Comptage de représentations cuspidales congruentes

Abstract : Let $\F$ be a non-Archimedean locally compact field of residue characteristic $p$, $\G$ be an inner form of $\GL_n(\F)$, $n\>1$, and $\ell$ be a prime number different from $p$. We give a numerical criterion for an integral $\ell$-adic irreducible cuspidal representation $\rt$ of $\G$ to have a super\-cuspidal irreducible reduction mod $\ell$, by counting inertial classes of cuspidal representations that are congruent to the inertial class of $\rt$, generalizing results by Vignéras and Dat. In the case the reduction mod $\ell$ of $\rt$ is not super\-cuspidal irreducible, we show that this counting argument allows us to compute its length and the size of the supercuspidal support of its irreducible components. We define an invariant $w(\rt)\>1$ | the product of this length by this size | which is expected to behave nicely through the local Jacquet-Langlands correspondence. Given an $\ell$-modular irreducible cuspidal representation $\rho$ of $\G$ and a positive integer $a$, we give a criterion for the existence of an integral $\ell$-adic irreducible cuspidal representation $\rt$ of $\G$ such that its reduction mod $\ell$ contains $\rho$ and has length $a$. This allows us to obtain a formula for the cardinality of the set of reductions mod $\ell$ of inertial classes of $\ell$-adic irreducible cuspidal representations $\rt$ with given depth and invariant $w$. These results are expected to be useful to prove that the local Jacquet-Langlands correspondence preserves congruences mod $\ell$.
Document type :
Preprints, Working Papers, ...
Liste complète des métadonnées

Cited literature [10 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01174800
Contributor : Vincent Sécherre <>
Submitted on : Thursday, July 9, 2015 - 6:02:09 PM
Last modification on : Wednesday, January 23, 2019 - 2:39:26 PM
Document(s) archivé(s) le : Wednesday, April 26, 2017 - 2:37:48 AM

Files

comptage.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01174800, version 1
  • ARXIV : 1507.02634

Citation

Vincent Sécherre. Comptage de représentations cuspidales congruentes. 2015. ⟨hal-01174800⟩

Share

Metrics

Record views

436

Files downloads

51