Uniform bounds and ultraproducts of cycles

This paper is about the question whether a cycle in the l-adic cohomology of a smooth projective variety over the rational numbers, which is algebraic over almost all finite fields, is also algebraic over the rationals. We use ultraproducts respectively nonstandard techniques in the sense of A. Robinson, which the authors applied systematically to algebraic geometry. We give a reformulation of the question in form of uniform bounds for the complexity of algebraic cycles over finite fields.