P-adic L-functions for GL(3)

TL;DR

The paper constructs p-adic L-functions for GL_3 RACARs Π in the general-type setting (not arising from functorial lifts) by developing a Betti Euler system via Beilinson’s Eisenstein classes and a spherical-varieties framework. It proves p-adic interpolation in the left half of the critical strip under p_1-near-ordinarity and, separately, in the right half under p_2-near-ordinarity, ultimately producing a full p-adic L-function when both conditions hold. A key innovation is controlling denominators of Eisenstein classes to yield a uniquely determined bounded p-adic measure that interpolates critical L-values L(Π×η, -j) across varying η and j, including the archimedean factors to match Coates–Perrin-Riou. The results advance p-adic interpolation beyond GL_2 and GL_{2n} special lifts, providing a first construction for RACARs of GL_3 of general type and offering pathways to GL_n over Q or totally real fields, with potential extensions to non-ordinary settings and higher rank via motivic-Eisenstein classes and spherical-variety techniques.

Abstract

Let $Π$ be a regular algebraic cuspidal automorphic representation (RACAR) of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$. When $Π$ 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 $Π$. More precisely, we construct a (single) bounded measure $L_p(Π)$ on $\mathbb{Z}_p^\times$ attached to $Π$, and show it interpolates all the critical values $L(Π\timesη,-j)$ 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 $\mathrm{GL}_3$. 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 $n > 2$.

Theorems & Definitions (100)

• Conjecture 1.1: Algebraicity Conjecture for $\mathrm{GL}_{2m+1}$
• Theorem A
• Conjecture 1.2: $p$-adic $L$-functions Conjecture for $\mathrm{GL}_{2m+1}$
• Theorem B
• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Proposition 2.4
• Remark 2.5
• Definition 2.6