Serre’s conjecture over F9

Abstract

In this paper we show that an odd Galois representation ˉρ:Gal(ˉQ/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.

Authors

Jordan S. Ellenberg

Department of Mathematics, University of Wisconsin, Madison, WI 53706, United States