# Introduction to Riemann Surfaces and Algebraic 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.

Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √z. For the imaginary part of √z, rotate the plot 180° around the vertical axis.
Not yet rated

### Video Lectures & Study Materials

Visit the official course website for more study materials: http://nptel.ac.in/syllabus/syllabus.php?subjectId=111106044

# Lecture Play Lecture
I. Definitions and Examples of Riemann Surfaces
1 The Idea of a Riemann Surface (0057:13) Play Video
2 Simple Examples of Riemann Surfaces (0057:47) Play Video
3 Maximal Atlases and Holomorphic Maps of Riemann Surfaces (0050:58) Play Video
4 Riemann Surface Structure on a Cylinder (0054:56) Play Video
5 Riemann Surface Structure on a Torus (0048:26) Play Video
II. Classification of Riemann Surfaces
6 Riemann Surface Structures on Cylinders and Tori via Covering Spaces (0056:44) Play Video
7 Möbius Transformations Make up Fundamental Groups of Riemann Surfaces (0048:34) Play Video
8 Homotopy and the First Fundamental Group (0053:35) Play Video
9 A First Classification of Riemann Surfaces (0049:03) Play Video
III. Universal Covering Space Theory
10 The Importance of the Path-lifting Property (0057:49) Play Video
11 Fundamental groups as Fibres of the Universal covering Space (0056:52) Play Video
12 The Monodromy Action (0053:33) Play Video
13 The Universal covering as a Hausdorff Topological Space (1:01:02) Play Video
14 The Construction of the Universal Covering Map (0055:26) Play Video
15 Universality of the Universal Covering (0037:29) Play Video
16 The Fundamental Group of the base as the Deck Transformation Group (0043:47) Play Video
IV. Classifying Moebius Transformations and Deck Transformations
17 The Riemann Surface Structure on the Topological Covering of a Riemann Surface (0059:12) Play Video
18 Riemann Surfaces with Universal Covering the Plane or the Sphere (1:18:54) Play Video
19 Classifying Complex Cylinders Riemann Surfaces (1:01:21) Play Video
20 Möbius Transformations with a Single Fixed Point (0056:08) Play Video
21 Möbius Transformations with Two Fixed Points (1:01:38) Play Video
22 Torsion-freeness of the Fundamental Group of a Riemann Surface (0046:25) Play Video
23 Characterizing Riemann Surface Structures on Quotients of the Upper Half (1:12:59) Play Video
24 Classifying Annuli up to Holomorphic Isomorphism (0045:18) Play Video
V. 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 (1:15:20) Play Video
26 Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions (1:05:23) Play Video
27 Local Actions at the Region of Discontinuity of a Kleinian Subgroup (1:11:25) Play Video
28 Quotients by Kleinian Subgroups give rise to Riemann Surfaces (0050:51) Play Video
29 The Unimodular Group is Kleinian (1:06:11) Play Video
VI. Doubly-Periodic Meromorphic (or) Elliptic Functions
30 The Necessity of Elliptic Functions for the Classification of Complex Tori (0048:15) Play Video
31 The Uniqueness Property of the Weierstrass Phe-function (1:08:15) Play Video
32 The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function (1:06:12) Play Video
33 The Values of the Weierstrass Phe function at the Zeros of its Derivative (0049:24) Play Video
VII. 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 (0055:50) Play Video
35 The Fundamental Functional Equations satisfied by the Modular Form of Weight (0054:34) Play Video
36 The Weight Two Modular Form assumes Real Values on the Imaginary Axis (0056:55) Play Video
37 The Weight Two Modular Form Vanishes at Infinity (0050:47) Play Video
38 The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity (0043:36) Play Video
39 Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane (0050:24) Play Video
VIII. 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 (0059:17) Play Video
41 Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant (0051:28) Play Video
42 The Fundamental Region in the Upper Half-Plane for the Unimodular Group (1:16:24) Play Video
43 A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once (0049:46) Play Video
44 Moduli of Elliptic Curves (1:08:12) Play Video
IX. Complex 1-dimensional Tori are Projective Algebraic Elliptic Curves
45 Punctured Complex Tori are Elliptic Algebraic Affine Plane (1:00:32) Play Video
46 The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve (1:09:14) Play Video
47 Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two (0043:29) Play Video
48 Complex Tori are the same as Elliptic Algebraic Projective Curves (0036:19) Play Video