An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

IIT Madras Course , Prof. T.E. Venkata Balaji

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Lecture 23: Characterizing Riemann Surface Structures on Quotients of the Upper Half

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        An Introduction to Riemann Surfaces and Algebraic Curves Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit httpwww.nptel.iitm.ac.insyllabus111106044Goals To show that any Riemann Surface with nonzero abelian fundamental group and universal covering the upper half-plane has fundamental group isomorphic to the additive group of integers i.e., that it is cyclic of infinite order To classify the Riemann surface structures naturally inherited by annuli in the complex plane, and to show that there is a family of such distinct (i.e., non-isomorphic) structures parametrized by a real parameter To deduce that if a Riemann surface has fundamental group isomorphic to the product of the additive group of integers with itself, then it has to be isomorphic to a complex torus, and hence in particular that it has to necessarily be compactKeywords Upper half-plane, unit disc, annulus, torus, simply connected, abelian fundamental group, additive group, translation, deck transformation, Moebius transformation, universal covering, holomorphic automorphism, parabolic, elliptic, hyperbolic, loxodromic, fixed point, commuting Moebius transformations, conjugation, translation, universal covering, discrete subgroup, discrete submodule, generator of a group

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