P-adic L-functions for GL(3)

Let Π Π be a regular algebraic cuspidal automorphic representation (RACAR) of GL₃(AQ) GL 3 ( A Q ) . When Π Π is p -nearly-ordinary for the maximal standard parabolic with Levi GL₁ × GL₂ GL 1 × GL 2 , we construct a p -adic L -function for Π Π . More precisely, we construct a (single) bounded measure Lₚ(Π ) L p ( Π ) on Zₚ× Z p × attached to Π Π , and show it interpolates all the critical values L(Π × η ,-j) L ( Π × η , - j ) at p in the left-half of the critical strip for Π Π (for varying η 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 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 GLₙ(AQ) GL n ( A Q )