Seminar of Algebra

In this talk we will first make a short introduction to the local Langlands program. Then we will prove that the local Langlands transfer is compatible with congruences in the following sense. Let $\pi$ and $\pi'$ be two $\ell$-adic irreducible cuspidal representations of a quasi-split classical $p$-adic group $G$, $\ell \neq p$, and let $\sigma$ and $\sigma'$ be their respective Langlands's transfers to $GL(N)$. Assume $\pi$ and $\pi'$ are integral and denote $r_\ell(\pi)$ and $r_\ell(\pi')$ their reduction modulo $\ell$. We will show that, if $r_\ell(\pi) \leq r_\ell(\pi')$, then $r_\ell(\sigma)$ and $r_\ell(\sigma')$ have a unique generic irreducible component in common. The proof is local-global and we will use the latest results in the trace formula, some modularity theorems and Vignéras' mod $\ell$ local Langlands correspondence. This is joint work with Vincent Sécherre.