Ekedahl-Oort Types of Double Covers in Characteristic Two

Abstract: Let X be a smooth projective curve over a perfect field of characteristic two and Y be a ramified double cover of X. If X is ordinary, we compute the Ekedahl-Oort type of Y in terms of the ramification of the cover generalizing results of Elkin and Pries and of Voloch. If X is not ordinary, we bound the Ekedahl-Oort type in terms of the ramification of the cover. We do so by analyzing the de Rham cohomology of the cover with its Frobenius and Verschiebung and establishing a local-to-global principle for the effects of each point of ramification. This is joint work with Joe Kramer-Miller and Steven Groen.