Algebraic Curves Abelian Threefolds Period Matrices Moduli Spaces Shimura Surface Siegel Domain Complex Unit Ball Uniformization Braid Group Monodromy Group Modular Group Gundamental Groups Picard-Fuchsian Groups Symplectic Group Aritmetic Group Representation Quadratic Number Field
Issue Date:
1997
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 23, No 2, (1997), 143p-164p
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.
