# Serre’s modularity conjecture (I)

@article{Khare2009SerresMC,
title={Serre’s modularity conjecture (I)},
author={Chandrashekhar Khare and Jean-Pierre Wintenberger},
journal={Inventiones mathematicae},
year={2009},
volume={178},
pages={485-504}
}
• Published 2009
• Inventiones mathematicae
AbstractThis paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases $p\not=2$ and odd conductor, and p=2 and weight 2, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see Khare and Wintenberger (Invent. Math., doi:10.1007/s00222-009-0206-6, 2009). We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin… Expand
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