Nearly ordinary deformations of irreducible residual representations. (English) Zbl 1024.11036

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 §5 and depends on the study of certain deformation problems and associated deformation rings and Hecke rings (§2–§4). The main ideas are the same as in [C. Skinner and 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 \(F\).


11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight


Zbl 1005.11030
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