P-adic L-functions for GL ( 3 ).

Abstract

Let $\Pi$ be a regular algebraic cuspidal automorphic representation (RACAR) of ${\text{GL}}_3\left(A_Q\right)$ . When $\Pi$ is p-nearly-ordinary for the maximal standard parabolic with Levi ${\text{GL}}_1\times {\text{GL}}_2$ , we construct a p-adic L-function for $\Pi$ . More precisely, we construct a (single) bounded measure $L_p\left(\Pi \right)$ on $Z_p^{\times }$ attached to $\Pi$ , and show it interpolates all the critical values $L(\Pi \times \eta ,-j)$ at p in the left-half of the critical strip for $\Pi$ (for varying $\eta$ and j). This proves conjectures of Coates-Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a "Betti Euler system", a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for ${\text{GL}}_3$ . We work in arbitrary cohomological weight, allow arbitrary ramification at p along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of p-adic L-functions for RACARs of ${\text{GL}}_n\left(A_Q\right)$ of 'general type' (i.e. those that do not arise as functorial lifts) for any $n>2$ .