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arXiv · 0909.2569

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

David Helm
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Abstract

We compute the universal deformations of cuspidal representations $π$ 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 $π$ is supercuspidal there is an irreducible, two-dimensional representation $ρ$ of $G_F$ that corresponds to $π$ by the mod $l$ local Langlands correspondence of Vign{é}ras; we show there is a natural isomorphism between the universal deformation rings of $π$ and $ρ$ that induces the usual local Langlands correspondence on characteristic zero points. Our work establishes certain cases of a conjecture of Emerton.

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David Helm. 2009-09-14. On $l$-adic families of cuspidal representations of $\GL_2(\Q_p)$. https://arxiv.org/abs/0909.2569

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