On Serre's conjecture for 2-dimensional mod p representations of Gal((Q)over-bar/Q)

• 成果类型: Article
• 署名作者: Khare, Chandrashekhar; Wintenberger, Jean-Pierre
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.169.229
• 发表日期: 2009
• 页码: 229-253
• 关键词: hilbert modular-forms; Galois representations; crystalline representations; deformation rings; skolem-problems; fontaine-mazur; hecke algebras; picard-groups; nonexistence; families

摘要

We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod p representations (rho) over bar of the absolute Galois group of Q. It is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instal-ice we prove that there are no irreducible (p,p) type group schemes over Z. We prove that a (rho) over bar as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function. In the last part of the paper we present arguments that reduce Serre's conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.