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 34: The Construction of a Modular Form of Weight Two on 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 construct an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 subgroup of the unimodular groupKeywords Upper half-plane, 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) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, universal covering, fundamental group, deck transformation group, holomorphic lifting, conjugation by a Moebius transformation, conjugate subgroup, additive group of translations, isomorphism of lattices, generator of a group, locally invertible map, locally biholomorphic map, conformal map, zeros of the derivative of the Weierstrass phe-function

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