TY - JOUR

T1 - Elements of given order in tate–shafarevich groups of abelian varieties in quadratic twist families

AU - Bhargava, Manjul

AU - Klagsbrun, Zev

AU - Lemke Oliver, Robert J.

AU - Shnidman, Ari

N1 - Funding Information:
The authors would like to thank the anonymous referees for their many detailed and thoughtful comments. Bhargava was supported by a Simons Investigator Grant and NSF grant DMS-1001828. Lemke Oliver was partially supported by NSF grant DMS-1601398.
Publisher Copyright:
© 2021, Mathematical Science Publishers. All rights reserved.

PY - 2021

Y1 - 2021

N2 - 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 p ≡ 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.

AB - 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 p ≡ 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.

KW - Abelian varieties

KW - Elliptic curves

KW - Selmer groups

KW - Tate–Shafarevich groups

UR - http://www.scopus.com/inward/record.url?scp=85108641909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85108641909&partnerID=8YFLogxK

U2 - 10.2140/ant.2021.15.627

DO - 10.2140/ant.2021.15.627

M3 - Article

AN - SCOPUS:85108641909

VL - 15

SP - 627

EP - 655

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 3

ER -