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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)
Published electronically: May 23, 2011
Keywords:Supersingular, $mod p$ Langlands correspondence, Serre weights
MSC: Primary 22E50, 11F80, 11F70

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Representation theory of $\Gamma $ over $\bar {\mathbb {F}}_p$ I
  • Chapter 3. Representation theory of $\Gamma $ over $\bar {\mathbb {F}}_p$ II
  • Chapter 4. Representation theory of $\Gamma $ over $\bar {\mathbb {F}}_p$ III
  • Chapter 5. Results on $K$-extensions
  • Chapter 6. Hecke algebra
  • Chapter 7. Computation of $\mathbb {R}^1 \mathcal {I}$ for principal series
  • Chapter 8. Extensions of principal series
  • Chapter 9. General theory of diagrams and representations of $\mathrm {GL}_2$
  • Chapter 10. Examples of diagrams
  • Chapter 11. Generic Diamond weights
  • Chapter 12. The unicity Lemma
  • Chapter 13. Generic Diamond diagrams
  • Chapter 14. The representations $D_0(\rho )$ and $D_1(\rho )$
  • Chapter 15. Decomposition of generic Diamond diagrams
  • Chapter 16. Generic Diamond diagrams for $f \in \{1,2\}$
  • Chapter 17. The representation $R(\sigma )$
  • Chapter 18. The extension lemma
  • Chapter 19. Generic Diamond diagrams and representations of $\mathrm {GL}_2$
  • Chapter 20. The case $F = \mathbb {Q}_p$


We construct new families of smooth admissible -representations of , where is a finite extension of . When is unramified, these representations have the -socle predicted by the recent generalizations of Serre's modularity conjecture. Our motivation is a hypothetical mod Langlands correspondence.

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