On $p$-adic $L$-functions for symplectic representations of GL(N) over number fields

Chris Williams

Published 2023-05-13Version 1

Let $F$ be a number field, and $\pi$ a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_N(\mathbb{A}_F)$ of symplectic type. We construct a $p$-adic $L$-function attached to any non-critical $p$-refinement $\tilde\pi$ of $\pi$ to $(n,n)$-parahoric level. More precisely, we construct a distribution $L_p(\tilde\pi)$ on the Galois group $\mathrm{Gal}_p$ of the maximal abelian extension of $F$ unramified outside $p\infty$, and show that it interpolates all the standard critical $L$-values of $\pi$ at $p$ (including, for example, cyclotomic and anticyclotomic variation when $F$ is imaginary quadratic). We show that $L_p(\tilde\pi)$ satisfies a natural growth condition; in particular, when $\tilde\pi$ is ordinary, $L_p(\tilde\pi)$ is a (bounded) measure on $\mathrm{Gal}_p$. As a corollary, when $\pi$ is unitary, has regular weight, and admits an ordinary refinement, we deduce non-vanishing $L(\pi\times(\chi\circ N_{F/\mathbb{Q}}),1/2) \neq 0$ of the twisted central value for all but finitely many Dirichlet characters $\chi$ of $p$-power conductor.