The nonvanishing of Selmer groups for certain symplectic Galois representations

Sam Mundy, Princeton University
Fine Hall 214

In-Person and Online Talk 

Given an automorphic representation \pi of SO(n,n+1) with certain nice properties at infinity, one can nowadays attach to \pi a p-adic Galois representation R of dimension 2n. The Bloch--Kato conjectures then predict in particular that if the L-function of R vanishes at its central value, then the Selmer group attached to a particular twist of R is nontrivial.I will explain work in progress proving the nonvanishing of these Selmer groups for such representations R, assuming the L-function of R vanishes to odd order at its central value, under some hypotheses on \pi at the prime p. The proof constructs a nontrivial Selmer class using p-adic deformations of Eisenstein series attached to \pi, and I will highlight the key new input coming from local representation theory which allows us to check the Selmer conditions for this class at primes for which \pi is ramified.