Let K be a number field, let OK be the ring of integers, let K be an algebraic closure of K and let OK be the ring of integers of K. Let M 0 K be the set of finite places and let M∞ K be the set of infinite places. Let Kv be the completion of K at v and let Ov be the ring of integers of Kv. Let ℘v, kv, qv be the maximal ideal of Ov, the residue field Ov/℘v and the size of the residue field |kv|...

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...

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 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 ...

The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Mani...

We establish a congruence formula between p-adic logarithms of Heegner points for two elliptic curves with the same mod p Galois representation. As a first application, we use the congruence formula when p = 2 to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves E. We show that the number of twists of E up to twisting discriminant X...

We improve Kolyvagin’s upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.

Let G be an algebraic group defined over a number field k. By choosing a lifting of G to a group scheme over 6' s c k, the ring of S-integers for some finite set of places S of k, we may define G(C,~), where (5~, c k~ is the ring of integers in the vadic completion of k for all non-archimedean places vr In this way, we can define the adelic points G(Ak). Since different choices of lifting will ...