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 "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 ρ by χ.

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