Superspecial rank of supersingular abelian varieties and Jacobians

Abstract

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines information about the decomposition of a supersingular abelian variety up to isomorphism; namely it is a bound for the maximal number of supersingular elliptic curves appearing in such a decomposition.

Proportion of ordinarity in some families of curves over finite fields

“…For the purpose of this paper, we will only be interested in G = Jac(C) [p]. In this case, it is well known that 0 ≤ r(G) ≤ g(C) and 0 ≤ a(G) + r(G) ≤ g. The Jacobian is called ordinary if r(G) = g or equivalently, when a(G) = 0 [1]. By abuse of notation, we will denote the a(C) and r(C) to be the corresponding invariants of Jac(C) [p].…”