Serre's conjecture over F₉

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In this paper, we show that an odd Galois representation rhobar: Gal(Qbar/Q) --> GL₂(F₉) satisfying certain local conditions at 3 and 5 is modular. Our main tool is an idea of Taylor, which reduces the problem to that of exhibiting points on a Hilbert modular surface which are defined over a solvable extension of Q, and which satisfy certain reduction properties. As a corollary, we show that Hilbert-Blumenthal abelian surfaces over Q with good ordinary reduction at 3 and 5 are modular.