L-invariants for cohomological representations of PGL(2) over arbitrary number fields

TL;DR

This paper constructs automorphic L-invariants for cohomological automorphic representations of PGL(2) over arbitrary number fields and shows their equivalence with Fontaine-Mazur invariants in totally real cases, generalizing previous results.

Contribution

It introduces a method to define automorphic L-invariants for a broad class of automorphic representations and proves their equality with Galois-theoretic invariants in totally real fields.

Findings

Automorphic L-invariants constructed for cohomological representations.

Equivalence with Fontaine-Mazur invariants in totally real fields.

Generalization from parallel weight 2 to arbitrary cohomological weights.

Abstract

Let $\pi$ be a cuspidal, cohomological automorphic representation of an inner form $G$ of $\mathrm{PGL}_2$ over a number field $F$ of arbitrary signature. Further, let $\mathfrak{p}$ be a prime of $F$ such that $G$ is split at $\mathfrak{p}$ and the local component $\pi_\mathfrak{p}$ of $\pi$ at $\mathfrak{p}$ is the Steinberg representation. Assuming that the representation is non-critical at $\mathfrak{p}$ we construct automorphic $\mathcal{L}$-invariants for the representation $\pi$. If the number field $F$ is totally real, we show that these automorphic $\mathcal{L}$-invariants agree with the Fontaine-Mazur $\mathcal{L}$-invariant of the associated $p$-adic Galois representation. This generalizes a recent result of Spiess respectively Rosso and the first named author from the case of parallel weight $2$ to arbitrary cohomological weights.