## Derived Hecke algebra and automorphic ${\mathcal {L}}$-invariants

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- by Lennart Gehrmann PDF
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**372**(2019), 7767-7784 Request permission

## Abstract:

Let $\pi$ be a cohomological cuspidal automorphic representation of PGL$_2$ over a number field of arbitrary signature. Under the assumption that the local component of $\pi$ at a prime ${\mathfrak {p}}$ is the Steinberg representation, the automorphic ${\mathcal {L}}$-invariant of $\pi$ at ${\mathfrak {p}}$ has been defined using the lowest degree cohomology in which the system of Hecke eigenvalues associated with $\pi$ occurs.

In this article we define automorphic ${\mathcal {L}}$-invariants for each cohomological degree and show that they behave well with respect to the action of Venkatesh’s derived Hecke algebra. As a corollary, we show that these ${\mathcal {L}}$-invariants are (essentially) the same if the following conjecture of Venkatesh holds: the $\pi$-isotypic component of the cohomology is generated by the minimal degree cohomology as a module over the $p$-adic derived Hecke algebra.

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## Additional Information

**Lennart Gehrmann**- Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
- MR Author ID: 1209875
- Email: lennart.gehrmann@uni-due.de
- Received by editor(s): February 1, 2019
- Published electronically: June 5, 2019
- Additional Notes: The ideas presented in this article emerged during a stay at the Bernoulli Center (CIB) in the course of the semester-long program on Euler systems and Special Values of $L$-functions. It is the author’s pleasure to thank the organizers of the program as well as the local staff for a pleasant and scientifically stimulating stay.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 7767-7784 - MSC (2010): Primary 11F41; Secondary 11F67, 11F75, 11F85
- DOI: https://doi.org/10.1090/tran/7815
- MathSciNet review: 4029680