Duke Mathematical Journal

Serre's modularity conjecture: The level one case

Chandrashekhar Khare

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Abstract

We prove the level one case of Serre's conjecture. Namely, we prove that any continuous, odd, irreducible representation $\overline{\rho}:G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}_{p}})$ which is unramified outside $p$ arises from a cuspidal eigenform in $S_{k}(\mathrm{SL}_2(\mathbb{Z}))$ for some integer $k \geq 2$. The proof relies on the methods introduced in an earlier joint work with J.-P. Wintenberger [31] together with a new method of weight reduction

Article information

Source
Duke Math. J. Volume 134, Number 3 (2006), 557-589.

Dates
First available in Project Euclid: 28 August 2006

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1156771903

Digital Object Identifier
doi:10.1215/S0012-7094-06-13434-8

Mathematical Reviews number (MathSciNet)
MR2254626

Subjects
Primary: 11F80: Galois representations 11F11: Holomorphic modular forms of integral weight
Secondary: 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55]

Citation

Khare, Chandrashekhar. Serre's modularity conjecture: The level one case. Duke Math. J. 134 (2006), no. 3, 557--589. doi:10.1215/S0012-7094-06-13434-8. http://projecteuclid.org/euclid.dmj/1156771903.


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