Ranks of twists of elliptic curves and Hilbert's Tenth Problem

TL;DR

This paper explores the distribution of 2-Selmer ranks in quadratic twists of elliptic curves over number fields, linking these properties to the solvability of Hilbert's Tenth Problem over rings of integers.

Contribution

It provides conditions for elliptic curves to have twists with arbitrary 2-Selmer ranks and establishes lower bounds for their frequency, connecting these results to Hilbert's Tenth Problem.

Findings

Many twists with trivial Mordell-Weil group exist under certain conditions.

Assuming the Shafarevich-Tate conjecture, many twists have infinite cyclic Mordell-Weil group.

If the conjecture holds, Hilbert's Tenth Problem is unsolvable over the ring of integers of every number field.

Abstract

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 many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth Problem has a negative answer over the ring of integers of every number field.