Geodesic
Serre’s conjecture over $\mathbb F_9$
Jordan S. Ellenberg1
1 Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States
Annals of mathematics, Tome 161 (2005) no. 3, pp. 1111-1142.

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In this paper we show that an odd Galois representation $\bar{\rho}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_2(\mathbb{F}_9)$ having nonsolvable image and satisfying certain local conditions at $3$ and $5$ is modular. Our main tools are ideas of Taylor [21] and Khare [10], which reduce the problem to that of exhibiting points on a Hilbert modular surface which are defined over a solvable extension of $\mathbb{Q}$, and which satisfy certain reduction properties. As a corollary, we show that Hilbert-Blumenthal abelian surfaces with ordinary reduction at $3$ and $5$ are modular.
DOI : 10.4007/annals.2005.161.1111

Jordan S. Ellenberg 1

1 Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States 
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Jordan S. Ellenberg. Serre’s conjecture over $\mathbb F_9$. Annals of mathematics, Tome 161 (2005) no. 3, pp. 1111-1142. doi : 10.4007/annals.2005.161.1111. http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.161.1111/

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