# 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 $\ell$.
Keywords :
Type de document :
Pré-publication, Document de travail
2015
Domaine :

Littérature citée [10 références]

https://hal.archives-ouvertes.fr/hal-01174800
Contributeur : Vincent Sécherre <>
Soumis le : jeudi 9 juillet 2015 - 18:02:09
Dernière modification le : jeudi 11 janvier 2018 - 06:25:42
Document(s) archivé(s) le : mercredi 26 avril 2017 - 02:37:48

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### Identifiants

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

### Citation

Vincent Sécherre. Comptage de représentations cuspidales congruentes. 2015. 〈hal-01174800〉

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