R\'esum\'e
We compute Benois \({\mathscr {L}}\)-invariants of weight 1 cuspforms and of their adjoint representations and show how this extends Gross’ p-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois’ construction depends on the choice of a regular submodule which is well understood when the representation is p-regular, as it then amounts to the choice of a “motivic” p-refinement. The situation is dramatically different in the p-irregular case, where the regular submodules are parametrized by a flag variety and thus depend on continuous parameters. We are nevertheless able to show in some examples, how Hida theory and the geometry of the eigencurve can be used to detect a finite number of choices of arithmetic and “mixed-motivic” significance.
Résumé
Nous calculons les \({\mathscr {L}}\)-invariants de Benois attachés aux formes paraboliques de poids 1 ainsi qu’à leurs représentations adjointes, et nous montrons que cela étend le régulateur p-adique de Gross aux motifs d’Artin qui ne sont pas critiques au sens de Deligne. La construction de Benois dépend du choix d’un sous-module régulier, qui est bien compris lorsque la représentation est p-régulière, puisque cela revient au choix d’un p-raffinement “motivique”. La situation est radicalement différente dans le cas p-irrégulier, où les sous-modules réguliers sont paramétrés par une variété de drapeaux et dépendent ainsi de paramètres continus. Nous arrivons néanmoins à montrer dans quelques exemples comment la théorie de Hida et la géométrie de la courbe de Hecke permettent de détecter un nombre fini de choix de nature arithmétique et “mixte-motivique”.
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Notes
The definition of \(L_p(V,D,s)\), resp. \(L_p(W,D^\perp ,s)\), depends on the choice of a Galois stable lattice in V, resp. D, but \({\mathscr {L}}(V,D)\), resp. \({\mathscr {L}}(W,D^\perp )\), does not, hence our abuse of notation.
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Acknowledgements
The authors are mostly indebted to D. Benois for numerous discussions related to this project. We would like also to thank A. Betina and J. Bellaïche for helpful discussions, as well as the the anonymous referee for the carefully reading of the manuscript and numerous comments. The research leading to this article is jointly funded by the Agence Nationale de Recherche ANR-18-CE40-0029 and the Fonds National de Recherche Luxembourg INTER/ANR/18/12589973 in the project Galois representations, automorphic forms and their L-functions (GALF).
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To Bernadette Perrin-Riou on the occasion of her 65th birthday.
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Dimitrov, M., Maksoud, A. \(\pmb {\mathscr {L}}\)-invariants of Artin motives. Ann. Math. Québec 47, 49–71 (2023). https://doi.org/10.1007/s40316-022-00201-0
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DOI: https://doi.org/10.1007/s40316-022-00201-0