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Gelfand-Kirillov dimension and mod p cohomology for GL2

Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, Benjamin Schraen

Abstract

Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$ and $\overline{r} : {\rm Gal}(\overline F/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ an irreducible modular continuous Galois representation which, at the place $v$, is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of $\mathrm{GL}_2(F_v)$ over $\overline{\mathbb{F}}_p$ associated to $\overline{r}$ in the corresponding Hecke-eigenspaces of the mod $p$ cohomology have Gelfand--Kirillov dimension $[F_v:\mathbb{Q}]$, as well as several related results.

Gelfand-Kirillov dimension and mod p cohomology for GL2

Abstract

Let be a prime number, a totally real number field unramified at places above and a quaternion algebra of center split at places above and at no more than one infinite place. Let be a fixed place of above and an irreducible modular continuous Galois representation which, at the place , is semisimple and sufficiently generic (and satisfies some weak genericity conditions at a few other finite places). We prove that many of the admissible smooth representations of over associated to in the corresponding Hecke-eigenspaces of the mod cohomology have Gelfand--Kirillov dimension , as well as several related results.

Paper Structure

This paper contains 49 sections, 99 theorems, 361 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Torsion in cohomology and Gelfand--Kirillov dimension
  3. The main theorem and its consequences
  4. The proof
  5. Smooth representations
  6. Patching: the setup
  7. Lattices in locally algebraic representations
  8. Deformation rings, and conclusion
  9. Notation
  10. Acknowledgements
  11. Preliminaries
  12. Group theoretic preliminaries
  13. The inertial local Langlands correspondence and Serre weights
  14. Tame inertial types
  15. Combinatorics of types and Serre weights
  16. ...and 34 more sections

Key Result

Theorem 1.1

Keep all the above assumptions on $F$, $D$, and assume that $\overline{r}$ is generic and that $\overline{r} |_{G_{F(\!\sqrt[p]{1})}}$ is absolutely irreducible. Let $V^v=\prod_{w\ne v}V_w$ with $V_w=\mathop{\mathrm{GL}}\nolimits_2({\mathcal{O}}_{F_w})$ if neither $D$ nor $\overline{r}$ ramifies at

Figures (2)

  • Figure 1: Extension graph
  • Figure 2: Change of variables between the tables

Theorems & Definitions (216)

  • Theorem 1.1: Corollary \ref{['mainmain']}
  • Theorem 1.2: Corollary \ref{['cor:platitude_Hecke']}
  • Theorem 1.3: Corollary \ref{['padiclanglands']}
  • Theorem 1.4: Theorem \ref{['thm:GKdim-criterion']}
  • Theorem 1.5: Proposition \ref{['prop-W3topi=dim1']}
  • Theorem 1.6: Corollary \ref{['cor:GKdim']}
  • Theorem 1.7: Corollary \ref{['rpr']}
  • Theorem 1.8: Corollary \ref{['HT102-1']}
  • Theorem 1.9: Theorem \ref{['largest']}
  • Theorem 1.10: Theorem \ref{['mainpatching2']}
  • ...and 206 more