, Volume 178, Issue 3, pp 485-504
Date: 04 Jul 2009

Serre’s modularity conjecture (I)

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Abstract

This 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 (Invent. Math., doi:10.1007/s00222-009-0207-5, 2009).

Dedicated to Jean-Pierre Serre
CK was partially supported by NSF grants DMS 0355528 and DMS 0653821, the Miller Institute for Basic Research in Science, University of California Berkeley, and a Guggenheim fellowship.
JPW is member of the Institut Universitaire de France.