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 40: The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic 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 To associate to each complex 1-dimensional torus a complex number, called the j-invariant of the complex torus, which depends only on the holomorphic isomorphism class of the torus. This j-invariant will be shown in the forthcoming lectures to completely classify all complex tori In the previous unit of lectures, we constructed a weight two modular form on the upper half-plane and studied its mapping properties. In this lecture we use this weight two modular form to define a full modular form, i.e., a holomorphic function on the upper half-plane that is invariant under the action of the full unimodular group. It is this modular form that goes down to give the j-invariant function on the Riemann surface of holomorphic isomorphism classes of complex tori with underlying set consisting of the orbits of the unimodular group in the upper half-planeKeywords Upper half-plane, quotient by the unimodular group, orbits of the unimodular group, 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) weight two modular form, full modular function (or) full modular form, congruence-mod-2 normal subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, meromorphic functions are holomorphic functions to the Riemann Sphere, j-invariant of a complex torus (or) j-invariant of an algebraic elliptic curve

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