Skinner, C. M.; Wiles, A. J. Ordinary representations and modular forms. (English) Zbl 0924.11044 Proc. Natl. Acad. Sci. USA 94, No. 20, 10520-10527 (1997). Let \(p\) be a prime, and fix on embedding of \(\overline{\mathbb{Q}}\) into \(\overline{\mathbb{Q}}_p\). Let \(f\) be a new form of weight \(k\geq 2\), level \(N\) and character \(\psi\). Let \(\rho_f: \text{Gal} (\overline{\mathbb{Q}}/ \mathbb{Q})\to \text{GL}_2 (\overline{\mathbb{Q}}_p)\) be a continuous representation attached to \(f\) by Eichler and Deligne. The authors study criteria for a representation \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2 (\overline{\mathbb{Q}}_p)\) to be “modular” in the sense that there is a new form \(f\) such that \(\rho\cong \rho_f\). They establish many new cases of the conjecture of Fontaine and Mazur (Theorem 1.2) for \(\rho\) residually reducible and ordinary. This result is a consequence of a more general one (Theorem 6.1), identifying certain universal deformation rings with Hecke rings. Reviewer: A.Dabrowski (Szczecin) Cited in 6 ReviewsCited in 16 Documents MSC: 11F80 Galois representations 11F11 Holomorphic modular forms of integral weight Keywords:modular representation; ordinary representation; new form; universal deformation rings; Hecke rings PDFBibTeX XMLCite \textit{C. M. Skinner} and \textit{A. J. Wiles}, Proc. Natl. Acad. Sci. USA 94, No. 20, 10520--10527 (1997; Zbl 0924.11044) Full Text: DOI References: [1] ANN MATH 141 pp 443– (1995) · Zbl 0823.11029 · doi:10.2307/2118559 [2] ANN MATH 144 pp 136– (1996) [3] ANN MATH 142 pp 553– (1995) [4] INVENT MATH 76 pp 179– (1984) · Zbl 0545.12005 · doi:10.1007/BF01388599 [5] COMPOSITIO MATH 87 pp 269– (1993) [6] INVENT MATH 85 pp 545– (1985) [7] ANN MATH 128 pp 295– (1988) · Zbl 0658.10034 · doi:10.2307/1971444 [8] INVENT MATH 94 pp 529– (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.