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An Introduction to Riemann Surfaces and Algebraic Curves

An Introduction to Riemann Surfaces and Algebraic Curves. Instructor: Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. 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 Teichmuller. 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. (from nptel.ac.in)

The Idea of a Riemann Surface


Definitions and Examples of Riemann Surfaces
Lecture 01 - The Idea of a Riemann Surface
Lecture 02 - Simple Examples of Riemann Surfaces
Lecture 03 - Maximal Atlases and Holomorphic Maps of Riemann Surfaces
Lecture 04 - A Riemann Surface Structure on a Cylinder
Lecture 05 - A Riemann Surface Structure on a Torus
Classification of Riemann Surfaces
Lecture 06 - Riemann Surface Structures on Cylinders and Tori via Covering Spaces
Lecture 07 - Moebius Transformations Make Up Fundamental Groups of Riemann Surfaces
Lecture 08 - Homotopy and the First Fundamental Group
Lecture 09 - A First Classifications of Riemann Surfaces
Universal Covering Space Theory
Lecture 10 - The Importance of the Path-lifting Property
Lecture 11 - Fundamental Groups as Fibres of the Universal Covering Space
Lecture 12 - The Monodromy Action
Lecture 13 - The Universal Covering as a Hausdorff Topological Space
Lecture 14 - The Construction of the Universal Covering Map
Lecture 15 - Completion of the Construction of the Universal Covering: Universality of the Universal Covering
Lecture 15B - Completion of the Construction of the Universal Covering: The Fundamental Group of the Base as the Deck Transformation Group
Classifying Moebius Transformations and Deck Transformations
Lecture 16 - Riemann Surface Structure on the Topological Covering of a Riemann Surface
Lecture 17 - Riemann Surfaces with Universal Covering the Plane or the Sphere
Lecture 18 - Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane
Lecture 19 - Characterizing Moebius Transformations with a Single Fixed Point
Lecture 20 - Characterizing Moebius Transformations with Two Fixed Point
Lecture 21 - Torsion-freeness of the Fundamental Group of a Riemann Surface
Lecture 22 - Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups
Lecture 23 - Classifying Annuli up to Holomorphic Isomorphism
The Riemann Surface Structure on the Quotient of the Upper Half-Plane by the Unimodular Group
Lecture 24 - Orbits of the Integral Unimodular Group in the Upper Half-Plane
Lecture 25 - Galois Coverings are Precisely Quotients by Properly Discontinuous Free Actions
Lecture 26 - Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations
Lecture 27 - Quotients by Kleinian Subgroups Give Rise to Riemann Surfaces
Lecture 28 - The Unimodular Group is Kleinian
Doubly-Periodic Meromorphic (or) Elliptic Functions
Lecture 29 - The Necessity of Elliptic Functions for the Classification of Complex Tori
Lecture 30 - The Uniqueness Property of the Weierstrass Phe-function Associated to a Lattice in the Plane
Lecture 31 - The First Order Degree Two Cubic Ordinary Differential Equation Satisfied by the Weierstrass Phe-function
Lecture 32 - The Values of the Weierstrass Phe-function at the Zeros of its Derivative are Nonvanishing Analytic Functions on the Upper Half-Plane
A Form Modular for the Congruence-Mod-2 Subgroup of the Unimodular Group on the Half-Plane
Lecture 33 - The Construction of a Modular Form of Weight Two on the Half-Plane
Lecture 34 - The Fundamental Functional Equations Satisfied by the Modular Form of Weight Two on the Half-Plane
Lecture 35 - The Weight Two Modular Form Assumes Real Values on the Imaginary Axis in the Upper Half-Plane
Lecture 36 - The Weight Two Modular Form Vanishes at Infinity
Lecture 37 - The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
Lecture 37B - A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
The Elliptic Modular J-invariant and the Moduli of Complex 1-Dimensional Tori (or) Elliptic Curves
Lecture 38 - The J-invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
Lecture 39 - A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-invariant
Lecture 40 - The Fundamental Region in the Upper Half-Plane for the Unimodular Group
Lecture 41 - A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
Lecture 42 - Moduli of Elliptic Curves
Complex 1-Dimensional Tori are Projective Algebraic Elliptic Curves
Lecture 43 - Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space
Lecture 44 - The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
Lecture 45 - Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
Lecture 45B - Complex Tori are the same as Elliptic Algebraic Projective Curves

References
An Introduction to Riemann Surfaces and Algebraic Curves
Instructor: Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. The course begins by introducing the notion of a Riemann surface followed by examples.