Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Abstract
We define Picard cycles on each smooth three-sheeted Galois
cover C of the Riemann sphere. The moduli space of all these algebraic
curves is a nice Shimura surface, namely a symmetric quotient of the projective
plane uniformized by the complex two-dimensional unit ball. We show that
all Picard cycles on C form a simple orbit of the Picard modular group
of Eisenstein numbers. The proof uses a special surface classification in
connection with the uniformization of a classical Picard-Fuchs system. It
yields an explicit symplectic representation of the braid groups (coloured or
not) of four strings
