On P-Refined Friedberg–Jacquet Integrals and the Classical Symplectic Locus in the gl2n Eigenvariety

Friedberg–Jacquet proved that if π is a cuspidal automorphic representation of GL2n(A), then π is a functorial transfer from GSpin2n+1 if and only if a global zeta integral ZH over H=GLn×GLn is non-vanishing on π. We conjecture a p-refined analogue: that any P-parahoric p-refinement π~P is a functorial transfer from GSpin2n+1 if and only if a P-twisted version of ZH is non-vanishing on the π~P-eigenspace in π. This twisted ZH 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 GL2n eigenvariety, and—by proving upper bounds on the dimensions of such families—obtain various results towards the conjecture. © The Author(s) 2025.