### Lecture Description

Goals: - To show that there exists a complex torus with j-invariant any prescribed complex number; in other words, to show that the j-invariant is surjective as a map onto the complex numbers - To use the functional equations satisfied by the weight two modular form as well as the mapping properties of that form, as studied in the previous unit of lectures, to find a suitable region in the upper half-plane where the mapping properties of the full modular form given by the j-invariant can be clearly studied Topics: Upper half-plane, quotient by the unimodular group, orbits of the unimodular group, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, 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, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group.

### Course Index

- The Idea of a Riemann Surface
- Simple Examples of Riemann Surfaces
- Maximal Atlases and Holomorphic Maps of Riemann Surfaces
- Riemann Surface Structure on a Cylinder
- Riemann Surface Structure on a Torus
- Riemann Surface Structures on Cylinders and Tori via Covering Spaces
- Möbius Transformations Make up Fundamental Groups of Riemann Surfaces
- Homotopy and the First Fundamental Group
- A First Classification of Riemann Surfaces
- The Importance of the Path-lifting Property
- Fundamental groups as Fibres of the Universal covering Space
- The Monodromy Action
- The Universal covering as a Hausdorff Topological Space
- The Construction of the Universal Covering Map
- Universality of the Universal Covering
- The Fundamental Group of the base as the Deck Transformation Group
- The Riemann Surface Structure on the Topological Covering of a Riemann Surface
- Riemann Surfaces with Universal Covering the Plane or the Sphere
- Classifying Complex Cylinders Riemann Surfaces
- Möbius Transformations with a Single Fixed Point
- Möbius Transformations with Two Fixed Points
- Torsion-freeness of the Fundamental Group of a Riemann Surface
- Characterizing Riemann Surface Structures on Quotients of the Upper Half
- Classifying Annuli up to Holomorphic Isomorphism
- Orbits of the Integral Unimodular Group in the Upper Half-Plane
- Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
- Local Actions at the Region of Discontinuity of a Kleinian Subgroup
- Quotients by Kleinian Subgroups give rise to Riemann Surfaces
- The Unimodular Group is Kleinian
- The Necessity of Elliptic Functions for the Classification of Complex Tori
- The Uniqueness Property of the Weierstrass Phe-function
- The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function
- The Values of the Weierstrass Phe function at the Zeros of its Derivative
- The Construction of a Modular Form of Weight Two on the Upper Half-Plane
- The Fundamental Functional Equations satisfied by the Modular Form of Weight
- The Weight Two Modular Form assumes Real Values on the Imaginary Axis
- The Weight Two Modular Form Vanishes at Infinity
- The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
- Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
- The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
- Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
- The Fundamental Region in the Upper Half-Plane for the Unimodular Group
- A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
- Moduli of Elliptic Curves
- Punctured Complex Tori are Elliptic Algebraic Affine Plane
- The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
- Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
- Complex Tori are the same as Elliptic Algebraic Projective 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.