On p-refined Friedberg-Jacquet integrals and the classical symplectic locus in the eigenvariety

Friedberg-Jacquet proved that if $$ is a cuspidal automorphic representation of $$ , then $$ is a functorial transfer from $$ if and only if a global zeta integral $$ over $$ is non-vanishing on $$ . We conjecture a p-refined analogue: that any P-parahoric p-refinement $$ is a functorial transfer from $$ if and only if a P-twisted version of $$ is non-vanishing on the $$ -eigenspace in $$ . This twisted $$ appears in all constructions of p-adic L-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the $$ eigenvariety, and-by proving upper bounds on the dimensions of such families-obtain various results towards the conjecture.