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

Ari Shnidman, Ariel Weiss

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article numbere98
JournalForum of Mathematics, Sigma
StatePublished - 4 Nov 2022

ASJC Scopus subject areas

  • Analysis
  • Theoretical Computer Science
  • Algebra and Number Theory
  • Statistics and Probability
  • Mathematical Physics
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


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