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2021 Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families
Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver, Ari Shnidman
Algebra Number Theory 15(3): 627-655 (2021). 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 III(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 p1(mod9), examples of CM abelian threefolds with a positive proportion of quadratic twists having elements of order p in their Tate–Shafarevich groups.

Citation

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Manjul Bhargava. Zev Klagsbrun. Robert J. Lemke Oliver. Ari Shnidman. "Elements of given order in Tate–Shafarevich groups of abelian varieties in quadratic twist families." Algebra Number Theory 15 (3) 627 - 655, 2021. https://doi.org/10.2140/ant.2021.15.627

Information

Received: 29 March 2019; Revised: 13 July 2020; Accepted: 20 September 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/ant.2021.15.627

Subjects:
Primary: 11G05
Secondary: 11G10

Keywords: abelian varieties , Elliptic curves , Selmer groups , Tate–Shafarevich groups

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 3 • 2021
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