Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$

Ari Shnidman, Ariel Weiss

Published 2021-06-26Version 1

For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, such that the Tate-Shafarevich group of $\widehat B_d$ has non-trivial $p$-torsion, is $\gg X/\log X$, where $\widehat B_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties of $\mathrm{GL}_2$-type with a $p$-isogeny satisfying a mild technical condition.