Mod-08 Lec-38 The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve

Course Lectures

• Mod-01 Lec-01 The Idea of a Riemann Surface

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• Mod-01 Lec-02 Simple Examples of Riemann Surfaces

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• Mod-01 Lec-03 Maximal Atlases and Holomorphic Maps of Riemann Surfaces

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• Mod-01 Lec-04 A Riemann Surface Structure on a Cylinder

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• Mod-01 Lec-05 A Riemann Surface Structure on a Torus

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• Mod-02 Lec-06 Riemann Surface Structures on Cylinders and Tori via Covering Spaces

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• Mod-02 Lec-07 Moebius Transformations Make up Fundamental Groups of Riemann Surfaces

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• Mod-02 Lec-08 Homotopy and the First Fundamental Group

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• Mod-02 Lec-09 A First Classification of Riemann Surfaces

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• Mod-03 Lec-10 The Importance of the Path-lifting Property

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• Mod-03 Lec-11 Fundamental groups as Fibres of the Universal covering Space

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• Mod-03 Lec-12 The Monodromy Action

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• Mod-03 Lec-13 The Universal covering as a Hausdorff Topological Space

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• Mod-03 Lec-14 The Construction of the Universal Covering Map

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• Mod-03 Lec-15A Completion of the Construction of the Universal Coveringl

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• Mod-03 Lec-15B Completion of the Construction of the Universal Covering: The Fundamental Group

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• Mod-04 Lec-16 The Riemann Surface Structure on the Topological Covering of a Riemann Surface

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• Mod-04 Lec-17 Riemann Surfaces with Universal Covering the Plane or the Sphere

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• Mod-04 Lec-18 Classifying Complex Cylinders Riemann Surfaces

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• Mod-04 Lec-19 Characterizing Moebius Transformations with a Single Fixed Point

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• Mod-04 Lec-20 Characterizing Moebius Transformations with Two Fixed Points

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• Mod-04 Lec-21 Torsion-freeness of the Fundamental Group of a Riemann Surface

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• Mod-04 Lec-22 Characterizing Riemann Surface Structures on Quotients of the Upper Half

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• Mod-04 Lec-23 Classifying Annuli up to Holomorphic Isomorphism

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• Mod-05 Lec-24 Orbits of the Integral Unimodular Group in the Upper Half-Plane

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• Mod-05 Lec-25 Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions

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• Mod-05 Lec-26 Local Actions at the Region of Discontinuity of a Kleinian Subgroup

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• Mod-05 Lec-27 Quotients by Kleinian Subgroups give rise to Riemann Surfaces

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• Mod-05 Lec-28 The Unimodular Group is Kleinian

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• Mod-06 Lec-29 The Necessity of Elliptic Functions for the Classification of Complex Tori

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• Mod-06 Lec-30 The Uniqueness Property of the Weierstrass Phe-function

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• Mod-06 Lec-31 The First Order Degree Two Cubic Ordinary Differential Equation satisfied

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• Mod-06 Lec-32 The Values of the Weierstrass Phe function at the Zeros of its Derivative

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• Mod-07 Lec-33 The Construction of a Modular Form of Weight Two on the Upper Half-Plane

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• Mod-07 Lec-34 The Fundamental Functional Equations satisfied by the Modular Form of Weight

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• M0d-07 Lec-35 The Weight Two Modular Form assumes Real Values on the Imaginary Axis

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• Mod-07 Lec-36 The Weight Two Modular Form Vanishes at Infinity

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• Mod-07 Lec-37A The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity

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• Mod-07 Lec-37B A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal

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• Mod-08 Lec-38 The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve

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• Mod-08 Lec-39 A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant

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• Mod-08 Lec-40 The Fundamental Region in the Upper Half-Plane for the Unimodular Group

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• Mod-08 Lec-41 A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once

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• Mod-08 Lec-42 Moduli of Elliptic Curves

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• Mod-09 Lec-43 Punctured Complex Tori are Elliptic Algebraic Affine Plane

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• Mod-09 Lec-44 The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve

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• Mod-09 Lec-45A Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two

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• Mod-09 Lec-45B Complex Tori are the same as Elliptic Algebraic Projective Curves

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