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Transactions of the American Mathematical Society

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


Author: Lennart Gehrmann
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F41; Secondary 11F67, 11F75, 11F85
DOI: https://doi.org/10.1090/tran/7815
Published electronically: June 5, 2019
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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 $ {\ensuremath {\mathfrak{p}}}$ is the Steinberg representation, the automorphic $ {\mathcal {L}}$-invariant of $ \pi $ at $ {\ensuremath {\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
Email: lennart.gehrmann@uni-due.de

DOI: https://doi.org/10.1090/tran/7815
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.
Article copyright: © Copyright 2019 American Mathematical Society