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 \epsilon and weight k , where N is the conductor, \epsilon 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.