Deformation rings and images of Galois representations

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

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