The Ekedahl–Oort type of Jacobians of Hermitian curves

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 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 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.

Superspecial rank of supersingular abelian varieties and Jacobians

“…In [27], for all q = p n , the authors determine the Dieudonné module D * (X q ) = D * (Jac(X q )[p]), complementing earlier work in [3,4]. In particular, [27,Theorem 5.13] states that the distinct indecomposable factors of Dieudonné module D * (X q ) are in bijection with orbits of Z/(2 n + 1) − {0} under ×2.…”

“…In [27], for all q = p n , the authors determine the Dieudonné module D * (X q ) = D * (Jac(X q )[p]), complementing earlier work in [3,4]. In particular, [27,Theorem 5.13] states that the distinct indecomposable factors of Dieudonné module D * (X q ) are in bijection with orbits of Z/(2 n + 1) − {0} under ×2. Each factor's structure is determined by the combinatorics of the orbit, which depends only on n and not on p. The multiplicities of the factors do depend on p. For example, when n = 2, the Dieudonné module of X p 2 is M g/2 2,2 , which has superspecial rank 0 (Lemma 3.9).…”

The a-numbers of Fermat and Hurwitz curves

“…Since the Hermitian curve is F p 2(r+1) -maximal, C r+1 = 0 by Theorem 2.1. It should be noticed that the ranks of C n for n ≤ r were determined in [16].…”

The de Rham cohomology of the Suzuki curves

“…Notation. We begin by establishing some notation regarding p-torsion group schemes, mod p Dieudonné modules, and Ekedahl-Oort types, taken directly from [22,Section 2].…”