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 analyze what the conditions of loxodromicity, ellipticity or hyperbolicity imply for an automorphism of the upper half-plane, i.e., to characterize the automorphisms of the upper half-plane. This is required for the classification of Riemann surfaces with universal covering the upper half-plane

* To show that the fundamental group of a Riemann surface is torsion free i.e., that it has no non-identity elements of finite order

* To show that the Deck transformations of the universal covering of a Riemann surface have to be either hyperbolic or parabolic in nature

* To deduce that the fundamental group of a Riemann surface is torsion free

Keywords: Moebius transformation, special linear group, projective special linear group, parabolic, elliptic, hyperbolic, loxodromic, fixed point, conjugation, translation, Riemann sphere, extended complex plane, upper half-plane, square of the trace (or trace square) of a Moebius transformation, torsion-free group, element of finite order of a group, torsion element of a group, universal covering, fundamental group, Deck transformations