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

Abstract

Let $\pi $ be a cuspidal, cohomological automorphic representation of an inner form G of $\operatorname {{PGL}}_2$ 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 noncritical 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.