## Duke Mathematical Journal

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

### On the modularity of certain 2-adic Galois representations

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 5 August 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1091737274

**Digital Object Identifier**

doi:10.1215/S0012-7094-01-10923-X

**Mathematical Reviews number (MathSciNet)**

MR1845182

**Zentralblatt MATH identifier**

1015.11020

**Subjects**

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]

#### Citation

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