We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.

In [12] and [13], Silverman discusses the problem of bounding the Mordell-Weil ranks of elliptic curves over towers of function fields. We first prove generalizations of the theorems of those two papers by a different method, allowing non-abelian Galois groups and removing the dependence on Tate’s conjectures. We then prove some theorems about the growth of Mordell-Weil ranks in towers of funct...

by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

The goal of this paper is to define an analogue the Weil-pairing for Drinfeld modules using explicit formulas and deduce its main properties from these formulas. Our result generalizes formula given rank 2 by van der Heiden works as a more explicit, elementary proof Weil-pairing’s existence Heiden.

Very rough notes for a lecture to be given October 5, 2013 at the Quebec/Maine Number Theory Conference. I’ll discuss diophantine questions that take on a somewhat different flavor when one deals with varying number fields rather than restricts to Q as a base field: an on-going joint project with Maarten Derickx and Sheldon Kamienny regarding Mordell-Weil torsion, and some recent work with Zev ...

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...