Vol. 72, No. 1 DUKE MATHEMATICAL JOURNAL (C) October 1993 TRANSCENDENTAL CYCLES ON ORDINARY K3 SURFACES OVER FINITE FIELDS YURI G. ZARHIN 1. Introduction. Let Z be a complex algebraic K3 surface, and V(Z) the -lattice of transcendental cycles on Z. By definition, V(Z) is the orthogonal complement of NS(Z) (R) of the second rational cohomology group H2(Z, ) with respect to the intersection pairing. Here NS(Z) is the Neron-Severi group of Z. It is well known that V(Z) carries a natural rational Hodge structure of weight 2. In [28] we have proven that this structure is irreducible and its endomorphism algebra is a number field. Now let Y be an ordinary K3 surface over a finite field k of characteristic p. We write Y for Y x k(a) where k(a) is an algebraic closure of k. For each rational prime different from p, let us consider the second twisted /-adic cohomology group n2(Ya, t)(1) of Y. The Galois group G(k) of k acts on n2(Ya, t)(1) in a natural way. One may identify NS(Y)t NS(Ya)(R) t with a certain Galois-invariant subspace of H2(y, )(1), and a theorem of Nygaard [12] asserts that this subspace coincides with G(k)-invariants H2(y, )(1) tk if k is "sufficiently large". (This theorem proves a special case of a general conjecture due to Tate 1-19].) We define the t-lattice V(Y) as the orthogonal complement of NS(Ya) in H2(y, )(1) with respect to the intersection pairing. Recall that this pairing and its restriction to NS(Ya) are nondegenerate. This gives us a canonical splitting H2(Ya, Qt)(1)= NS(Y)t ) Vt(Y). Since the intersection pairing is Galois-invariant, Vt(Y) is a Galois-invariant sub- space and the splitting above is also Galois-invariant. Recall that G(k) is procyclic and has a canonical generator, namely, the arithmetic Frobenius automorphism trk: k(a) k(a), x --. x where q is the number of elements of k. Clearly, q is an integral power of p. Another canonical generator of G(k) is the geometric Frobenius automorphism tPk tr -1. In this paper we examine the characteristic polynomial P,,tr(t) .’= det(id trPk, V(Y)). Received 28 December 1992. Author supported by the Netherlands Organization for Scientific Research (N.W.O.). 65