Let C be an algebraic curve of genus two over a finite field F, (p > 3, q = pf), and T = ZZ + n’ be a divisor on C x C composed of the graph ZZ of the qth power Frobenius morphism of C and its transpose n’. The purpose of this paper is to give some sufficient conditions for the liftability of (C, T) to characteristic zero. Examples of such liftings have been known for the Hecke correspondences of Shimura curves associated to quaternion algebras over totally real number fields. On the other hand, Y. Ihara looks at them from a new standpoint. In [4, 71, he developed some basic theory for the liftability of (C, 7’) (in arbitrary genus, including p = 2). He proved among others that the obstruction to the existence of infinitesimal liftings of (C, r) can be regarded as a linear form over the set of regular differentials of degree q + 1 on C with additional conditions, and he obtained many examples of C, such that (C, T) lifts to characteristic zero. In this paper, we shall apply his method to curves of genus two. Our main result (Theorem 1) claims that, ifthe action of thefth power y’ of the Cat-tier operator on the dlflerentials of the first kind on C is not identically 0, then there exists a lifting of (C, T) to the ring of Witt vectors over F,. When genus two and p > 3, we can regard the obstruction as an element of the dual space of the fixed one-dimensional space