Dihedral Galois representations and Katz modular forms

Abstract

We show that any two-dimensional odd dihedral representation $\rho$ over a finite field of characteristic $p>0$ of the absolute Galois group of the rational numbers can be obtained from a Katz modular form of level $N$, character $ϵ$ and weight $k$, where $N$ is the conductor, $ϵ$ is the prime-to-$p$ part of the determinant and $k$ is the so-called minimal weight of $\rho$. In particular, $k=1$ if and only if $\rho$ is unramified at $p$. Direct arguments are used in the exceptional cases, where general results on weight and level lowering are not available.