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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).

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Author notes

  1. Chandrashekhar Khare

    Present address: Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

Authors and Affiliations

  1. Department of Mathematics University of Utah, 155 South 1400 East, Room 233, Salt Lake City, UT, 84112-0090, USA

    Chandrashekhar Khare

  2. Département de Mathématique, Université de Strasbourg, Membre de l’Institut Universitaire de France 7, rue René Descartes, 67084, Strasbourg Cedex, France

    Jean-Pierre Wintenberger

Authors
  1. Chandrashekhar Khare
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  2. Jean-Pierre Wintenberger
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Corresponding author

Correspondence to Chandrashekhar Khare.

Additional information

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.

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Khare, C., Wintenberger, JP. Serre’s modularity conjecture (I). Invent. math. 178, 485–504 (2009). https://doi.org/10.1007/s00222-009-0205-7

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