A congruence property of the local Langlands correspondence

Abstract

Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \ne p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of ${\text{GL}}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.

14 pages

References in corpus (2)

• Théorie de Lubin-Tate non abélienne l-entière
• Intertwining of simple characters in GL(n)

Cited by in corpus (1)

• The local Jacquet--Langlands correspondence and congruences modulo {\ell}