The multiplication-by-two map on the integers modulo 2n + 1 gives rise to intriguing combinatorial structures. For example, the number of orbits of maximal length 2n equals the number of self-reciprocal polynomials of degree 2n over GF(2). The orbits of this map play a surprising role in recent work of myself and Colin Weir about the Jacobians of Hermitian curves.
For a prime power q = pn, the Hermitian curve Xq : yq + y = xq+1 has special arithmetic significance. It is maximal over GF(q2) which implies that its Jacobian is supersingular, namely the Jacobian is isogenous to a product of supersingular elliptic curves. Moreover, Xq is the Deligne-Lusztig variety of dimension 1 associated with the group PGU(3,q). In this work, we find new results about the decomposition of the Jacobians of the Hermitian curves up to isomorphism by determining their Ekedahl-Oort types (equivalently, their Dieudonne modules or the module structure of the de Rham cohomology). There are very few curves for which this structure has been analyzed. We find that the indecomposable factors of the de Rham cohomology have a surprising combinatorial structure; while their multiplicities depend on p, the structure of each indecomposable factor does not.