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

TL;DR

This paper computes universal deformations of cuspidal representations of al_2(F) over fields of characteristic l, linking them to Galois representations and confirming cases of Emerton's conjecture.

Contribution

It establishes an isomorphism between deformation rings of cuspidal representations and Galois representations, connecting mod l local Langlands correspondence with deformation theory.

Findings

Universal deformation rings of cuspidal representations are isomorphic to those of associated Galois representations.

The isomorphism induces the local Langlands correspondence at characteristic zero points.

The work confirms specific cases of Emerton's conjecture.

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.