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