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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.

Unit Number / Title

Lecture Number / Title

Unit 1: Definitions and Examples of Riemann Surfaces



Lecture 1:
The Idea of a Riemann Surface

Lecture 2:
Simple Examples of Riemann Surfaces

Lecture 3:
Maximal Atlases and Holomorphic Maps of Riemann Surfaces

Lecture 4:
A Riemann Surface Structure on a Cylinder

Lecture 5:
A Riemann Surface Structure on a Torus

Unit 2: Classification of Riemann Surfaces



Lecture 6:
Riemann Surface Structures on Cylinders and Tori via Covering Spaces

Lecture 7:
Moebius Transformations Make up Fundamental Groups of Riemann Surfaces

Lecture 8:
Homotopy and the First Fundamental Group

Lecture 9:
A First Classification of Riemann Surfaces

Unit 3: 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-Part A:
Completion of the Construction of the Universal Covering: Universality of the Universal Covering

Lecture 15-Part B:
Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group

Unit 4: Classifying Moebius Transformations and Deck Transformations



Lecture 16:
The 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 Points

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

Unit    5: 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

Unit   6: 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

Unit   7: A Form Modular for the Congruence-mod-2 Subgroup of the Unimodular Group on the Upper Half-Plane



Lecture 33: The Construction of a Modular Form of Weight Two on the Upper Half-Plane
Lecture 34: The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper 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 37A: 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

Unit   8: 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

Unit   9: 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 45A: Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
Lecture 45B: Complex Tori are the same as Elliptic Algebraic Projective Curves
 

First courses in Topology and Complex Analysis. However, to help the students taking the course, most of the needed material from topology, complex analysis and algebra is recalled when necessary and their details, including exercises and suggestions for further reading, are given in the supplementary slides at the end of each lecture. These slides are also available separately as PDF files, apart from the lecture notes, for download below.


  1. H. M. Farkas, I. Kra: Riemann Surfaces, Graduate Texts in Mathematics 71, Springer-Verlag, 1980
  2. R. Miranda: Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics vol.5, American Mathematical Society, 1995
  3. O. Forster: Lectures on Riemann Surfaces, Graduate Texts in Mathematics 81 / Springer-Verlag, 1981
  4. R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag,
  5. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris: Geometry of Algebraic Curves - I, Grundlehren der mat.Wissen.267, Springer-Verlag, 1985
  6. T. E. Venkata Balaji: An Introduction to Families, Deformations and Moduli, available at
  7. http://webdoc.sub.gwdg.de/univerlag/2010/balaji.pdf



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