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Minimally ramified deformations when $\ell \neq p$

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  12 November 2018

Jeremy Booher
Jeremy Booher*
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA email jeremybooher@math.arizona.edu
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Abstract

Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.

Keywords

Galois representationsgeometric representationsdeformation theoryreductive groups

MSC classification

Primary: 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 1 , January 2019 , pp. 1 - 37
Copyright
© The Author 2018 

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