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

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