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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.

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11G18 Arithmetic aspects of modular and Shimura varieties
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