2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61876In this paper, we show that an odd Galois representation rhobar: Gal(Qbar/Q) --> GL_2(F_9) 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.14 pagesNumber TheoryAlgebraic Geometry11F80 (Primary) 11G18 14K15 (Secondary)text