Duke Mathematical Journal

On the modularity of certain 2-adic Galois representations

Mark Dickinson

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We prove some results of the form "r residually irreducible and residually modular implies r is modular," where r is a suitable continuous odd 2-dimensional 2-adic representation of the absolute Galois group of ℚ. These results are analogous to those obtained by A. Wiles, R. Taylor, F. Diamond, and others for p-adic representations in the case when p is odd; some extra work is required to overcome the technical difficulties present in their methods when p=2. The results are subject to the assumption that any choice of complex conjugation element acts nontrivially on the residual representation, and the results are also subject to an ordinariness hypothesis on the restriction of r to a decomposition group at 2. Our main theorem (Theorem 4) plays a major role in a programme initiated by Taylor to give a proof of Artin's conjecture on the holomorphicity of L-functions for 2-dimensional icosahedral odd representations of the absolute Galois group of ℚ some results of this programme are described in a paper that appears in this issue, jointly authored with K. Buzzard, N. Shepherd-Barron, and Taylor.

Article information

Duke Math. J., Volume 109, Number 2 (2001), 319-382.

First available in Project Euclid: 5 August 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F80: Galois representations
Secondary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11R34: Galois cohomology [See also 12Gxx, 19A31] 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]


Dickinson, Mark. On the modularity of certain 2-adic Galois representations. Duke Math. J. 109 (2001), no. 2, 319--382. doi:10.1215/S0012-7094-01-10923-X. https://projecteuclid.org/euclid.dmj/1091737274

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