P-adic L-functions for GL(3)

2021/11/08 by Loeffler, David, Williams, Chris
#11F67 #11R23 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2111.04535

Abstract

Let Π be a regular algebraic cuspidal automorphic representation (RACAR) of GL₃(\mathbbA_ℚ). 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 L_p(Π) on ℤ_p^× 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 GL_n(\mathbbA_ℚ) of 'general type' (i.e., those that do not arise as functorial lifts) for any n > 2.

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