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 To associate to each complex 1-dimensional torus a complex number, called the j-invariant of the complex torus, which depends only on the holomorphic isomorphism class of the torus. This j-invariant will be shown in the forthcoming lectures to completely classify all complex tori In the previous unit of lectures, we constructed a weight two modular form on the upper half-plane and studied its mapping properties. In this lecture we use this weight two modular form to define a full modular form, i.e., a holomorphic function on the upper half-plane that is invariant under the action of the full unimodular group. It is this modular form that goes down to give the j-invariant function on the Riemann surface of holomorphic isomorphism classes of complex tori with underlying set consisting of the orbits of the unimodular group in the upper half-planeKeywords Upper half-plane, quotient by the unimodular group, orbits of the unimodular group, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, 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, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, congruence-mod-2 normal subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, meromorphic functions are holomorphic functions to the Riemann Sphere, j-invariant of a complex torus (or) j-invariant of an algebraic elliptic curve

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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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