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The Fontaine-Mazur conjecture for \(\text{GL}_2\). (English) Zbl 1251.11045
Summary: We prove new cases of the Fontaine-Mazur conjecture, that a 2-dimensional \( p\)-adic representation \( \rho\) of \( G_{\mathbb{Q}, S}\) which is potentially semi-stable at \( p\) with distinct Hodge-Tate weights arises from a twist of a modular eigenform of weight \( k\geq 2\). Our approach is via the Breuil-Mézard conjecture, which we prove (many cases of) by combining a global argument with recent results of Colmez and Berger-Breuil on the \( p\)-adic local Langlands correspondence.

MSC:
11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
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