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

TL;DR

This paper proves the algebraicity of critical values of standard $L$-functions for vector-valued Siegel cusp forms, using integral representations and representation theory to connect analytic and arithmetic properties.

Contribution

It establishes the algebraicity of these $L$-values for a broad class of Siegel cusp forms, extending previous results to vector-valued cases with explicit methods.

Findings

Proves algebraicity of critical $L$-values for vector-valued Siegel cusp forms.

Uses integral representations and differential operators to analyze $L$-values.

Employs a representation-theoretic approach to establish arithmetic properties.

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 $(k_1, k_2, \ldots, k_n)$, where $k_n \geq n+1$ 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.