Lecture series on Vanishing of Selmer groups for Siegel modular forms
Dr. Samuel Mundy
2024-07-12 10:30-11:30
MCM110
Speaker: Dr. Samuel Mundy (Princeton University)Time: 10:30-11:30 June 18, June 20, June 21, June 25, June 27, July 9, July 11, July 12, 2024
Venue: MCM110 & Online (Zoom ID: 3329836068 Password: mcm1234)
Title: Lecture series on Vanishing of Selmer groups for Siegel modular forms
Abstract:
Let π be a cuspidal automorphic representation of Sp_2n over Q which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that π is ordinary at p, and that p does not divide the conductor of χ. I will describe work in progress which aims to prove that the Bloch--Kato Selmer group attached to the twist of ρ by χ vanishes, under some mild ramification assumptions on π; this is what is predicted by the Bloch--Kato conjectures.
The proof uses a variety of techniques from the theory of automorphic representations as well as Galois theory, and I will explain the relevant machinery as the lectures progress. The main point is the construction of "ramified Eisenstein congruences;" we construct p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of nonclassical weight and high level at an auxiliary prime. These families are then used to construct Galois cohomology classes, which are ramified for this auxiliary prime, for the Tate dual of the twist of ρ by χ.
