Punctured Complex Tori are Elliptic Algebraic Affine Plane 
Punctured Complex Tori are Elliptic Algebraic Affine Plane
by IIT Madras / T.E. Venkata Balaji
Video Lecture 45 of 48
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Date Added: January 12, 2015

Lecture Description

Goals of the Lecture: - 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 satisfy Topics: 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

Course Index

  1. The Idea of a Riemann Surface
  2. Simple Examples of Riemann Surfaces
  3. Maximal Atlases and Holomorphic Maps of Riemann Surfaces
  4. Riemann Surface Structure on a Cylinder
  5. Riemann Surface Structure on a Torus
  6. Riemann Surface Structures on Cylinders and Tori via Covering Spaces
  7. Möbius 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. Universality of the Universal Covering
  16. The Fundamental Group of the base as the Deck Transformation Group
  17. The Riemann Surface Structure on the Topological Covering of a Riemann Surface
  18. Riemann Surfaces with Universal Covering the Plane or the Sphere
  19. Classifying Complex Cylinders Riemann Surfaces
  20. Möbius Transformations with a Single Fixed Point
  21. Möbius Transformations with Two Fixed Points
  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 by the Weierstrass Phe-function
  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
  38. The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
  39. Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
  40. The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
  41. 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
  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

Course Description

The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmueller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra---especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory. The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given.

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