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Serre’s conjecture over $$\mathbb F_9$$. (English) Zbl 1153.11312
Summary: In this paper we show that an odd Galois representation $$\bar{\rho}: \text{Gal }\mathbb Q \to \text{GL}_2(\mathbb F_9)$$ having nonsolvable image and satisfying certain local conditions at $$3$$ and $$5$$ is modular. Our main tools are ideas of Taylor and Khare, 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.

##### MSC:
 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms 11G18 Arithmetic aspects of modular and Shimura varieties
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