Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$

Abstract

For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb{Q}$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1(\mathrm{mod}p)$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B=A_f/⟨P⟩$. Then the number of positive integers $d\le X$, such that the Tate-Shafarevich group of ${\hat{B}}_d$ has non-trivial $p$-torsion, is $\gg X/\log X$, where ${\hat{B}}_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties of ${\mathrm{GL}}_2$-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 $\mathrm{Sha}(E_d)[p]\ne 0$ for an explicit positive proportion of integers $d$.

arXiv comment10 pages. Final version, with improved exposition and a new section with explicit calculations for elliptic curves. To appear in Forum of Mathematics, Sigma

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Cited by in corpus (1)

• Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians