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

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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 of Lecture 44 To show that the graph of a holomorphic function is naturally a Riemann surface embedded in complex affine 2-space To use the Implicit Function Theorem to show that the zero locus of a nonsingular polynomial in two complex variables is naturally a Riemann surface embedded in complex affine 2-space To show that the elliptic algebraic affine cubic plane curve associated to a punctured complex torus, as described in the previous lecture, has a natural Riemann surface structure which is holomorphically isomorphic to the natural Riemann surface structure on the punctured complex torus (inherited from the complex torus)Keywords for Lecture 44 Upper half-plane, complex torus associated to a lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, 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, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, triple pole (or) pole of order three, cubic equation, elliptic algebraic cubic curve, zeros of a polynomial equation, bicontinuous map (or) homeomorphism, open map, order of an elliptic function, elliptic integral, Argument principle, even function, odd function, analytic branch of the square root, simply connected, punctured torus, elliptic algebraic affine cubic plane curve, projective plane cubic curve, complex affine two-space, complex projective two-space, one-point compactification by adding a point at infinity, Implicit function theorem, graph of a holomorphic function, nonsingular (or) smooth polynomial in two variables, zero locus of a polynomial, solving an implicit equation locally for an explicit function, nonsingular (or) smooth point of the zero locus of a polynomial in two variables, Hausdorff, second countable, connected component, nonsingular cubic polynomial, discriminant of a polynomial, cubic discriminant

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