On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety

TL;DR

This paper explores a conjecture linking $p$-refined automorphic representations of $ ext{GL}_{2n}$ to functorial transfers from $ ext{GSpin}_{2n+1}$, using twisted zeta integrals and eigenvariety symplectic loci, with partial results supporting the conjecture.

Contribution

It formulates a $p$-refined conjecture relating twisted zeta integrals to functorial transfers and connects this to classical symplectic loci in the eigenvariety, providing bounds on symplectic family dimensions.

Findings

Proposes a $p$-refined conjecture for automorphic transfers.

Establishes upper bounds on classical symplectic family dimensions.

Provides evidence supporting the conjecture through these bounds.

Abstract

Friedberg--Jacquet proved that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A})$, then $\pi$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H = \mathrm{GL}_n \times \mathrm{GL}_n$ 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 $\mathrm{GSpin}_{2n+1}$ 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 $\mathrm{GL}_{2n}$ eigenvariety, and -- by proving upper bounds on the dimensions of such families -- obtain various results towards the conjecture.