The local $p$-refined Friedberg–Jacquet conjecture

Let $\pi$ be symplectic, let $P$ be a spin parabolic, and let $\tilde\pi^P$ be a $P$-refinement. Let $\chi$ be finite order of conductor $p^\beta>1$, and suppose $L(\pi\times\chi,s+1/2)\neq0$. Let $\zeta_p$ denote the local Friedberg–Jacquet zeta integral, and let $P$-spin mean the refinement condition defined in the paper. Local $p$-refined Friedberg–Jacquet conjecture. The following are equivalent:

• There exists $\varphi_p\in\tilde\pi_p^P$ such that

$$\zeta_p(ut_P^\beta\cdot\varphi_p,\chi_p,s+1/2)\neq0.$$

• $P$ is contained in the $(n,n)$-parabolic and $\tilde\pi^P$ is $P$-spin.

This is stated as an equivalent local form of the global $p$-refined Friedberg–Jacquet conjecture. The global-to-local reduction is proved in the surrounding lemmas, but the local equivalence itself remains conjectural in the source.