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 Moebius transformations with more than one fixed point in the extended complex plane

* To continue with the classification of Moebius transformations begun in the previous lecture by defining the notions of loxodromic, elliptic and hyperbolic Moebius transformations using the values of the square of the trace of the transformation

* To characterize geometrically the loxodromic, elliptic and hyperbolic Moebius transformations by showing that they can be conjugated by suitable Moebius transformations to multiplication by a complex number

* To show that the elliptic Moebius transformations are precisely those that are conjugate to a rotation about the origin

* To show that the hyperbolic Moebius transformations are precisely those that are conjugate to a real scaling

Keywords: Parabolic, elliptic, hyperbolic and loxodromic Moebius transformations, fixed point of a Moebius transformation, square of the trace of a Moebius transformation, translation, conjugation by a Moebius transformation, special linear group, projective special linear group