Abstract. Given a complete local Noetherian ring (A, m A ) with finite residue field and a subfield k k k of A/m A , we show that every closed subgroup G ofunder some small restrictions on k k k. Here W (k k k) A is the closed subring of A generated by the Teichmüller lifts of elements of the subfield k k k.
We prove modularity for a huge class of rigid Calabi-Yau threefolds over Q. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular.
We study generalisations to totally real fields of the methods originating with Wiles and Taylor and Wiles [A. Wiles, Modular elliptic curves and Fermat's Last ] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over Q( √ 2 ).
Abstract. We show the inverse deformation problem has an affirmative answer: given a complete local noetherian ring A with finite residue field k k k, we show that there is a topologically finitely generated profinite group Γ and an absolutely irreducible continuous representation ρ : Γ → GLn(k k k) such that A is the universal deformation ring for Γ, ρ.
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