On surfaces of general type with p_g=q=1 isogenous to a product of curves

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C x F)/G. In this article, we classify the surfaces of general type with p(g) = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces m(I), m(II), m(IV) are irreducible, whereas m(III) is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples.