The de Rham cohomology of the Suzuki curves

Abstract

For a natural number m, let Sm/F 2 be the mth Suzuki curve. We study the mod 2 Dieudonné module of Sm, 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 Sm. For all m, we determine the structure of the de Rham cohomology as a 2-modular representation of the mth Suzuki group and the structure of a submodule of the mod 2 Dieudonné module. For m = 1 and 2, we determine the complete structure of the mod 2 Dieudonné module.Corollary 1.1. (Corollary 3.10) If 2 m ≡ 2 e mod 2 e+1 + 1, then the E-module E/E(V e+1 + F e+1 ) occurs as an E-submodule of the mod 2 Dieudonné module D m of S m . In particular,