Some Results on Surfaces with pg = q=1 and K2=2

Following an idea of Ishida, we develop polynomial equations for certain unramified double covers of surfaces with p(g) = q = 1 and K-2 = 2. Our first main result provides an explicit surface X with these invariants defined over Q that has Picard number rho(X) = 2, which is the smallest possible for these surfaces. This is done by giving equations for the double cover (X) over tilde of X, calculating the zeta function of the reduction of (X) over tilde to F-3, and extracting from this the zeta function of the reduction of X to F-3; the basic idea used in this process may be of independent interest. Our second main result is a big monodromy theorem for a family that contains all surfaces with p(g) = q = 1, K-2 = 2, and K ample. It follows from this that a certain Hodge correspondence of Kuga and Satake, between such a surface and an abelian variety, is motivated (and hence absolute Hodge). This allows us to deduce our third main result, which is that the Tate Conjecture in characteristic zero holds for all surfaces with p(g) = q = 1, K-2 = 2, and K ample.