| In 1976, F. Hirzebruch [9] showed that the smooth model Y 2 of the Hilbert modular surface of level 2 for the field Q ( 5 ) is isomorphic to the Klein surface of the icosahedron obtained from a blowing-up of P2 C . Later, T. Schmidt [16] showed that under this isomorphism, the Cohen-Wolfart embedding of the non-arithmetic Hecke group of signature (2, 5, infinity) has a lift in Y2 that corresponds to six specific exceptional divisors of the Klein surface.; In this dissertation, we consider the Klein A5 -Invariants of P2C as seen in the Klein surface. We show, using Hirzebruch's isomorphism, that their images in the Hilbert modular surface X of level one are curves uniformized by non-compact and non-arithmetic triangle groups contained in the Hilbert modular group. We also give a correspondence between the A5-orbits of P2C and the elliptic singularities of the surface X. |
