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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 $\pi$ issupercuspidal there is an irreducible, two-dimensional representation $\rho$ of$G_F$ that corresponds to $\pi$ by the mod $l$ local Langlands correspondenceof Vign{\'e}ras; we show there is a natural isomorphism between the universaldeformation rings of $\pi$ and $\rho$ that induces the usual local Langlandscorrespondence on characteristic zero points. Our work establishes certaincases of a conjecture of Emerton.

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