The special values of the standard 𝐿-functions for 𝐺𝑆𝑝_{2𝑛}×𝐺𝐿₁

Horinaga, Shuji; Pitale, Ameya; Saha, Abhishek; Schmidt, Ralf

doi:10.1090/tran/8617

The special values of the standard $L$-functions for $\operatorname {GSp}_{2n} \times \operatorname {GL}_1$
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by Shuji Horinaga, Ameya Pitale, Abhishek Saha and Ralf Schmidt PDF

Trans. Amer. Math. Soc. 375 (2022), 6947-6982 Request permission

Abstract:

We prove the expected algebraicity property for the critical values of character twists of the standard $L$-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type $(k_1$, $k_2$, …, $k_n)$, where $k_n \ge n+1$ and all $k_i$ are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.

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