Reduction of Weil-Deligne Representations

Abstract

Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil--Deligne representations of the Weil group $W_F$. Given two $2$-dimensional, semisimple, continuous representations $\rho_1, \rho_2$ of $W_F$ with images in $\text{GL}_2(\mathcal{O}_K)$, where $K/\mathbb{Q}_\ell$ is a finite extension with maximal ideal $\lambda \subset \mathcal{O}_K$, a natural question to ask is when their mod $\lambda$ reductions $\overline{\rho}_1$ and $\overline{\rho}_2$ are isomorphic. In this paper, we give a complete characterization of when the reductions are isomorphic. For this, we first give a description of when the reductions of these representations are decomposable or irreducible, utilizing the correspondence between continuous $\ell$-adic representations of $W_F$ and $\ell$-adic Weil--Deligne representations. We conclude with some examples which arise in the modular method for solving generalized Fermat equations.