Modular Local Langlands Correspondence for GLn

Let F be a non-Archimedean local field of residual characteristic p and l be a prime number, l not equal p. We consider the Langlands correspondence between irreducible, n-dimensional, smooth representations of the Weil group of F and irreducible cuspidal representations of GL(n)( F). We 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 l. We indicate a parallel application to the Jacquet-Langlands correspondence.