The non-abelian Leopoldt conjecture and equalities of $\mathcal{L}$-invariants

TL;DR

This paper explores the non-abelian Leopoldt conjecture's implications for the structure of eigenvarieties and the equality of Fontaine--Mazur and automorphic -invariants, with applications to automorphic representations and functoriality.

Contribution

It establishes a link between the NALC and the etaleness of eigenvarieties, and proves the equality of -invariants under certain conditions, extending previous work.

Findings

Eigenvariety is -etale over weight space at non-critical points.

Automorphic -invariants equal Fontaine--Mazur invariants under tangent vector hypotheses.

Functoriality results for symmetric power lifts of modular forms.

Abstract

Let $G$ be a reductive group quasi-split at $p$. Using arguments of Hansen--Thorne, we show that under the non-abelian Leopoldt conjecture (NALC), Hansen's $p$-adic overconvergent cohomology eigenvariety for $G$ is \'etale over its image in weight space at any non-critical classical tempered cuspidal point of `cohomological multiplicity one'. This applies to all non-critical classical cuspidal points if $G = \mathrm{Res}_{F/\mathbb{Q}}\mathrm{GL}_n$. We then let $\pi$ be a $p$-ordinary regular algebraic cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ such that $\pi_p$ is Steinberg. Combining the above \'etaleness result for the classical point attached to $\pi$, and a local-global compatibility result from our earlier work, we deduce -- under a tangent vector hypothesis that is true for at least half the simple roots -- the equality of Fontaine--Mazur and automorphic $\mathcal{L}$-invariants for $\pi$. Where this assumption is satisfied, we deduce the NALC implies a conjecture of Gehrmann: that automorphic $\mathcal{L}$-invariants are independent of cohomological degree. Our approach is inspired by (and generalises) previous work of Gehrmann--Rosso. When $\pi = \operatorname{Sym}^{n-1} \pi_f$ is the symmetric power lift of a modular form, we verify all assumptions other than the NALC, and deduce a functoriality result for the automorphic $\mathcal{L}$-invariants.