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Towards a Modulo \(p\) Langlands Correspondence for GL\(_2\)
Christophe Breuil, CNRS, Bures-sur-Yvette, France, and IHES, Bures-sur-Yvette, France, and Vytautas Paškūnas, Universität Bielefeld, Germany

Memoirs of the American Mathematical Society
2012; 114 pp; softcover
Volume: 216
ISBN-10: 0-8218-5227-2
ISBN-13: 978-0-8218-5227-9
List Price: US$70
Individual Members: US$42
Institutional Members: US$56
Order Code: MEMO/216/1016
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The authors construct new families of smooth admissible \(\overline{\mathbb{F}}_p\)-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. The authors' motivation is a hypothetical mod \(p\) Langlands correspondence.

Table of Contents

  • Introduction
  • Representation theory of \(\Gamma\) over \(\overline{\mathbb{F}}_p\thinspace \mathrm{I}\)
  • Representation theory of \(\Gamma\) over \(\overline{\mathbb{F}}_p\thinspace \mathrm{II}\)
  • Representation theory of \(\Gamma\) over \(\overline{\mathbb{F}}_p\thinspace \mathrm{III}\)
  • Results on \(K\)-extensions
  • Hecke algebra
  • Computation of \(\mathbb{R}^1\mathcal{I}\) for principal series
  • Extensions of principal series
  • General theory of diagrams and representations of \(\mathrm{GL}_2\)
  • Examples of diagrams
  • Generic Diamond weights
  • The unicity lemma
  • Generic Diamond diagrams
  • The representations \(D_{0}(\rho)\) and \(D_1(\rho)\)
  • Decomposition of generic Diamond diagrams
  • Generic Diamond diagrams for \(f\in \{1,2\}\)
  • The representation \(R(\sigma)\)
  • The extension Lemma
  • Generic Diamond diagrams and representations of \({\mathrm{GL}}_2\)
  • The case \(F=\mathbb Q_{p}\)
  • References
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