Algebra Seminar - dr. Timo Keller University of Bayreuth

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

Abstract: Let X be 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. We prove that for all but two such J, the Shafarevich-Tate group of J is trivial and satisfies the strong Birch-Swinnerton-Dyer conjecture. (To prove this also for the remaining two abelian surfaces, we are currently performing descent.)

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. Because many ingredients are involved in the proof, we will give an overview and focus on the computation of the mod-\mathfrak{p} Galois representation in our talk.