The Ekedahl–Oort type of Jacobians of Hermitian curves

The Ekedahl–Oort type is a combinatorial invariant of a principally polarized abelian variety $A$ defined over an algebraically closed field of characteristic $p \gt 0$. It characterizes the $p$-torsion group scheme of $A$ up to isomorphism. Equivalently, it characterizes (the $\mathrm{mod} \: p$ reduction of) the Dieudonné module of $A$ or the de Rham cohomology of $A$ as modules under the Frobenius and Vershiebung operators.

There are very few results about which Ekedahl–Oort types occur for Jacobians of curves. In this paper, we consider the class of Hermitian curves, indexed by a prime power $q = p^n$, which are supersingular curves well-known for their exceptional arithmetic properties. We determine the Ekedahl–Oort types of the Jacobians of all Hermitian curves. An interesting feature is that their indecomposable factors are determined by the orbits of the multiplication-by-two map on $\mathbb{Z} / (2^n + 1)$, and thus do not depend on $p$. This yields applications about the decomposition of the Jacobians of Hermitian curves up to isomorphism.

Keywords

Hermitian curve, maximal curve, Jacobian, supersingular, Dieudonné module, $p$-torsion, de Rham cohomology, Ekedahl-Oort type, $a$-number, Selmer group

2010 Mathematics Subject Classification

11G20, 14G50, 14H40

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Published 20 November 2015