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

Lecture Details

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 37 Part B In the last few lectures, we constructed 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 group. We saw how the effect of a general element of the unimodular group on this weight two modular form can be understood by just computing the effect under each of five unimodular elements representing pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 subgroup. We saw that these computations resulted in five simple and beautiful functional equations satisfied by the weight two modular form. Using these functional equations, we want to find a suitable region in the upper half-plane on which the mapping properties of this weight two modular form may be easily studied In the last couple of lectures we saw that the weight two modular form assumes only real values on the imaginary axis, which will turn out to be a boundary for such a region, and further that the weight two modular form vanishes at infinity. In part A of this lecture we estimated, by calculating the Fourier coefficient that mattered most, that this vanishing at infinity is in fact an exponential decay. This estimation is critical for the study of the mapping properties which we complete in part B of this lecture. We show that the weight two modular form assumes every value on the upper half-plane, and that when restricted to a suitable region it actually gives a holomorphic conformal isomorphism onto the upper half-plane with a continuous monotonic conformal extension to the boundary on the Riemann Sphere so that every real value and the point at infinity is also assumed precisely onceKeywords for Lecture 37 Part B 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, zeros of the derivative of the Weierstrass phe-function, monotonic function, contour, winding number, Fourier coefficient, Fourier development