P-adic L-functions for $$\textrm{GL}(3)$$

Abstract Let $$\Pi $$ Π be a regular algebraic cuspidal automorphic representation (RACAR) of $$\textrm{GL}_3(\mathbb {A}_{\mathbb {Q}})$$ GL 3 ( A Q ) . When $$\Pi $$ Π is p -nearly-ordinary for the maximal standard parabolic with Levi $$\textrm{GL}_1 \times \textrm{GL}_2$$ GL 1 × GL 2 , we construct a p -adic L -function for $$\Pi $$ Π . More precisely, we construct a (single) bounded measure $$L_p(\Pi )$$ L p ( Π ) on $$\mathbb {Z}_p^{\times }$$ Z p × attached to $$\Pi $$ Π , and show it interpolates all the critical values $$L(\Pi \times \eta ,-j)$$ L ( Π × η , - j ) at p in the left-half of the critical strip for $$\Pi $$ Π (for varying $$\eta $$ 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 $$\textrm{GL}_3$$ 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 $$\textrm{GL}_n(\mathbb {A}_{\mathbb {Q}})$$ GL n ( A Q )