English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

For each prime $p$, we show that there exist geometrically simple abelianvarieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevichgroups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be anoptimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B= A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, suchthat 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 $\mathrm{GL}_2$-typewith a $p$-isogeny satisfying a mild technical condition. In the special caseof elliptic curves, we give stronger results, including many examples where$\mathrm{Sha}(E_d)[p] \neq 0$ for an explicit positive proportion of integers$d$.

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