Mathematical Research Letters

Volume 17 (2010)

Number 5

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

Pages: 805 – 822



David Helm (Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712)


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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