IIT Madras Course , Prof. T.E. Venkata Balaji

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IIT Madras Course , Prof. T.E. Venkata Balaji

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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 of Lecture 43 In this and the forthcoming lectures, our aim is to show that complex tori are algebraic, i.e., that they are actually elliptic algebraic projective curves. This is the reason that complex tori exhibit a rich geometry which involves a beautiful interplay between their complex analytic properties and the algebraic geometric and number theoretic properties of the elliptic curves they are associated to. It is a deep and nontrivial theorem that any compact Riemann surface is algebraic, so such Riemann surfaces exhibit a rich geometry as in the case of complex tori Towards the above end, in this lecture we begin by identifying any punctured complex torus with a plane curve in complex 2-space. This plane curve is called the associated elliptic algebraic affine plane cubic curve. For this identification we make use of the Weierstrass phe-function associated to the complex torus, its derivative, their properties and the first order degree two cubic ordinary differential equation that they satisfyKeywords for Lecture 43 Upper half-plane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, triple pole (or) pole of order three, cubic equation, elliptic algebraic cubic curve, zeros of a polynomial equation, bicontinuous map (or) homeomorphism, open map, order of an elliptic function, elliptic integral, Argument principle, even function, odd function, analytic branch of the square root, simply connected, punctured torus, elliptic algebraic affine cubic plane curve, projective plane cubic curve, complex affine two-space, complex projective two-space, one-point compactification by adding a point at infinity

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- 1.The Idea of a Riemann Surface
- 2.Simple Examples of Riemann Surfaces
- 3.Maximal Atlases and Holomorphic Maps of Riemann Surfaces
- 4.A Riemann Surface Structure on a Cylinder
- 5.A Riemann Surface Structure on a Torus I
- 6.Riemann Surface Structures on Cylinders and Tori via Covering Spaces II
- 7.Moebius Transformations Make up Fundamental Groups of Riemann Surfaces
- 8.Homotopy and the First Fundamental Group
- 9.A First Classification of Riemann Surfaces
- 10.The Importance of the Path-lifting Property
- 11.Fundamental groups as Fibres of the Universal covering Space
- 12.The Monodromy Action
- 13.The Universal covering as a Hausdorff Topological Space
- 14.The Construction of the Universal Covering Map
- 15.Completion of the Construction of the Universal Coveringl
- 16.Completion of the Construction of the Universal Covering The Fundamental Group I
- 17.The Riemann Surface Structure on the Topological Covering of a Riemann Surface
- 18.Riemann Surfaces with Universal Covering the Plane or the Sphere I
- 19.Classifying Complex Cylinders Riemann Surfaces
- 20.Characterizing Moebius Transformations with a Single Fixed Point
- 21.Characterizing Moebius Transformations with Two Fixed Points I
- 22.Torsion-freeness of the Fundamental Group of a Riemann Surface
- 23.Characterizing Riemann Surface Structures on Quotients of the Upper Half
- 24.Classifying Annuli up to Holomorphic Isomorphism
- 25.Orbits of the Integral Unimodular Group in the Upper Half-Plane
- 26.Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
- 27.Local Actions at the Region of Discontinuity of a Kleinian Subgroup
- 28.Quotients by Kleinian Subgroups give rise to Riemann Surfaces
- 29.The Unimodular Group is Kleinian
- 30.The Necessity of Elliptic Functions for the Classification of Complex Tori
- 31.The Uniqueness Property of the Weierstrass Phe-function
- 32.The First Order Degree Two Cubic Ordinary Differential Equation satisfied
- 33.The Values of the Weierstrass Phe function at the Zeros of its Derivative
- 34.The Construction of a Modular Form of Weight Two on the Upper Half-Plane
- 35.The Fundamental Functional Equations satisfied by the Modular Form of Weight
- 36.The Weight Two Modular Form assumes Real Values on the Imaginary Axis
- 37.The Weight Two Modular Form Vanishes at Infinity I
- 38.The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity III
- 39.A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal
- 40.The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
- 41.A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
- 42.The Fundamental Region in the Upper Half-Plane for the Unimodular Group I
- 43.A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
- 44.Moduli of Elliptic Curves
- 45.Punctured Complex Tori are Elliptic Algebraic Affine Plane
- 46.The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
- 47.Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
- 48.Complex Tori are the same as Elliptic Algebraic Projective Curves I

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