The Ekedahl-Oort type of Jacobians of Hermitian curves

TL;DR

This paper determines the Ekedahl-Oort types of Jacobians of Hermitian curves, revealing their structure and decomposition properties, and showing these types depend on the orbits of a specific map on a finite cyclic group.

Contribution

It provides the first explicit classification of Ekedahl-Oort types for Jacobians of Hermitian curves, linking their structure to combinatorial orbits and independence from the characteristic.

Findings

Ekedahl-Oort types are explicitly determined for all Hermitian curve Jacobians.

Decomposition factors are characterized by orbits of multiplication-by-two on a finite cyclic group.

Results show types do not depend on the characteristic p.

Abstract

The Ekedahl-Oort type is a combinatorial invariant of a principally polarized abelian variety $A$ defined over an algebraically closed field of characteristic $p > 0$. It characterizes the $p$-torsion group scheme of $A$ up to isomorphism. Equivalently, it characterizes (the mod $p$ reduction of) the Dieudonn\'e 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.