Families of Galois representations - increasing the ramification (Q1196390)

Let \(k\) be a finite field of characteristic \(l \neq 2\), let \(S\) be a finite set of primes and \(G_{\mathbb{Q},S}\) the maximal algebraic extension of \(\mathbb{Q}\) unramified outside \(S\). Following \textit{B. Mazur} [Compos. Math. 74, 115-133 (1990)] the author considers the lifting of representations \(\bar\rho: G_{\mathbb{Q},S} \to GL_ 2(k)\) to \(W(k)\)- algebras. In particular he studies the effect of enlarging the set \(S\) to a set \(S'\). His main result (Theorem 1) describes the relation between the corresponding universal deformation rings \(R_ S\) and \(R_{S'}\).