A Higher-Dimensional Generalization of the Gross Zeta Function Original Research Article

Let X be a smooth projective variety over the finite field Fq. Let X superset of Xn − 1 superset of Xn − 2 superset of · · · superset of X0 be a complete flag of smooth irreducible subvarieties, dim Xi = i, such that Xi − 1 is ample in Xi. In this set-up we are able to use an idea of Parshin to "localize at the flag" and obtain the completion k∞ = Fq((t1)) · · · ((tn)), as well as the exponent space S = (image*∞)n × Zp. We then define a zeta function and prove it to be a family of entiren-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 − Xn − 1) and have good congruences at the closed points of A.