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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
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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

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.

• 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