Ball and Surface Arithmetics (Aspects of Mathematics)

The main purpose of this book is to introduce into surface theory new kinds of rational discrete invariants (hights) explicitly and to demonstrate their power for solving actual problems. These hights are postulated to be invariant with respect to finite coverings up to a well-documented factor. As a first immediate application one gets, explicitly and quite generally, formulae of Hurwitz type for the classical basic invariants of surfaces. These Hurwitz formulae are explicitly known in general only for special coverings. Another field of application is the theory of Picard modular surfaces. The book also presents the new results of these.