P-adic L-functions for GL(3)

TL;DR

This paper constructs a new p-adic L-function for certain automorphic representations of GL(3), proving conjectures and extending the theory to more general cases beyond previous work.

Contribution

It introduces the first construction of p-adic L-functions for RACARs of GL(n) of general type for any n > 2, using spherical varieties and Betti Euler systems.

Findings

Constructed a p-adic L-function for GL(3) RACARs under near-ordinarity.

Proved conjectures of Coates-Perrin-Riou and Panchishkin in this context.

Extended p-adic L-function theory to non-functorial, higher-rank automorphic representations.

Abstract

Let $\Pi$ be a regular algebraic cuspidal automorphic representation (RACAR) of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$. When $\Pi$ is $p$-nearly-ordinary for the maximal standard parabolic with Levi $\mathrm{GL}_1 \times \mathrm{GL}_2$, we construct a $p$-adic $L$-function for $\Pi$. More precisely, we construct a (single) bounded measure $L_p(\Pi)$ on $\mathbb{Z}_p^\times$ 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 $\mathrm{GL}_3$. 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 $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ of 'general type' (i.e., those that do not arise as functorial lifts) for any $n > 2$.