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

We compute the universal deformations of cuspidal representations $\pi$ of $\GL_2(F)$ over $\overline{\FF}_l$, where $F$ is a local field of residue characteristic $p$ and $l$ is an odd prime different from $p$. When $\pi$ is supercuspidal there is an irreducible, two dimensional representation $\rho$ of $G_F$ that corresponds to $\pi$ via the mod $l$ local Langlands correspondence of~\cite{viglanglands}; we show that there is a natural isomorphism between the universal deformation rings of $\rho$ and $\pi$ that induces the usual (suitably normalized) local Langlands correspondence on characteristic zero points. Our work establishes certain cases of a conjecture on the existence of families of admissible representations.

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Published 1 January 2010