The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstras

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 show that the Weierstrass phe-function associated to a lattice satisfies a first order degree two cubic ODE

* The ODE mentioned above is the key to studying the geometry of the complex torus associated to the lattice and eventually leads to the classification (moduli) theory of complex tori


Keywords: Upper half-plane, invariants for complex tori, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, isolated double pole, singular part of the Laurent expansion, analytic part of the Laurent expansion, antiderivative for the Weierstrass phe-function, Identity theorem for power series or Laurent series, differentiating term-by-term and integrating term-by-term under uniform convergence, even function, odd function, entire elliptic functions are constants, algebraic elliptic cubic curve