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...

Let E be an elliptic curve over Q and ` be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at `. Under certain assumptions, we prove the vanishing of the Galois cohomology group H1(Gal(K(E[`i])/K), E[`i]) for all i ≥ 1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyv...

We study zeros of the L-functions L(f, s) of primitive weight two forms of level q. Our main result is that, on average over forms f of level q prime, the order of the L-functions at the central critical point s = 1 2 is absolutely bounded. On the Birch and Swinnerton-Dyer conjecture, this provides an upper bound for the rank of the Jacobian J0(q) of the modular curve X0(q), which is of the sam...

Let F be a number field with ring of integers OF , and let E/OF be an abelian scheme of arbitrary dimension. In this paper, we study the class invariant homomoprhisms on E with respect to powers of a prime p of ordinary reduction of E. Our main result implies that if the p-adic Birch and Swinnerton-Dyer conjecture holds for E, then the kernels of these homomorphisms are of bounded order. It fol...

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y =N. We first study 3-adic valuation of algebraic part value Hasse–Weil L-function L(C ,s) over ℚ at s=1, and exhibit a relation between 3-part its Tate–Shafarevich group number distinct divisors which are inert in imaginary quadratic field K=ℚ(-3). In case where ,1)≠0 is product split primes K, show that order as predi...

Suppose E is an elliptic curve over Q and χ a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via Birch Swinnerton-Dyer conjecture, this gives heuristic Mordell–Weil rank grows in abelian extensions Q. Using heuristic, we find large class infinite F where expect E(F) be finitely generated. Our work was inspired by ...

We study the Galois groups of polynomials arising from a compatible family representations with big orthogonal monodromy. show that are usually as large possible given constraints imposed on them by functional equation and discriminant considerations. As an application, we consider Frobenius middle étale cohomology hypersurfaces in [Formula: see text] degree at least three. also text]-functions...

We prove many new cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Ω2n+1(p) and PΩ4n(p) both occur as the Galois group of a Galois extension of the rationals for all integers n ≥ 2 and all primes p ≥ 5. We obtain our representations by studying families of twists of elliptic curves a...

We report on a computation of congruent numbers, which subject to the Birch and Swinnerton-Dyer conjecture is an accurate list up to 10. The computation involves multiplying large theta series as per Tunnell (1983). The first method, which we describe in some detail, uses a multimodular disk based technique for multiplying polynomials out-of-core which minimises expensive disk access by keeping...

We take an approach toward counting the number of n for which the curve En : y = x3−n2x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown [6, 7]. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is...