Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

Rolf-Peter, Holzapfel. "Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere." Serdica Mathematical Journal 23.2 (1997): 143-164. <http://eudml.org/doc/11609>.

@article{Rolf1997,
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.},
author = {Rolf-Peter, Holzapfel},
journal = {Serdica Mathematical Journal},
keywords = {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; Shimura surface; Siegel domain; complex unit ball; braid group; modular group; Picard-Fuchsian groups; quadratic number field},
language = {eng},
number = {2},
pages = {143-164},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere},
url = {http://eudml.org/doc/11609},
volume = {23},
year = {1997},
}

TY - JOUR
AU - Rolf-Peter, Holzapfel
TI - Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere
JO - Serdica Mathematical Journal
PY - 1997
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 23
IS - 2
SP - 143
EP - 164
AB - 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.
LA - eng
KW - 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; Shimura surface; Siegel domain; complex unit ball; braid group; modular group; Picard-Fuchsian groups; quadratic number field
UR - http://eudml.org/doc/11609
ER -