Vol. 15, No. 3, 2021

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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
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 (As) should contain an element of order p for a positive proportion of quadratic twists As 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 X0(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 X0(3p) has only finitely many rational points. For example, we find an elliptic curve over 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 1(mod9), examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order p in their Tate–Shafarevich groups.

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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