On p-refined Friedberg-Jacquet integrals and the classical symplectic locus in the GL 2 n eigenvariety.

Abstract

Friedberg-Jacquet proved that if $\pi$ is a cuspidal automorphic representation of ${\text{GL}}_{2n}\left(A\right)$ , then $\pi$ is a functorial transfer from ${\text{GSpin}}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H={\text{GL}}_n\times {\text{GL}}_n$ is non-vanishing on $\pi$ . We conjecture a p-refined analogue: that any P-parahoric p-refinement ${\tilde{\pi }}^P$ is a functorial transfer from ${\text{GSpin}}_{2n+1}$ if and only if a P-twisted version of $Z_H$ is non-vanishing on the ${\tilde{\pi }}^P$ -eigenspace in $\pi$ . This twisted $Z_H$ appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the ${\text{GL}}_{2n}$ eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.