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Big principal series, $p$-adic families and $\mathcal {L}$-invariants

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  25 April 2022

Lennart Gehrmann  and
Giovanni Rosso
Lennart Gehrmann
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germanylennart.gehrmann@uni-due.de
Giovanni Rosso
Affiliation:
Departments of Mathematics and Statistics, Concordia University, 1455 De Maisonneuve Blvd. W., Montréal, Québec, Canada H3G 1M8 giovanni.rosso@concordia.ca
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Abstract

In earlier work, the first named author generalized the construction of Darmon-style $\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal {L}$-invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur $\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur $\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

Keywords

automorphic formsGalois representationsL-invariantsp-adic families

MSC classification

Primary: 11F55: Other groups and their modular and automorphic forms (several variables)
Secondary: 11F33: Congruences for modular and $p$-adic modular forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F75: Cohomology of arithmetic groups 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 2 , February 2022 , pp. 409 - 436
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

While working on this paper the first named author was visiting McGill University, supported by Deutsche Forschungsgemeinschaft, and he would like to thank these institutions. The second named author was partly funded by FRQNT grant 2019-NC-254031 and NSERC grant RGPIN-2018-04392. In addition, the authors would like to thank Henri Darmon, Michael Lipnowski, Vytautas Pas̆kūnas and Chris Williams for several intense and stimulating discussions. We also thank the referee for their careful reading of the paper and many suggestions.

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