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 Dieudonne 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 = pn, 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 Z/(2n + 1), and thus do not depend on p. This yields applications about the decomposition of the Jacobians of Hermitian curves up to isomorphism.
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