ON THE UNIFORMIZATION OF CERTAIN CURVES
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The uniformization theorem of Poincar 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 S = {z C s.t. Im z > 0}. When the universal covering space is the upper half plane S, we can regard the fundamental group 1(X) as a subgroup of SL2(R) acting as covering transformations via linear fractional transformation. We shall focus on the case 1(X) SL2(Z). 1983 by Pacific Journal of Mathematics.