Geodesic
Elements of prime order in Tate–Shafarevich groups of abelian varieties over ${\mathbb Q}$
Forum of Mathematics, Sigma, Tome 10 (2022)

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For each prime p, we show that there exist geometrically simple abelian varieties A over ${\mathbb Q}$ with . 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 $d \leq X$ with is $ \gg X/\log X$, where $\widehat B_d$ is the dual of the dth quadratic twist of B. We prove this more generally for abelian varieties of $\operatorname {\mathrm {GL}}_2$-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 for an explicit positive proportion of integers d. 
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     author = {Ari Shnidman and Ariel Weiss},
     title = {Elements of prime order in {Tate{\textendash}Shafarevich} groups of abelian varieties over ${\mathbb Q}$},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {10},
     year = {2022},
     doi = {10.1017/fms.2022.80},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2022.80/}
}
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Ari Shnidman; Ariel Weiss. Elements of prime order in Tate–Shafarevich groups of abelian varieties over ${\mathbb Q}$. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.80

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