Goals: - In Part A (Lec 47) of this lecture, we define complex projective 2-space and show how it can be turned into a two-dimensional complex manifold. In Part B (Lec 48), we show that any complex torus is holomorphically isomorphic to the natural Riemann surface structure on the associated elliptic algebraic cubic plane projective curve embedded in complex projective 2-space 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, Implicit function theorem, graph of a holomorphic function, nonsingular (or) smooth polynomial in two variables, zero locus of a polynomial, solving an implicit equation locally for an explicit function, nonsingular (or) smooth point of the zero locus of a polynomial in two variables, Hausdorff, second countable, connected component, nonsingular cubic polynomial, discriminant of a polynomial, cubic discriminant, homogeneous coordinates on projective 2-space, punctured complex 3-space, quotient topology, open map, complex two-dimensional manifold (or) complex surface, complex one-dimensional manifold (or) Riemann surface, complex coordinate chart in two complex variables, holomorphic (or) complex analytic function of two complex variables, glueing of Riemann surfaces, glueing of complex planes, zero set of a homogeneous polynomial in projective space, degree of homogeneity, Euler's formula for homogeneous functions, homogenisation, dehomogenisation, a complex curve is a real surface, a complex surface is a real 4-manifold

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

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.