Nearly ordinary deformations of irreducible residual representations (Q1612805)

Let \(E\) denote a finite extension of \(\mathbb{Q}_p\), let \({\mathcal O}_E\) be its ring of integers and \(\lambda\) a uniformizer. Let \[ \rho:\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\to\text{GL}_2(E) \] be a continuous representation. Picking a stable lattice in \(E^2\) and reducing \(\rho\) modulo \(\lambda\), we obtain a representation \[ \overline\rho: \text{Gal}(\overline \mathbb{Q}/\mathbb{Q})\to\text{GL}_2({\mathcal O}_{E/\lambda}). \] The authors establish the modularity of \(\rho\) under certain conditions (i.e. \(\rho\) is irreducible and unramified outside a finite set of primes, the semisimplification \(\overline\rho^{ss}\) of \(\overline\rho\) is irreducible and comes from a modular form). The proof is contained in \S 5 and depends on the study of certain deformation problems and associated deformation rings and Hecke rings (\S 2--\S 4). The main ideas are the same as in [\textit{C. Skinner} and \textit{A. Wiles}, Publ. Math., Inst. Hautes Étud. Sci. 89, 5-126 (1999; Zbl 1005.11030)], where the residual representation is reducible. The authors also prove a similar theorem with \(\mathbb{Q}\) replace by a general totally real number field \(F\).