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

Abstract

Let $F$ be a number field, and $\pi$ a regular algebraic cuspidal automorphic representation of ${\mathrm{GL}}_N({\mathbb{A}}_F)$ of symplectic type. When $\pi$ is spherical at all primes $\mathfrak{p}|p$, we construct a $p$-adic $L$-function attached to any regular non-critical spin $p$-refinement $\tilde{\pi }$ of $\pi$ to $Q$-parahoric level, where $Q$ is the $(n,n)$-parabolic. 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 very regular weight, and is $Q$-ordinary at all $\mathfrak{p}|p$, we deduce non-vanishing $L(\pi \times (\chi \circ N_{F/\mathbb{Q}}),1/2)\ne 0$ of the twisted central value for all but finitely many Dirichlet characters $\chi$ of $p$-power conductor.

arXiv comment27 pages. Final version, to appear in Res. Math. Sci

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