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In mathematics , a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field .. mathematics. mathematics. ring. ring ( mathematics ). Galois group. Galois group. finite field. finite field. local field. local field. In particular for any such lifting problem there is often a universal deformation ring that classifies all such liftings , and whose spectrum is the universal deformation space .. ring. ring ( mathematics ). spectrum. spectrum of a ring. A key step in Wiles 's proof of the modularity theorem was to study the relation between universal deformation rings and Hecke algebras .. modularity theorem. modularity theorem