An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

Lecture Details

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 httpwww.nptel.iitm.ac.insyllabus111106044Goals 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 freeKeywords 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