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 see how the topological quotient of the universal covering of a space by the deck transformation group (which is isomorphic to the fundamental group of the space) gives back the space

* In particular, the topological universal covering of a Riemann surface (which inherits a unique Riemann surface structure as shown in the previous lecture) modulo (or quotiented by or divided by) the fundamental group gives back the Riemann surface

* To see that nontrivial deck transformations are fixed-point free

* To see why any Riemann surface with universal covering the Riemann sphere is isomorphic to the Riemann sphere itself

* To get a characterisation of discrete subgroups of the additive group of complex numbers

* To use the above characterisation to deduce that a Riemann surface with universal covering the plane has to be isomorphic to either the plane itself, or to a complex cylinder, or to a complex torus

Keywords: Holomorphic covering, holomorphic universal covering, group action on a topological space, orbit of a group action, equivalence relation defined by a group action, quotient by a group, topological quotient, quotient topology, quotient map, transitive action, deck transformation, open map, Riemann sphere, one-point compactification, stereographic projection, Moebius transformation, unique lifting property, group of translations, admissible neighborhood, module, submodule, subgroup, discrete submodule, discrete subgroup