A HIGHER-DIMENSIONAL GENERALIZATION OF THE GROSS ZETA-FUNCTION

Let X be a smooth projective variety over the finite field F(q). Let X superset-of X(n-1) superset-of X(n-2) superset-of ... superset-of X0 be a complete flag of smooth irreducible subvarieties, dim X(i) = i, such that X(i-1) is ample in X(i). In this set-up we are able to use an idea of Parshin to ''localize at the flag'' and obtain the completion k(infinity) = F(q)((t1))...((t(n)), as well as the exponent space S = (kBAR*(infinity)n x Z(p). We then define a zeta function and prove it to be a family of entire n-dimensional power series. Finally, we study the values of this zeta function at negative integers and show that they are integral (i.e., in the affine ring A of X - X(n-1) and have good congruences at the closed points of A. (C) 1995 Academic Press, Inc.