The special values of the standard $L$-functions for ${\mathrm{GSp}}_{2n}\times {\mathrm{GL}}_1$

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 $(k_1,k_2,\dots ,k_n)$, 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.

This is the accepted version. There are significant improvements over the previous version, thanks to the referee's comments. To appear in Transactions of the American Mathematical Society

References in corpus (3)

• Doubling constructions: Global functoriality for non-generic cuspidal representations
• The doubling Archimedean zeta integrals for p-adic interpolation
• Integral representation and critical $L$-values for holomorphic forms on $GS{p}_{2n}\times G{L}_1$