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arXiv · 2010.02328

G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range

Jeremy Booher·Brandon Levin
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Abstract

Let $G$ be a split reductive group over the ring of integers in a $p$-adic field with residue field $\mathbf{F}$. Fix a representation $\overline{\rho}$ of the absolute Galois group of an unramified extension of $\mathbf{Q}_p$, valued in $G(\mathbf{F})$. We study the crystalline deformation ring for $\overline{\rho}$ with a fixed $p$-adic Hodge type that satisfies an analog of the Fontaine-Laffaille condition for $G$-valued representations. In particular, we give a root theoretic condition on the $p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.

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BibTeXRIS

Jeremy Booher, Brandon Levin. 2020-10-05. G-Valued Crystalline Deformation Rings in the Fontaine-Laffaille Range. https://arxiv.org/abs/2010.02328

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