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# Dr. Timo Keller: Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

## November 26 at 13:30 – 15:00 CET

Let $X$ be (1) a quotient of the modular curve $X_0(N)$ by a subgroup generated by Atkin-Lehner involutions such that its Jacobian $J$ is a $\mathbf{Q}$-simple modular abelian surface, or, more generally, (2) an $\mathbf{Q}$-simple factor of $J_0(N)$ isomorphic to the Jacobian $J$ of a genus-$2$ curve $X$. We prove that for all such $J$ from (1), the Shafarevich-Tate group of $J$ is trivial and satisfies the strong Birch-Swinnerton-Dyer conjecture. We further indicate how to verify strong BSD in the cases (2) in principle and in many cases in practice.

To achieve this, we compute the image and the cohomology of the mod-$\mathfrak{p}$ Galois representations of $J$, show effectively that almost all of them are irreducible and have maximal image, make Kolyvagin-Logachev effective, compute the Heegner points and Heegner indices, compute the $\mathfrak{p}$-adic $L$-function, and perform $\mathfrak{p}$-descents.