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

Bhargava, Manjul; Klagsbrun, Zev; Lemke Oliver, Robert J.; Shnidman, Ari

doi:10.2140/ant.2021.15.627

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

Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver and Ari Shnidman

Vol. 15 (2021), No. 3, 627–655

DOI: 10.2140/ant.2021.15.627

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 $X_{0} (3 p)$ has infinitely many $F$ -rational points, the method applies to “most” elliptic curves $E$ having a cyclic $3 p$ -isogeny. It also applies in certain cases when $X_{0} (3 p)$ has only finitely many rational points. For example, we find an elliptic curve over $ℚ$ 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 $p \equiv 1 (mod 9)$ , examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order $p$ in their Tate–Shafarevich groups.

Keywords

elliptic curves, Tate–Shafarevich groups, Selmer groups, abelian varieties

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11G10

Milestones

Received: 29 March 2019

Revised: 13 July 2020

Accepted: 20 September 2020

Published: 20 May 2021

Authors

Manjul Bhargava
Department of Mathematics Princeton University Princeton, NJ United States

Zev Klagsbrun
Center for Communications Research San Diego, CA United States

Robert J. Lemke Oliver
Department of Mathematics Tufts University Medford, MA United States

Ari Shnidman
Einstein Institue of Mathematics Hebrew University Jerusalem Israel

Vol. 15, No. 3, 2021