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 extend the theory of topological coverings to that of holomorphic (complex analytic) coverings

* To show that any Riemann surface structure on the base space of a topological covering induces a Riemann surface structure on the covering space in such a way that the covering projection map is holomorpic. To achieve this using the technique of “pulling back charts from below”

* To see why the Riemann surface structure induced above is essentially unique

* In particular, we get a unique Riemann surface structure on the topological covering of a Riemann surface. The deck transformations therefore become holomorphic automorphisms of this Riemann surface structure

Keywords: Topological covering, holomorphic covering, admissible neighborhood, chart, pulling back charts by local homeomorphisms, locally biholomorphic, pulling back Riemann surface structures, holomorphicity or complex analyticity of continuous liftings, deck transformation