On the canonical map of some surfaces isogenous to a product

We construct several families (indeed, connected components of the moduli space) of surfaces S of general type with p(g) = 5, 6 whose canonical map has image Sigma of very high degree, d = 48 for p(g) = 5, d = 56 for p(g) = 6. And a connected component of the moduli space consisting of surfaces S with K-S(2) = 40, p(g) = 4, q = 0 whose canonical map has always degree >= 2, and, for the general surface, of degree 2 onto a canonical surface Y with K-Y(2), = 12, p(g) = 4, q = 0. The surfaces we consider are SIP's, i.e. surfaces S isogenous to a product of curves (C-1 x C-2)/G; in our examples the group G is elementary abelian, G congruent to (Z/m)(k). We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory. Our methods and results are a first step towards answering the question of existence of SIP 's S with p(g) = 6, q = 0 whose canonical map embeds S as a surface of degree 56 in P-5.