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- W753610305 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 Vigneras 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 computeits 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 $ell$." @default.
- W753610305 created "2016-06-24" @default.
- W753610305 creator A5059740571 @default.
- W753610305 date "2015-07-09" @default.
- W753610305 modified "2023-10-18" @default.
- W753610305 title "Comptage de représentations cuspidales congruentes" @default.
- W753610305 hasPublicationYear "2015" @default.
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