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

Abstract

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 $$G = \textrm{Res}_{F/\mathbb {Q}}{{\,\textrm{GL}\,}}_n$$ G = Res F / Q GL n . We then let $$\pi $$ π be a p -ordinary regular algebraic cuspidal automorphic representation of $$\textrm{GL}_n(\mathbb {A}_{\mathbb {Q}})$$ GL n ( A Q ) such that $$\pi _p$$ π p is Steinberg. Combining the above étaleness result for the classical point attached to $$\pi $$ 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 $$\mathcal {L}$$ L -invariants for $$\pi $$ π . Where this assumption is satisfied, we deduce the NALC implies a conjecture of Gehrmann: that automorphic $$\mathcal {L}$$ L -invariants are independent of cohomological degree. Our approach is inspired by (and generalises) previous work of Gehrmann–Rosso. When $$\pi = \operatorname {Sym}^{n-1} \pi _f$$ π = 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 $$\mathcal {L}$$ L -invariants.