The Fontaine-Mazur conjecture for {GL}_2

Introduction 641 1. Breuil-Mezard conjecture and the p-adic local Langlands 644 (1.1) The Breuil-Mezard conjecture 644 (1.2) Review of Colmez’s functor 647 (1.3) Hilbert-Samuel multiplicities 650 (1.4) Representations and pseudo-representations 653 (1.5) GL2(Qp)-representations mod p. 657 (1.6) Local patching and multiplicities 661 (1.7) From pseudo-representations to representations 669 2. Modularity via the Breuil-Mezard conjecture 675 (2.1) Quaternionic forms 675 (2.2) Global patching and multiplicities 677 (2.3) The Breuil-Mezard conjecture 686 Addendum to [Ki 1] 687 References 689

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