TY - JOUR

T1 - Elements of prime order in Tate-Shafarevich groups of abelian varieties over

AU - Shnidman, Ari

AU - Weiss, Ariel

N1 - Funding Information:
The authors thank Manjul Bhargava, Brendan Creutz and Robert Lemke Oliver for helpful conversations. They also thank the referees for their careful reading and helpful suggestions to improve the exposition. The first author was supported by the Israel Science Foundation (grant No. 2301/20). The second author was supported by an Emily Erskine Endowment Fund postdoctoral fellowship at The Hebrew University of Jerusalem, by the Israel Science Foundation (grant No. 1963/20) and by the Binational Science Foundation (grant No. 2018250).
Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press.

PY - 2022/11/4

Y1 - 2022/11/4

N2 - For each prime p, we show that there exist geometrically simple abelian varieties A over with. Specifically, for any prime, let be an optimal quotient of with a rational point P of order p, and let. Then the number of positive integers with is, where is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of-type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.

AB - For each prime p, we show that there exist geometrically simple abelian varieties A over with. Specifically, for any prime, let be an optimal quotient of with a rational point P of order p, and let. Then the number of positive integers with is, where is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of-type with a p-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where for an explicit positive proportion of integers d.

UR - http://www.scopus.com/inward/record.url?scp=85142223673&partnerID=8YFLogxK

U2 - 10.1017/fms.2022.80

DO - 10.1017/fms.2022.80

M3 - Article

AN - SCOPUS:85142223673

SN - 2050-5094

VL - 10

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e98

ER -