A birational model of the Cartwright-Steger surface

A Cartwright-Steger surface is a complex ball quotient by a certain arithmetic cocompact group associated to the cyclotomic field $Q(e^{2\pi i/12})$, its numerical invariants are with $c_1^2 = 3c_2 = 9, p_g = q = 1$. It is a cyclic degree 3 cover of a simply connected surface of general type with $c_1^2 = 2, p_g = 1$. A similar construction in the case of the cyclotomic fields $Q(e^{2\pi i/5})$ (resp. $Q(e^{2\pi i/7})$) leads to the beautiful geometry of a del Pezzo surface of degree 5 and its K3 double cover branched along the union of lines (resp. the Naruki K3 surface and its finite group cover by the elliptic modular surface of level 7). In my talk I will discuss a possible explicit birational model of the Cartwright-Steger surface and its degree 3 quotient.