Elements of given order in TateShafarevich 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. CohenLenstraDelaunaystyle heuristics predict that the TateShafarevich 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 TateShafarevich 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 TateShafarevich 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 TateShafarevich groups.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 DOI:
 10.48550/arXiv.1904.00116
 arXiv:
 arXiv:1904.00116
 Bibcode:
 2019arXiv190400116B
 Keywords:

 Mathematics  Number Theory
 EPrint:
 Alg. Number Th. 15 (2021) 627655