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Let X be a smooth projective variety over a field. Let ~f~ denote the Zariski sheaf associated to the presheaf U ~-, Kj(F(U, Cv)) of Quillen K-groups. The collection of Zariski cohomology groups H~(X, ~Fj) will be referred to as the K-cohomology of X. The groups contain a great deal of information about the geometry of X. For example, Bloch's formula [11] says that Hi(X, ~ ) is isomorphic to the group Ctt~(X) of codimension i cycles on X modulo rational equivalence. In this paper, the K-cohomology of a rational surface over a finite field will be calculated up to p-torsion, where p is the characteristic. Recall that a surface X over a field F is called rational if Z = X | is birational with I'3 over the algebraic closure F. Motifs play a central role in the computation. The idea that motifs could be used effectively to study the K-theory of varieties over finite fields is due to Soulr, whose paper [15] is a fertile source of inspiration. He introduced the K-cohomology of motifs and proved several important finiteness results. In particular, he showed for a wide class of varieties, including rational surfaces, that Ki(X) is torsion for j > 0. A second source of ideas is the fundamental work of Merkurjev and Suslin [I 8], especially as applied in [4] to codimension two cycles and in [17] to the K2-cohomology of rational surfaces. The paper is organized as follows. The first section contains a brief review of the theory of motifs. From the motivic point of view, all the relevant data about a rational surface is contained in the action of the Galois group on Pic(X). This action is a continuous, finite dimensional, integral Galois representation. The second section contains a computation of the prime-to-p K-theory of such representations when F is finite. In the final section, the orders of the K-cohomology groups are related to the values at integer points of the L-functions associated to X. Let # F = q = ig and let l be a prime different from p. Let 4j denote the endomorphism of the &ale C0homology group HJ(X ", r which is induced by the q-power Frobenius. Define .P~(t) = det(1 ~:) and Lj(s) = Pj(q~). Also, if a, b ~ ~, write a ~ b to mean that a/b is a power of p.

Motifs,L-functions, and theK-cohomology of rational surfaces over finite fields