Large Shafarevich-Tate groups over quadratic number fields

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 ${\dim }_{{\mathbf{F}}_2}(\mathrm{ST}(E^d/K)[2])>r$

References in corpus (1)

• 2-Selmer near-companion curves