The local Langlands correspondence for $GL(n)$, which was proved in 2001, connects representations of the Weil group of a nonarchimedean local field, $F$, to admissible representations of $GL_n(F)$. In this talk we will present recent results showing that a version of the correspondence respects congruences modulo $\ell$ between objects on either side ($\ell$ is a prime different from the residue characteristic of $F$), and works more broadly for representations whose coefficient rings are Galois deformation rings. We will describe how $\ell$-adically interpolated zeta integrals, gamma factors, and converse theorems are involved in the proof.