The Breuil–mézard Conjecture for Quaternion Algebras

The Breuil–mézard Conjecture for Potentially Barsotti–tate Representations

We prove the Breuil–Mézard conjecture for 2-dimensional potentially Barsotti–Tate representations of the absolute Galois group GK , K a finite extension of Qp, for any p > 2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Qp is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serr...

The Breuil–mézard Conjecture for Potentially Barsotti–tate Representations. Toby Gee and Mark Kisin

We prove the Breuil–Mézard conjecture for 2-dimensional potentially Barsotti–Tate representations of the absolute Galois group GK , K a finite extension of Qp, for any p > 2 (up to the question of determining precise values for the multiplicities that occur). In the case that K/Qp is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serr...

On Finiteness Conjectures for Modular Quaternion Algebras

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL2-type over Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbe...

Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture

The Fontaine-mazur Conjecture for Gl2

Introduction 641 1. Breuil-Mézard conjecture and the p-adic local Langlands 644 (1.1) The Breuil-Mézard 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. Mod...