Modularity of some potentially Barsotti–Tate Galois representations

David Savitt
2004 Compositio Mathematica  
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 ρ be a continuous odd irreducible l-adic Galois representation (l an odd prime) satisfying the hypotheses of the Fontaine-Mazur conjecture and such that ρ is modular. The notable additional hypotheses we must impose in order to conclude that ρ is modular are that ρ is potentially Barsotti-Tate, that the Weil-Deligne representation associated to ρ is
more » ... associated to ρ is irreducible and tamely ramified, and that ρ 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.
doi:10.1112/s0010437x03000289 fatcat:b2loqv2fpverzlwozq5cufqj4u