Vol. 107, No. 1, 1983

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On the uniformization of certain curves

Peter Frederick Stiller

Vol. 107 (1983), No. 1, 229–244

The uniformization theorem of Poincaré and Koebe tells us that every smooth connected algebraic curve X over the complex numbers (or any Riemann surface) has as its universal covering space either the complex projective line PC1, the complex numbers C, or the complex upper half plane H = {z C s.t. Imz > 0}. When the universal covering space is the upper half plane H, we can regard the fundamental group π1(X) as a subgroup of SL2(R) acting as covering transformations via linear fractional transformation. We shall focus on the case π1(X) SL2(Z).

Mathematical Subject Classification 2000
Primary: 14H15
Secondary: 14D05, 30F10
Received: 29 January 1981
Revised: 11 January 1982
Published: 1 July 1983
Peter Frederick Stiller