On $\mathscr L$-invariants associated to Hilbert modular forms

TL;DR

This paper proves the equivalence of two different definitions of $\\mathscr{L}$-invariants associated to certain Hilbert modular forms, linking cohomological and Galois-theoretic perspectives.

Contribution

It establishes the equality of cohomological and Galois-theoretic $\mathscr{L}$-invariants for Hilbert modular forms with Steinberg local components.

Findings

The two $\\mathscr{L}$-invariants coincide for the considered forms.

The result bridges cohomology and Galois representations in the context of Hilbert modular forms.

Abstract

Given a cuspidal Hilbert modular eigenform $\pi$ of parallel weight 2 and a nonarchimedian place $\mathfrak p$ of the underlying totally real field such that the local component of $\pi$ at $\mathfrak p$ is the Steinberg representation, one can associate two types of $\mathscr L$-invariants, one defined in terms of the cohomology of arithmetic groups and the other in terms of the Galois representation associated to $\pi$. We show that the $\mathscr L$-invariants are the same.