Vanishing of Selmer groups for Siegel modular forms

Abstract: Let $\pi$ be a cuspidal automorphic representation of $Sp_{2n}$ 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 $\rho\otimes\chi$ 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 $\rho\otimes\chi$.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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