A SURFACE OVER Q WITH p g = q = 1 , K 2 = 2 AND MINIMAL PICARD NUMBER

Following an idea of Ishida, we develop polynomial equations for certain unramified double covers of surfaces with pg = q = 1 and K 2 = 2. Our first main result is to show the existence of a surface X with these invariants defined over Q that has the smallest possible Picard number ρ(X) = 2. This is done by giving equations for the double cover X̃ of X, calculating the zeta function of the reduction of X̃ to F3, and inferring from this the zeta function of the reduction of X to F3; the basic idea used in this process may be of independent interest. Our second main result (which uses the first result in part) is a big monodromy theorem for the family of all surfaces with pg = q = 1 and K 2 = 2 such that K is ample, and from this we deduce the Tate Conjecture in characteristic zero for the members of this family.