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 ask for a description of the set of holomorphic isomorphism classes of complex tori

* To state the Theorem on the Moduli of Elliptic Curves that not only answers the question above but also shows that the set above has a beautiful God-given geometry

* To see how the upper half-plane and the unimodular group (integral projective special linear group) enter into the discussion

* To use the theory of covering spaces to prove a part of the Theorem on the Moduli of Elliptic Curves, namely

that the set of holomorphic isomorphism classes of complex 1-dimensional tori is in a natural bijective correspondence with the set of orbits of the unimodular group in the upper half-plane

Keywords: Real torus, complex torus, Moebius transformation, translation, abelian group, holomorphic universal covering, admissible neighborhood, fundamental group, deck transformation group, biholomorphism class (or) holomorphic isomorphism class, locally biholomorphic map, upper half-plane, projective special linear group, unimodular group, orbits of a group action, action of a subgroup, underlying fixed geometric structure, superimposed (or) overlying (or) extra geometric structure, variation of extra structure for a fixed underlying structure (or) moduli problem, quotient by a group, equivalence relation induced by a group action, universal property of the universal covering, unique lifting property, moduli of elliptic curves, forming the fundamental group is functorial