An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

Lectures

Definitions and Examples of Riemann Surfaces

1. The Idea of a Riemann Surface Video
2. Simple Examples of Riemann Surfaces Video
3. Maximal Atlases and Holomorphic Maps of Riemann Surfaces Video
4. A Riemann Surface Structure on a Cylinder Video
5. A Riemann Surface Structure on a Torus Video

Classification of Riemann Surfaces

6. Riemann Surface Structures on Cylinders and Tori via Covering Spaces Video
7. Moebius Transformations Make up Fundamental Groups of Riemann Surfaces Video
8. Homotopy and the First Fundamental Group Video
9. A First Classification of Riemann Surfaces Video

Universal Covering Space Theory

10. The Importance of the Path-lifting Property Video
11. Fundamental groups as Fibres of the Universal covering Space Video
12. The Monodromy Action Video
13. The Universal covering as a Hausdorff Topological Space Video
14. The Construction of the Universal Covering Map Video
15. Completion of the Construction of the Universal Covering: Universality of the Universal Covering Video
16. Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group Video

Classifying Moebius Transformations and Deck Transformations

17. The Riemann Surface Structure on the Topological Covering of a Riemann Surface Video
18. Riemann Surfaces with Universal Covering the Plane or the Sphere Video
19. Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane Video
20. Characterizing Moebius Transformations with a Single Fixed Point Video
21. Characterizing Moebius Transformations with Two Fixed Points Video
22. Torsion-freeness of the Fundamental Group of a Riemann Surface Video
23. Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups Video
24. Classifying Annuli up to Holomorphic Isomorphism Video

The Riemann Surface Structure on the Quotient of the Upper Half-Plane by the Unimodular Group

25. Orbits of the Integral Unimodular Group in the Upper Half-Plane Video
26. Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions Video
27. Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations Video
28. Quotients by Kleinian Subgroups give rise to Riemann Surfaces Video
29. The Unimodular Group is Kleinian Video

Doubly-Periodic Meromorphic (or) Elliptic Functions

30. The Necessity of Elliptic Functions for the Classification of Complex Tori Video
31. The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane Video
32. The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function Video
33. The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing Analytic Functions on the Upper Half-Plane Video

A Form Modular for the Congruence-mod-2 Subgroup of the Unimodular Group on the Upper Half-Plane

34. The Construction of a Modular Form of Weight Two on the Upper Half-Plane Video
35. The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half-Plane Video
36. The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane Video
37. The Weight Two Modular Form Vanishes at Infinity Video
38. The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity Video
39. A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane Video

The Elliptic Modular J-invariant and the Moduli of Complex 1-dimensional Tori (or) Elliptic Curves

40. The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve Video
41. A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant Video
42. The Fundamental Region in the Upper Half-Plane for the Unimodular Group Video
43. A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once Video
44. Moduli of Elliptic Curves Video

Complex 1-dimensional Tori are Projective Algebraic Elliptic Curves

45. Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space Video
46. The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve Video
47. Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two Video
48. Complex Tori are the same as Elliptic Algebraic Projective Curves Video