Goals: - 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-plane Topics: 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
The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmueller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra---especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory. The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given.