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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
Vytautas Paškūnas Universität Bielefeld, Bielefeld, Germany
Available Formats:
Electronic ISBN: 978-0-8218-8525-3
Product Code: MEMO/216/1016.E
List Price: $70.00 MAA Member Price:$63.00
AMS Member Price: $42.00 Click above image for expanded view Towards a Modulo$p$Langlands Correspondence for GL$_2$Christophe Breuil CNRS, Bures-sur-Yvette, France and IHES, Bures-sur-Yvette, France Vytautas Paškūnas Universität Bielefeld, Bielefeld, Germany Available Formats:  Electronic ISBN: 978-0-8218-8525-3 Product Code: MEMO/216/1016.E  List Price:$70.00 MAA Member Price: $63.00 AMS Member Price:$42.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2162012; 114 pp
MSC: Primary 22; 11;

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.

• 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 $F=\mathbb Q_{p}$
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Volume: 2162012; 114 pp
MSC: Primary 22; 11;

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.

• 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 $F=\mathbb Q_{p}$
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