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

IIT Madras Course , Prof. T.E. Venkata Balaji

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

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        An Introduction to Riemann Surfaces and Algebraic Curves Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit httpwww.nptel.iitm.ac.insyllabus111106044Goals To ask for a description of the set of holomorphic isomorphism classes of complex tori To state the Theorem on the Moduli of Elliptic Curves that not only answers the question above but also shows that the set above has a beautiful God-given geometry To see how the upper half-plane and the unimodular group (integral projective special linear group) enter into the discussion To use the theory of covering spaces to prove a part of the Theorem on the Moduli of Elliptic Curves, namely that the set of holomorphic isomorphism classes of complex 1-dimensional tori is in a natural bijective correspondence with the set of orbits of the unimodular group in the upper half-planeKeywords Real torus, complex torus, Moebius transformation, translation, abelian group, holomorphic universal covering, admissible neighborhood, fundamental group, deck transformation group, biholomorphism class (or) holomorphic isomorphism class, locally biholomorphic map, upper half-plane, projective special linear group, unimodular group, orbits of a group action, action of a subgroup, underlying fixed geometric structure, superimposed (or) overlying (or) extra geometric structure, variation of extra structure for a fixed underlying structure (or) moduli problem, quotient by a group, equivalence relation induced by a group action, universal property of the universal covering, unique lifting property, moduli of elliptic curves, forming the fundamental group is functorial

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