Even Galois representations and the Fontaine-Mazur conjecture. II (Q2879892)

Let \(G_{\mathbb Q}\) be the absolute Galois group of the field of rational numbers. The author shows the following striking result:NEWLINENEWLINELet \(p>7\) be an prime number and \(\rho:G_{\mathbb Q} \to \mathrm{GL}(2,\overline{\mathbb Q}_p)\) a continuous homomorphism which is unramified outside finitely many places. Without loss of generality, its image sits inside some \(\mathrm{GL}(2,{\mathcal O})\) where \(\mathcal O\) is the valuation ring of some finite extension of \({\mathbb Q}_p\). Assume that the residual representation is absolutely irreducible and not of dihedral type. Assume furthermore that the restriction of \(\rho\) to the decomposition group of \(p\) is potentially semi-stable with distinct Hodge-Tate weights and its residual representation is not a twist of a certain reducible representation. Then \(\rho\) is modular.NEWLINENEWLINEThis result generalizes a result of the author [Invent. Math. 185, No. 1, 1--16 (2011; Zbl 1231.11058)] in that it replaces the ordinarity condition there by the semi-stability condition here.NEWLINENEWLINEUsing a theorem of \textit{M. Kisin} [J. Am. Math. Soc. 22, No. 3, 641--690 (2009; Zbl 1251.11045)] the theorem is true if \(\rho\) is odd. Therefore the proof consists in showing that \(\rho\) has to be odd, thereby supporting the Fontaine-Mazur Conjecture.NEWLINENEWLINEThe methods used also produce results as the following: There exists a surjective even representation \(G_{\mathbb Q}\to \mathrm{SL}(2,{\mathbb F}_{11})\) with no geometric (= de Rham) deformations.