Deformation theory of the trivial mod \(p\) Galois representation for \(\mathrm{GL}_n\)

We study the rigid generic fiber \(\mathcal{X}^\square_{\overline\rho}\) of the framed deformation space of the trivial representation \(\overline\rho: G_K \to \text{GL}_n(k)\) where \(k\) is a finite field of characteristic \(p>0\) and \(G_K\) is the absolute Galois group of a finite extension \...

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Bibliographic Details
Published in:arXiv.org 2021-10
Main Author: Iyengar, Ashwin
Format: Article
Language:English
Subjects:
Deformation
Fields (mathematics)
Representations
Online Access:Get full text
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Summary:We study the rigid generic fiber \(\mathcal{X}^\square_{\overline\rho}\) of the framed deformation space of the trivial representation \(\overline\rho: G_K \to \text{GL}_n(k)\) where \(k\) is a finite field of characteristic \(p>0\) and \(G_K\) is the absolute Galois group of a finite extension \(K/\mathbf{Q}_p\). Under some mild conditions on \(K\) we prove that \(\mathcal{X}^\square_{\overline\rho}\) is normal. When \(p > n\) we describe its irreducible components, and show Zariski density of its crystalline points.
ISSN:2331-8422
DOI:10.48550/arxiv.1904.05996