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

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 étale 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 $π$ be a $p$-ordinary regular algebraic cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ such that $π_p$ is Steinberg. Combining the above étaleness result for the classical point attached to $π$, 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 $π$. 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 $π= \operatorname{Sym}^{n-1} π_f$ 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.

Theorems & Definitions (98)

• Theorem 1
• Remark 1.1
• Remark 1.2
• Theorem 2
• Remark 1.3
• Remark 1.4
• Corollary 3
• Theorem 4
• Remark 2.1
• Definition 2.2