The $p$-refined Friedberg–Jacquet conjecture

Let $G=\mathrm{GL}_{2n}$, let $P\subsetneq G$ be a proper spin parabolic with associated parabolic $\cP\subset\mathrm{GSpin}_{2n+1}$, and let $\tilde\pi^P=(\pi,\alpha^P)$ be a $P$-refined RACAR of $G(\mathbb{A})$. Let $\chi$ and $\eta$ be algebraic Hecke characters, with $\chi$ finite order of conductor $p^\beta>1$, and let $Z_H$ be the Friedberg–Jacquet global zeta integral for $H=\mathrm{GL}_n\times\mathrm{GL}_n$. $p$-refined Friedberg–Jacquet conjecture. For every $s\in\mathbb{C}$, there exists $\varphi\in\tilde\pi^P$ such that

$$Z_H(ut_P^\beta\cdot\varphi,\chi,s+1/2)\neq0$$

if and only if all of the following hold: $\tilde\pi^P$ is a functorial transfer of some $\tilde\Pi^{\cP}$ on $\mathrm{GSpin}_{2n+1}(\mathbb{A})$ with central character $\eta$; $L(\pi\times\chi,s+1/2)\neq0$; and $P$ is contained in the $(n,n)$-parabolic. The conjecture refines the classical Friedberg–Jacquet non-vanishing criterion to parahoric $p$-refinements. The source develops local reformulations involving the twisted local zeta integral, $P$-spin refinements, and the same parabolic-containment condition; no resolution is supplied.