's conjecture on rational points on surfaces of general type implies that there exist only finitely many elliptic curves, when the x-coordinates of n rational points are specified with n?8.     

Some Elementary Ramanujan Graphs

We give elementary constructions of two infinite families of Ramanujan graphs of unbounded degree. The first uses the geometry of buildings over finite fields, and the second uses triangulations of modular curves.Mathematics Subject Classiffications (2000). Primary: 05C25; secondary: 05C50, 51E24     

Heights of CM Points on Complex Affine Curves

In this note we show that, assuming the generalized Riemann hypothesis for quadratic imaginary fields, an irreducible algebraic curve in is modular if and only if it contains a CM point of sufficiently large height. This is an effective version of a theorem of Edixhoven.     

Twists of elliptic curves

If E is an elliptic curve over , then let E(D) denote theD-quadratic twist of E. It is conjectured that there are infinitely many primesp for which E(p) has rank 0, and that there are infinitely many primes for which has positive rank. For some special curvesE we show that there is a set S of primes p with density for which if is a squarefree integer where , then E(D) has rank 0. In particular E(p) has rank 0 for every . As an example let E1 denote the curve .Then its associated set of primes S₁ consists of the prime11 and the primes p for which the order of the reduction ofX₀(11) modulo p is odd. To obtain the general result we show for primes that the rational factor of L(E(p),1) is nonzero which implies thatE(p) has rank 0. These special values are related to surjective Galois representations that are attached to modularforms. Another example of this result is given, and we conclude with someremarks regarding the existence of positive rank prime twists via polynomialidentities.