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

## TL;DR

This paper develops a general method to demonstrate the existence of elements of specific orders in the Tate-Shafarevich groups of abelian varieties, especially elliptic curves, within quadratic twist families, supporting heuristic predictions.

## Contribution

It introduces a new approach leveraging isogenies to prove instances of Tate-Shafarevich group element existence in quadratic twists, applicable to various abelian varieties.

## Key findings

- Positive proportion of quadratic twists have nontrivial p-torsion in Tate-Shafarevich groups.
- Method applies to elliptic curves with cyclic 3p-isogenies and certain cases with finitely many rational points.
- Examples of CM abelian threefolds with elements of order p in Tate-Shafarevich groups for primes p ≡ 1 mod 9.

## 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 of $A_s$ 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 $X_0(3p)$ 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 $X_0(3p)$ has only finitely many 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 $p \equiv 1 \pmod 9$, examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate-Shafarevich groups.