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

Let F be a number field and $\pi$ a regular algebraic cuspidal automorphic representation of ${\text{GL}}_N\left(A_F\right)$ of symplectic type. When $\pi$ is spherical at all primes $p|p$ , 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 $L_p\left(\tilde{\pi }\right)$ on the Galois group ${\text{Gal}}_p$ 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 $L_p\left(\tilde{\pi }\right)$ satisfies a natural growth condition; in particular, when $\tilde{\pi }$ is ordinary, $L_p\left(\tilde{\pi }\right)$ is a (bounded) measure on ${\text{Gal}}_p$ . As a corollary, when $\pi$ is unitary, has very regular weight, and is Q-ordinary at all $p|p$ , we deduce non-vanishing $L(\pi \times (\chi \circ N_{F/Q}),1/2)\ne 0$ of the twisted central value for all but finitely many Dirichlet characters $\chi$ of p-power conductor.