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 http://www.nptel.iitm.ac.in/syllabus/111106044/

Goals: * 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 compact

Keywords: 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