A congruence property of the local Langlands correspondence

Details A congruence property of the local Langlands correspondence A congruence property of the local Langlands correspondence

2011-07-12

arxiv.org/abs/1107.2266v1

Mathematics

Number Theory

14 pages

Scientific paper

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.

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