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تسجيل مستخدم جديدDeformation rings and images of Galois representations
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Let $mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(mathbb{F})$ for $mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R$ be complete noetherian local $W(mathbb{F})$ -algebras with residue field $mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $mathcal{G}(R)$ with full residual image $mathcal{G}(mathbb{F})$ is a conjugate of a group $mathcal{G}(A)$ for $Asubset R$ a closed subring that is local and has residue field $mathbb{F}$ . (2) Every surjective homomorphism $mathcal{G}(R)tomathcal{G}(R)$ is, up to conjugation, induced from a ring homomorphism $Rto R$. (3) The identity map on $mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $mathcal{G}(R)$ given by the reduction map $mathcal{G}(R)tomathcal{G}(mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $mathcal{G}$, and we study in the case at hand in great detail what conditions on $mathbb{F}$ or on $p$ in relation to $mathcal{G}$ are necessary for the above results to hold.
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