On $l$-adic families of cuspidal representations of $GL 2Q p$ - Mathematics > Number Theory

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Abstract: We compute the universal deformations of cuspidal representations $\pi$ of$\GL 2F$ 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 $ho$ 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 $ho$ that induces the usual local Langlandscorrespondence on characteristic zero points. Our work establishes certaincases of a conjecture of Emerton.

Author: David Helm

Source: https://arxiv.org/