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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
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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  • [1] D. Barrera and C. Williams, Exceptional zeros and $ {\mathcal {L}}$-invariants of Bianchi modular forms, Trans. Amer. Math. Soc. (to appear).
  • [2] L. Barthel and R. Livné, Modular representations of 𝐺𝐿₂ of a local field: the ordinary, unramified case, J. Number Theory 55 (1995), no. 1, 1–27. MR 1361556,
  • [3] Massimo Bertolini, Henri Darmon, and Adrian Iovita, Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture, Astérisque 331 (2010), 29–64 (English, with English and French summaries). MR 2667886
  • [4] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403
  • [5] C. Breuil, Invariant $ L$ et série spéciale $ p$-adique, Annales scientifiques de l'École Normale Supérieure, 37 (2004), no. 4, 559-610 (French).
  • [6] C. Breuil, Série spéciale $ p$-adique et cohomologie étale complétée, Astérisque 331 (2010), 65-115.
  • [7] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
  • [8] Henri Darmon, Integration on ℋ_{𝓅}×ℋ and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589–639. MR 1884617,
  • [9] L. Gehrmann, Functoriality of automorphic L-invariants and applications, arXiv:1704.00619, 2017.
  • [10] G. Harder, Eisenstein cohomology of arithmetic groups. The case 𝐺𝐿₂, Invent. Math. 89 (1987), no. 1, 37–118. MR 892187,
  • [11] Shin-ichi Kato, On eigenspaces of the Hecke algebra with respect to a good maximal compact subgroup of a 𝑝-adic reductive group, Math. Ann. 257 (1981), no. 1, 1–7. MR 630642,
  • [12] Louisa Orton, An elementary proof of a weak exceptional zero conjecture, Canad. J. Math. 56 (2004), no. 2, 373–405. MR 2040921,
  • [13] Jean-Pierre Serre, Cohomologie des groupes discrets, Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 399, Springer, Berlin, 1971, pp. 337–350. Lecture Notes in Math., Vol. 244 (French). MR 0422504
  • [14] M. Spieß, On special zeros of $ p$-adic L-functions of Hilbert modular forms, Invent. Math. 196 (2014), no. 1, 69-138.
  • [15] A. Venkatesh, Derived Hecke algebra and cohomology of arithmetic groups, arXiv:1608.07234, 2016.

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Additional Information

Lennart Gehrmann
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

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