# Number Theory Seminar: Vanishing of Selmer groups for Siegel modular forms

SEMINARS, NUMBER THEORY

##### Speaker:

Sam Mundy *- Princeton University*

Let pi be a cuspidal automorphic representation of $\mathrm{Sp}_{2n}$ over $\mathbb{Q}$ which is holomorphic discrete series at infinity, and $\chi$ a Dirichlet character. Then one can attach to $\pi$ an orthogonal $p$-adic Galois representation $\rho$ of dimension $2n+1$. Assume $\rho$ is irreducible, that pi is ordinary at $p$, and that $p$ does not divide the conductor of $\chi$. I will describe work in progress which aims to prove that the Bloch--Kato Selmer group attached to the twist of $\rho$ by $\chi$ vanishes, under some mild ramification assumptions on $\pi$; this is what is predicted by the Bloch--Kato conjectures.

The proof uses "ramified Eisenstein congruences" by constructing $p$-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of nonclassical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of $\rho$ by $\chi$.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html