The de Rham cohomology of the Suzuki curves

TL;DR

This paper investigates the de Rham cohomology of Suzuki curves over finite fields to understand their $2$-torsion group schemes, providing explicit structures for small cases and a general description for all $m$.

Contribution

It determines the de Rham cohomology structure of Suzuki curves as a $2$-modular representation and describes the mod $2$ Dieudonné module for all $m$, with complete results for $m=1,2$.

Findings

Structure of de Rham cohomology as a $2$-modular representation.

Description of a submodule of the mod $2$ Dieudonné module.

Complete Dieudonné module structure for $m=1,2$.

Abstract

For a natural number $m$, let $\mathcal{S}_m/\mathbb{F}_2$ be the $m$th Suzuki curve. We study the mod $2$ Dieudonn\'{e} module of $\mathcal{S}_m$, which gives the equivalent information as the Ekedahl-Oort type or the structure of the $2$-torsion group scheme of its Jacobian. We accomplish this by studying the de Rham cohomology 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 mod $2$ Dieudonn\'{e} module. For $m=1$ and $2$, we determine the complete structure of the mod $2$ Dieudonn\'{e} module.