Some results on surfaces with p_g=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 surface X with these invariants defined over Q that has Picard number 2, which is the smallest possible for these surfaces. This is done by giving equations for the double cover Y of X, calculating the zeta function of the reduction of Y 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 also 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 is 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.