Serre’s conjecture over $\mathbb F_9$


In this paper we show that an odd Galois representation $\bar{\rho}:\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{GL}_2(\mathbb{F}_9)$ having nonsolvable image and satisfying certain local conditions at $3$ and $5$ is modular. Our main tools are ideas of Taylor [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 $\mathbb{Q}$, 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.


Jordan S. Ellenberg

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