Geodesic
L-invariants for cohomological representations of PGL(2) over arbitrary number fields
Forum of Mathematics, Sigma, Tome 12 (2024)

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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. 
@article{10_1017_fms_2024_51,
     author = {Lennart Gehrmann and Maria Rosaria Pati},
     title = {L-invariants for cohomological representations of {PGL(2)} over arbitrary number fields},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {12},
     year = {2024},
     doi = {10.1017/fms.2024.51},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.51/}
}
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Lennart Gehrmann; Maria Rosaria Pati. L-invariants for cohomological representations of PGL(2) over arbitrary number fields. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.51

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