Abstract
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.