Dihedral Galois Representations and Katz Modular Forms

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.

2000 Mathematics Subject Classification: 11F11, 11F80, 14G35

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