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MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

Part of: Representation theory of groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 May 2020

Daniel Le Open the ORCID record for Daniel Le [Opens in a new window] ,
Stefano Morra  and
Benjamin Schraen
Daniel Le
Affiliation:
University of Toronto, 40 St. George Street, Toronto, ONM5S 2E4 (daniel.le@math.toronto.edu)
Stefano Morra
Affiliation:
Université Paris 8, Laboratoire d’Analyse, Géométrie et Applications, LAGA, Université Sorbonne Paris Nord, CNRS, UMR 7539, F-93430, Villetaneuse, France (morra@math.univ-paris13.fr)
Benjamin Schraen
Affiliation:
Université Paris-Saclay, CNRS, Laboratoire de mathématiques dOrsay, 91405, Orsay, France (benjamin.schraen@math.u-psud.fr)
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Abstract

Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. We describe the $\mathfrak{m}$-torsion in the $\text{mod}\,p$ cohomology of Shimura curves with full congruence level at $v$ as a $\text{GL}_{2}(k_{v})$-representation. In particular, it only depends on $\overline{r}|_{I_{F_{v}}}$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $\text{GL}_{2}(\mathbf{F}_{q})$-projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math. 200(1) (2015), 1–96].

Keywords

multiplicity onecohomology of Shimura curvesmod p Langlands programmodular representations of GL2(Fq)

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11F80: Galois representations 20C33: Representations of finite groups of Lie type
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 2 , March 2022 , pp. 637 - 658
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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