Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families

Abstract

Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen–Lenstra–Delaunay-style heuristics predict that the Tate–Shafarevich group III$\left(A_s\right)$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate–Shafarevich groups. In particular, when the modular curve $X_0\left(3p\right)$ has infinitely many $F$-rational points, the method applies to “most” elliptic curves $E$ having a cyclic $3p$-isogeny. It also applies in certain cases when $X_0\left(3p\right)$ has only finitely many rational points. For example, we find an elliptic curve over $\mathbb{Q}$ for which a positive proportion of quadratic twists have an element of order $5$ in their Tate–Shafarevich groups.

The method applies to abelian varieties of arbitrary dimension, at least in principle. As a proof of concept, we give, for each prime $p\equiv 1\left(mod9\right)$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate–Shafarevich groups.