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


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


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


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 16 - Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group


Lecture 17 - The Riemann Surface Structure on the Topological Covering of a Riemann Surface


Lecture 18 - Riemann Surfaces with Universal Covering the Plane or the Sphere


Lecture 19 - Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane


Lecture 20 - Characterizing Moebius Transformations with a Single Fixed Point


Lecture 21 - Characterizing Moebius Transformations with Two Fixed Points


Lecture 22 - Torsion-freeness of the Fundamental Group of a Riemann Surface


Lecture 23 - Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups


Lecture 24 - Classifying Annuli up to Holomorphic Isomorphism


Lecture 25 - Orbits of the Integral Unimodular Group in the Upper Half-Plane


Lecture 26 - Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions


Lecture 27 - Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations


Lecture 28 - Quotients by Kleinian Subgroups give rise to Riemann Surfaces


Lecture 29 - The Unimodular Group is Kleinian


Lecture 30 - The Necessity of Elliptic Functions for the Classification of Complex Tori


Lecture 31 - The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane


Lecture 32 - The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function


Lecture 33 - The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing Analytic Functions on the Upper Half-Plane


Lecture 34 - The Construction of a Modular Form of Weight Two on the Upper Half-Plane


Lecture 35 - The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half-Plane


Lecture 36 - The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane


Lecture 37 - The Weight Two Modular Form Vanishes at Infinity


Lecture 38 - The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity


Lecture 39 - A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane


Lecture 40 - The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve


Lecture 41 - A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant


Lecture 42 - The Fundamental Region in the Upper Half-Plane for the Unimodular Group


Lecture 43 - A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once


Lecture 44 - Moduli of Elliptic Curves


Lecture 45 - Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space


Lecture 46 - The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve


Lecture 47 - Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two


Lecture 48 - Complex Tori are the same as Elliptic Algebraic Projective Curves