Abstract
Let Π be a regular algebraic cuspidal automorphic representation (RACAR) of GL3(AQ) . When Π is p-nearly-ordinary for the maximal standard parabolic with Levi GL1 × GL2 , we construct a p-adic L-function for Π . More precisely, we construct a (single) bounded measure Lp(Π) on Zp× attached to Π , and show it interpolates all the critical values L(Π × η, - j) at p in the left-half of the critical strip for Π (for varying η 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 GL3 . 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 GLn(AQ) of 'general type' (i.e. those that do not arise as functorial lifts) for any n > 2 .