On the canonical map of some surfaces isogenous to a product

We give new contributions to the existence problem of canonical surfaces of high degree. 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 $\mathrm{\Sigma }$ 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 $\geq 2$,and,forthegeneralsurface,ofdegree$2$ontoacanonicalsurface$Y$with$K^2_Y = 12, p_g=4, q=0$. The surfaces we consider are SIP 's, i.e. surfaces $S$ isogenous to a product of curves $(C_1\times C_2)\mathrm{/}G$; in our examples the group $G$ is elementary abelian, $G=(\mathbb{Z}\mathrm{/}m{)}^k$. We also establish some basic results concerning the canonical maps of any surface isogenous to a product, basing on elementary representation theory.