Space Filling Curves over Finite Fields

In this note, we construct curves over finite fields which have, in a certain sense, a “lot” of points, and give some applications to the zeta functions of curves and abelian varieties over finite fields. In fact, we found the basic construction, given in Lemma 1, of curves in A which go through every rational point, as part of an unsuccessful attempt to find curves of growing genus over a fixed finite field with lots of points in the sense of the Drinfield-Vladut bound [2]. The idea of applying that construction along the lines of this note grew out of an August 1996 conversation with Ofer Gabber about whether every abelian variety over a finite field was a quotient of a Jacobian, during which he constructed, on the fly, a proof of that fact. A variant of his proof appears here in Theorem 11. It is a pleasure to acknowledge my debt to him.