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Towards a modulo $p$ Langlands correspondence for ${\mathrm {GL}}_2$

About this Title

Christophe Breuil, C.N.R.S. & I.H.É.S., Le Bois-Marie, 35 route de Chartres, 91440 Bures-sur-Yvette, France and Vytautas Paškūnas, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 216, Number 1016
ISBNs: 978-0-8218-5227-9 (print); 978-0-8218-8525-3 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00623-4
Published electronically: May 23, 2011
Keywords: Supersingular, $\mathrm {mod}\, p$ Langlands correspondence, Serre weights
MSC: Primary 22E50, 11F80, 11F70

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Representation theory of $\Gamma$ over $\bar {\mathbb F}_p$ I
  • 3. Representation theory of $\Gamma$ over $\bar {\mathbb F}_p$ II
  • 4. Representation theory of $\Gamma$ over $\bar {\mathbb F}_p$ III
  • 5. Results on $K$-extensions
  • 6. Hecke algebra
  • 7. Computation of $\mathbb {R}^1\mathcal {I}$ for principal series
  • 8. Extensions of principal series
  • 9. General theory of diagrams and representations of ${\mathrm {GL}}_2$
  • 10. Examples of diagrams
  • 11. Generic Diamond weights
  • 12. The unicity Lemma
  • 13. Generic Diamond diagrams
  • 14. The representations $D_{0}(\rho )$ and $D_1(\rho )$
  • 15. Decomposition of generic Diamond diagrams
  • 16. Generic Diamond diagrams for $f\in \{1,2\}$
  • 17. The representation $R(\sigma )$
  • 18. The extension Lemma
  • 19. Generic Diamond diagrams and representations of ${\mathrm {GL}}_2$
  • 20. The case $F=\mathbb Q_{p}$

Abstract

We construct new families of smooth admissible $\bar {\mathbb {F}}_p$-representations of $\mathrm {GL}_2(F)$, where $F$ is a finite extension of $\mathbb {Q}_p$. When $F$ is unramified, these representations have the $\mathrm {GL}_2({\mathcal O}_F)$-socle predicted by the recent generalizations of Serre’s modularity conjecture. Our motivation is a hypothetical mod $p$ Langlands correspondence.

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