A congruence property of the local Langlands correspondence

TL;DR

This paper demonstrates that the local Langlands correspondence for GL_n over a non-Archimedean local field respects congruences modulo a prime , making the -modular correspondence as effective as the complex one.

Contribution

It proves that the local Langlands correspondence respects congruences modulo , using elementary methods and explicit descriptions, bridging the complex and -modular cases.

Findings

The correspondence respects congruences modulo .

-modular correspondence is as effective as the complex one.

Elementary methods suffice to establish the congruence property.

Abstract

Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. 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.