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

Let F be a number field and π π a regular algebraic cuspidal automorphic representation of GLN(AF) GL N ( A F ) of symplectic type. When π π is spherical at all primes p|p p | p , we construct a p -adic L -function attached to any regular non-critical spin p -refinement π ̃ π ~ of π π to Q -parahoric level, where Q is the ( n , n )-parabolic. More precisely, we construct a distribution Lₚ(π ̃) L p ( π ~ ) on the Galois group Galₚ Gal p of the maximal abelian extension of F unramified outside p∞ p ∞ and show that it interpolates all the standard critical L -values of π π at p (including, for example, cyclotomic and anticyclotomic variation when F is imaginary quadratic). We show that Lₚ(π ̃) L p ( π ~ ) satisfies a natural growth condition; in particular, when π ̃ π ~ is ordinary, Lₚ(π ̃) L p ( π ~ )