Ranks of Twists of Elliptic Curves and Hilbert’s Tenth Problem

Ranks of Twists of Elliptic Curves and Hilbert’s Tenth Problem

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...

Hilbert’s Tenth Problem and Elliptic Curves

Ranks of Quadratic Twists of Elliptic Curves over Fq(t)

Some notes on the analogy between number theory over Z and Fq[t] and an attempt to translate a paper of Gouvêa and Mazur on ranks of quadratic twists of elliptic curves over Q to elliptic curves over Fq(t).

Hilbert’s Tenth Problem for Function Fields of Characteristic Zero

In this article we outline the methods that are used to prove undecidability of Hilbert’s Tenth Problem for function fields of characteristic zero. Following Denef we show how rank one elliptic curves can be used to prove undecidability for rational function fields over formally real fields. We also sketch the undecidability proofs for function fields of varieties over the complex numbers of di...

Extensions of Hilbert's Tenth Problem

Lectures Two types of lectures were programmed into the schedule. First there were a number of introductory lectures which were were meant to introduce non-specialist into the area of Hilbert’s Tenth Problem. Since quite a few of the participants came from different backgrounds this was an ideal way to get a feeling of the problems which come up in this area of mathematics. The second type of l...