The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have n eigenvalues equal to 1, and investigate the first non-zero derivative of the characteristic polynomial at that point. The conn...

1997 2 Kevin Lee James On con gruences for the coefficients of modular forms and some applications (Under the direction of Andrew Granville) In this dissertation, we will study two different conjectures about elliptic curves and modular forms. First, we will exploit the theory developed by Shimura and Waldspurger to address Goldfeld's conjecture which states that the density of rank zero curves...

We study a subgroup of the Shafarevich-Tate group of an abelian variety known as the visible subgroup. We explain the geometric intuition behind this subgroup, prove its finiteness and describe several techniques for exhibiting visible elements. Two important results are proved one what we call the visualization theorem, which asserts that every element of the Shafarevich-Tate group of an abeli...

This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over Q viewed over the fields cut out by certain self-dual Artin representations of dimension at most 4. When the associated L-function vanishes (to even order ≥ 2) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be lin...

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of gen...

Let F be a global function field of characteristic p > 0, let F /F be a Galois extension with Gal(F /F) ≃ Z N p and let E/F be a non-isotrivial elliptic curve. We study the behaviour of Selmer groups Sel E (L) l (l any prime) as L varies through the subextensions of F via an appropriate version of Mazur's Control Theorem. In the case l = p we let F = F d where F d /F is a Z d p-extension. With ...

This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y = Q(x) where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. Our results are complete for unramified extensions ofQ2 and Q3 and for all p-adic fields for p > 5. Our primary motivation is to comp...

is the Hecke L-series attached to the eigenform f . Hecke’s theory shows that L(f, s) has an Euler product expansion identical to (2), and also that it admits an integral representation as a Mellin transform of f . This extends L(f, s) analytically to the whole complex plane and shows that it satisfies a functional equation relating its values at s and 2− s. The modularity of E thus implies tha...

For the integer D = pq of the product of two distinct odd primes, we construct an elliptic curve E2rD : y 2 = x3 − 2rDx over Q, where r is a parameter dependent on the classes of p and q modulo 8, and show, under Birch and Swinnerton-Dyer conjecture, that the elliptic curve has rank one and vp(x([k]Q)) 6= vq(x([k]Q)) for odd k and a generator Q of the free part of E2rD(Q). Thus we can get p or ...

Let E be an elliptic curve over Q, with L-function LE(s). For any primitive Dirichlet character χ, let LE(s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions LE(s, χ) at the central value s = 1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that LE(1, χ)...