Ekedahl-Oort types of $\Z/2\Z$-covers in characteristic $2$
Jeremy Booher, Steven R. Groen, Joe Kramer-Miller
TL;DR
This work analyzes the Ekedahl-Oort types of double covers $\pi:Y\to X$ with Galois group $\mathbf{Z}/2\mathbf{Z}$ in characteristic $2$, proving that when the base $X$ is ordinary the EO type of $Y$ is determined by the ramification together with $g_X$, while for non-ordinary $X$ it yields sharp bounds. A central development is the theory of enhanced differentials of the second kind, which provide a concrete, pole-bounded model for $\mathrm{H}_{\mathrm{dR}}^1$ as a Dieudonné module and enable precise control of $F$ and $V$-actions. In the ordinary-base case, the paper gives an explicit decomposition of $\mathrm{H}_{\mathrm{dR}}^1(Y)$ into ordinary and ramification-contributed blocks, leading to an exact EO-type description and the construction of families of curves with fixed EO-type. For non-ordinary bases, it develops bounding techniques for the final type via a careful analysis of $V$-stable subspaces and the symplectic pairing, including logarithmic and Tango-advantaged refinements, and applies these to examples such as supersingular elliptic curves. Overall, the results quantify how ramification and base EO-type control the EO-type of covers in characteristic two and provide tools for exploring EO-type stratifications in families of such covers.
Abstract
In this article we study the Ekedahl-Oort types of $\Z/2\Z$-Galois covers $π:Y \to X$ in characteristic two. When the base curve $X$ is ordinary, we show that the Ekedahl-Oort type of $Y$ is completely determined by the genus of $X$ and the ramification of $π$. For a general base curve $X$, we prove bounds on the Ekedahl-Oort depending on the Ekedahl-Oort type of $X$ and the ramification of $π$. Along the way, we develop a theory of \emph{enhanced differentials of the second kind}. This theory allows us to study algebraic de Rham cohomology in any characteristic by working directly with differentials, in contrast to the standard Čech resolution.