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Ordinary representations and modular forms. (English) Zbl 0924.11044

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.

MSC:

11F80 Galois representations
11F11 Holomorphic modular forms of integral weight
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References:

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