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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.
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
@article{10_4007_annals_2005_161_1111,
author = {Jordan S. Ellenberg},
title = {Serre{\textquoteright}s conjecture over $\mathbb F_9$},
journal = {Annals of mathematics},
pages = {1111--1142},
year = {2005},
volume = {161},
number = {3},
doi = {10.4007/annals.2005.161.1111},
mrnumber = {2180399},
zbl = {1153.11312},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.161.1111/}
}
TY - JOUR AU - Jordan S. Ellenberg TI - Serre’s conjecture over $\mathbb F_9$ JO - Annals of mathematics PY - 2005 SP - 1111 EP - 1142 VL - 161 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2005.161.1111/ DO - 10.4007/annals.2005.161.1111 LA - en ID - 10_4007_annals_2005_161_1111 ER -
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