Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$
| Autor: | Shnidman, Ari, Weiss, Ariel |
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| Rok vydání: | 2021 |
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| Zdroj: | Forum of Mathematics, Sigma, Volume 10, 2022, e98 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/fms.2022.80 |
| Popis: | 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 $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, 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 $\mathrm{Sha}(E_d)[p] \neq 0$ for an explicit positive proportion of integers $d$. Comment: 10 pages. Final version, with improved exposition and a new section with explicit calculations for elliptic curves. To appear in Forum of Mathematics, Sigma |
| Databáze: | arXiv |
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