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

Rolf-Peter, Holzapfel (1997)

Serdica Mathematical Journal

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....

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