Modular local Langlands correspondence for GL\(_n\) (Q2929644)

Let \(F\) be a non-Archimedean local field of residual characteristic \(p\) and \(\ell\) be a prime number, \(\ell\neq p\). The authors consider the Langlands correspondence between irreducible, \(n\)-dimensional, smooth representations of the Weil group of \(F\) and irreducible cuspidal representations of \(\mathrm{GL}_ n (F)\). They use an explicit description of the correspondence from an earlier paper to give a straightforward and transparent proof of the fact that it respects the relationship of congruence modulo \(\ell\). They indicate a parallel application to the Jacquet-Langlands correspondence.