Abstract
Let \(F\) be a non-archimedean local field, \(\pi \) an admissible irreducible \(\mathbf{GL }_{2}(F)\)-representation with complex coefficients. For a quadratic extension \(L/F\) and an \(L^{\times }\)-character \(\chi \) a classical result of Tunnell and Saito establish a precise connection between the dimension of the Hom-space \(\hbox {Hom}_{L^{\times }}\big (\pi \vert _{L^{\times }},\chi \big )\) and the normalized local factor of the pair \((\pi ,\chi )\). The study of analogous Hom-spaces for complex valued representations has recently been generalized to \(\mathbf{GL}_n\) in Aizenbud et al. (Ann Math 172:1407–1434, 2010) and their connections with local factors have been established by work of Moeglin and Waldspurger (La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux: le cas général, preprint. Available at http://www.math.jussieu.fr/moeglin/gp.pdf, 2009). In this paper we approach the analogous problem in the context of the \(p\)-modular Langlands correspondence for \(\mathbf{GL }_{2}({\mathbf{Q}}_p)\). We describe the restriction to Cartan subgroups of an irreducible \(p\)-modular representation \(\pi \) of \(\mathbf{GL }_{2}({\mathbf{Q}}_p)\) and deduce generalized multiplicity results on the dimension of the Ext-spaces \(\mathrm{Ext}_{{{{\fancyscript{O}}}}_L^{\times }}^{i}\big (\pi \vert _{{{{\fancyscript{O}}}}_L^{\times }},\chi \big )\) where \({{{\fancyscript{O}}}}_L^{\times }\) is the ring of integers of a quadratic extension of \({\mathbf{Q}}_p\) and \(\chi \) a smooth character of \({{{\fancyscript{O}}}}_L^{\times }\).
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Notes
We use the definition in [2]: a representation is uniserial if it admits a unique filtration whose graded pieces are irreducible.
According to [17], the morphisms \(T_n^{\pm }\) should be written as \((T_n^{\pm })^{\text {neg}}\). We decided to use here the lighter notation \(T_n^{\pm }\).
if we work with the categories of profinite \(k[[H]]\)-modules of finite type and discrete admissible \(k[[H]]\)-modules, the cohomological bifunctors \(\mathrm{Ext}_{H}^{i}\big (\circ ,\circ \big ), \mathrm{Tor}^{H}_{i}\big (\circ ,\circ \big )\) can be defined on the whole associated Pontryagin category. See for instance [32, Theorems 3.7.2 and 3.7.4].
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Acknowledgments
Part of this work is issued from the author’s Ph. D. thesis, undertaken at the Laboratoire de Mathématiques de Versailles. He would like to thank his advisor Ariane Mézard for her invaluable guidance and her constant encouragements. He wish to express his deepest gratitude to Michael Harris, for his constant interest and his many precious suggestions and questions, which greatly improved the insight of the first draft of this paper. He would like to thank Florian Herzig, Tony Ly and Dipendra Prasad for very profitable discussions, and the Fields Institute and the University of Toronto where part of this work has been accomplished.
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Morra, S. Multiplicity theorems modulo \(p\) for \(\mathbf{GL }_{2}({\mathbf{Q}}_p)\) . Math. Z. 276, 421–456 (2014). https://doi.org/10.1007/s00209-013-1207-0
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DOI: https://doi.org/10.1007/s00209-013-1207-0