P-adic L-functions for GL(3)

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 $n>2$.

44 pages. No changes to content, only updated funder credit. Final version, to appear in Math. Annalen

References in corpus (12)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Rankin--Eisenstein classes for modular forms
• Euler systems for GSp(4)
• Rankin--Eisenstein classes in Coleman families
• Higher Hida theory and p-adic L-functions for GSp(4)
• Topological polylogarithms and $p$-adic interpolation of $L$-values of totally real fields
• Iwasawa theory for the symmetric square of a modular form
• An Euler system for GU(2, 1)
• A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form
• On Shalika models and p-adic L-Functions
• P-adic Asai L-functions of Bianchi modular forms
• Poles of p-adic Asai L-functions and distinguished representations