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#### Seminar des SFB/TRR 326 GAUS

„Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces“
Dr. Timo Keller, Universität Bayreuth

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.

Freitag, den 26. November 2021 um 13:30 Uhr, in INF205, SR A Freitag, den 26. November 2021 at 13:30, in INF205, SR A

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Dr. Otmar Venjakob