The Number of Algebraic Cycles with Bounded Degree

Let X be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on X with bounded degree as it goes to the infinity. By this estimate, we can define a certain kind of zeta functions associated with groups of cycles. We also consider an analogue in Arakelov geometry. Introduction Let X be a projective scheme over a finite field Fq and H an ample line bundle on X. For a non-negative integer k, we denote by nk(X,H, l) the number of all effective l-dimensional cycles V on X with degH(V ) = k, where degH(V ) is the degree of V with respect to H given by degH(V ) = deg ( H ·l · V ) . One of the main results of this paper is to give an estimate of nk(X,H, l) as k goes to the infinity, namely, Theorem A (Geometric version). (1) If H is very ample, then there is a constant C depending only on l and dimFq H (X,H) such that logq nk(X,H, l) ≤ Ck for all k ≥ 0. (2) If l 6= dimX, then lim sup k→∞ logq nk(X,H, l) kl+1 > 0. As a consequence of the above theorem, we can define a certain kind of zeta functions of algebraic cycles as follows. We note that Weil’s zeta function ZX/Fq is given by ∞ ∑