| We prove a portion of a conjecture of Conrad-Diamond-Taylor, which yields proofs of some 2-dimensional cases of the Fontaine-Mazur conjectures. Let r be a continuous odd irreducible l-adic Galois representation (l an odd prime) satisfying the hypotheses of the Fontaine-Mazur conjecture and such that r is modular. The notable additional hypotheses we must impose in order to conclude that r is modular are that r is potentially Barsotti-Tate, that the Weil-Deligne representation associated to r is irreducible and tamely ramified, and that r is conjugate to a representation over F l which is reducible with scalar centralizer.;The proof follows techniques of Breuil, Conrad, Diamond, and Taylor, and in particular requires extensive calculation with Breuil's classification of l-torsion finite flat group schemes over base schemes with high ramification. |
