Resumen de Serre's conjecture over \Bbb F_9

Jordan S. Ellenberg

• In this paper we show that an odd Galois representation ¿P¿Ï : Gal( ¿PQ/Q) ¿¿ GL2(F9) 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 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.