The de Rham cohomology of the Suzuki curves

Abstract

For a natural number \(m\), let \(\mathcal{S}_m/\mathbb{F}_2\) be the \(m\)th Suzuki curve. We study the \(2\)-torsion group scheme and the Dieudonn\'{e} module of \(\mathcal{S}_m\). This is accomplished by studying the de Rham cohomology group of \(\mathcal{S}_m\). For all \(m\), we determine the structure of the de Rham cohomology as a \(2\)-modular representation of the \(m\)th Suzuki group and the structure of a submodule of the Dieudonn\'{e} module. We determine an explicit basis for the de Rham cohomology. For \(m=1\) and \(2\), we determine the Dieudonn\'{e} module completely.