The Necessity of Elliptic Functions for the Classification of Complex Tori

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: * In the last few lectures, we have shown that the quotient of the upper half-plane by the unimodular group has a natural Riemann surface structure. In order to show that this Riemann surface is isomorphic to the complex plane, we have to realize that we need to look for invariants for complex tori

* To motivate how the search for invariants for complex tori leads us to the study of doubly-periodic meromorphic functions (or) elliptic functions, the stereotype of which is given by the famous Weierstrass phe-function


Keywords: Upper half-plane, unimodular group, projective special linear group, set of orbits, quotient Riemann surface, lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, complex torus associated to a lattice, translation, j-invariant for a complex torus, Felix Klein, group-invariant function, bounded entire function, Liouville's theorem, singularity of an analytic function, poles, meromorphic function, doubly-periodic meromorphic function (or) elliptic function, Karl Weierstrass, algebraic curve, elliptic curve, Weierstrass phe-function, Residue theorem, double pole