Critical L-values and congruence primes for Siegel modular forms

Abstract: I will explain some recent joint work with Pitale and Schmidt where we obtain an explicit integral representation for the twisted standard L-function on GSp_{2n} \times GL_1 associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level, and a Dirichlet character. By combining this integral representation with a detailed arithmetic study of nearly holomorphic Siegel cusp forms (joint with Pitale, Schmidt, and Horinaga) we are able to prove an algebraicity result for the critical L-values on GSp_{2n} \times GL_1. To refine this result further, we prove that the pullback of the nearly holomorphic Eisenstein series that appears in our integral representation is actually cuspidal in each variable and has nice p-adic arithmetic properties. This directly leads to a result on congruences between Hecke eigenvalues of two Siegel cusp forms of degree 2 modulo primes dividing a certain quotient of L-values. Finally, I will describe a second, more refined version of our congruence theorem, that is obtained by looking at Arthur packets and the refined Gan-Gross-Prasad conjecture in this particular setup.

number theory

Audience: researchers in the topic

Number theory during lockdown

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