We study the growth of the Mordell-Weil groups E(Kn) of an elliptic curve E as Kn runs through the intermediate fields of a Zp-extension. We describe those Zp-extensions K∞/K where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms. 2000 Mathematics Subject Classification: Primary 11...

We give an algorithm to compute the canonical height on a Jacobian of a curve of genus 2. The computations involve only working with the Kummer surface and so lengthy computations with divisors in the Jacobian are avoided. We use this height algorithm to give an algorithm to perform the “infinite descent” stage of computing the Mordell-Weil group. This last stage is performed by a lattice enlar...

We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension a...

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find m...

In this paper we investigate divisibility properties of two families of sequences in the Mordell–Weil group of elliptic curves over number fields without complex multiplication. We also consider more general groups of Mordell–Weil type. M. Ward ([W], Theorem 1.) proved that a linear integral recurring sequence of order two which is not nontrivially degenerate has an infinite number of distinct ...

For an elliptic curve E defined over a fieldK, the Tate-Shafarevich group X(E/K) encodes important arithmetic and geometric information. An important conjecture of Tate and Shafarevich states X(E/K) is always finite. Supporting this conjecture is a cohomological analogy between Mordell-Weil groups of elliptic curves and unit groups of number fields. In this note, we follow the analogy to constr...

A technique of descent via 4-isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of Mordell-Weil groups in higher dimension. A selection of worked examples is included as illustration.

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many D ∈ K × with pairwise distinct image in K × /K × 2 , such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber p...

It is explained how the Mordell integral ∫ R e −2πzx cosh(πx) dx unifies the mock theta functions, partial (or false) theta functions, and some of Zagier’s quantum modular forms. As an application, we exploit the connections between q-hypergeometric series and mock and partial theta functions to obtain finite evaluations of the Mordell integral for rational choices of τ and z. 1. The Mordell In...