Large Shafarevich-Tate groups over quadratic number fields

TL;DR

This paper demonstrates that for a fixed elliptic curve over Q and a quadratic extension, there are infinitely many quadratic twists with arbitrarily large 2-torsion in the Shafarevich-Tate group over that extension.

Contribution

It proves the existence of infinitely many quadratic twists with arbitrarily large 2-torsion in the Shafarevich-Tate group over quadratic fields, under mild conditions.

Findings

Infinite quadratic twists with large 2-torsion in Sha over quadratic fields

Unbounded growth of 2-torsion in Sha for twists of elliptic curves

Results hold under mild assumptions on the elliptic curve

Abstract

Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for every $r>0$, there are infinitely many quadratic twists $E^d/\mathbf{Q}$ of $E/\mathbf{Q}$ such that $\mathrm{dim}_{\mathbf{F}_2}(\mathrm{ST}(E^d/K)[2]) > r$