On $l$-adic families of cuspidal representations of $\GL_2(\Q_p)$

Abstract

We compute the universal deformations of cuspidal representations $\pi$ of $\GL_2(F)$ over an algebraically closed field of characteristic $l$, where $F$ is a local field of residue characteristic $p$ not equal to $l$. When $\pi$ is supercuspidal there is an irreducible, two-dimensional representation $\rho$ of $G_F$ that corresponds to $\pi$ by the mod $l$ local Langlands correspondence of Vign{\'e}ras; we show there is a natural isomorphism between the universal deformation rings of $\pi$ and $\rho$ that induces the usual local Langlands correspondence on characteristic zero points. Our work establishes certain cases of a conjecture of Emerton.