Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

Keller, Timo; Stoll, Michael

Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 483-489.

Soit $X$ un des $28$ quotients d’Atkin–Lehner d’une courbe $X_0\left(N\right)$ tel que $X$ est de genre $2$ et sa jacobienne $J$ est absolument simple. On démontre que le groupe de Shafarevich–Tate $Ш(J/\mathbb{Q})$ est trivial. Ceci vérifie la conjecture BSD forte pour $J$.

Let $X$ be one of the $28$ Atkin–Lehner quotients of a curve $X_0\left(N\right)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich–Tate group $Ш(J/\mathbb{Q})$ is trivial. This verifies the strong BSD conjecture for $J$.

Classification : 11G40, 11-04, 11G10, 11G30, 14G35

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