Goals: - 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 that 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 previous lecture 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. In this lecture, we show that the weight two modular form vanishes at infinity Topics: 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, singular part of the Laurent expansion, pole of order two, uniform convergence, Weierstrass M-test, removable singularity, entire function, periodic function, period of a function, singly periodic function, Liouville's theorem
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.