Algebraicity of Spin $L$-functions for $\mathrm{GSp}_6$

TL;DR

This paper proves the algebraicity of critical values of certain Spin L-functions associated with Siegel modular forms on GSp_6, using Eisenstein series and Fourier coefficient analysis.

Contribution

It introduces a novel approach relating L-values to Eisenstein series on a group without a known moduli interpretation, expanding algebraicity results.

Findings

Proves algebraicity of specific Spin L-values for GSp_6.

Develops new techniques relating Eisenstein series to L-functions.

Analyzes Fourier coefficients of Eisenstein series in a novel context.

Abstract

We prove algebraicity of critical values of certain Spin $L$-functions. More precisely, our results concern $L(s, \pi \otimes \chi, Spin)$ for cuspidal automorphic representations $\pi$ associated to a holomorphic Siegel eigenform on $GSp_6$, real Dirichlet characters $\chi$, and critical points $s$ to the right of the center of symmetry. We use the strategy of relating the $L$-values to properties of Eisenstein series, and a significant portion of the paper concerns the Fourier coefficients of these Eisenstein series. Unlike in prior algebraicity results following this strategy, our Eisenstein series are on a group $G$ that has no known moduli problem, and the $L$-functions are related to the Eisenstein series through a non-unique model.