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

Abstract

We prove algebraicity of critical values of certain Spin $L$-functions. More precisely, our results concern $L(s,\ddot{I}€\otimes \ddot{I}‡,Spin)$ for cuspidal automorphic representations $\ddot{I}€$ associated to a holomorphic Siegel eigenform on $GS{p}_6$, real Dirichlet characters $\ddot{I}‡$, 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.

62 pages. Comments welcome