Goals: - In the last few lectures, we have shown that the quotient of the upper half-plane by the unimodular group has a natural Riemann surface structure. In order to show that this Riemann surface is isomorphic to the complex plane, we have to realize that we need to look for invariants for complex tori - To motivate how the search for invariants for complex tori leads us to the study of doubly-periodic meromorphic functions (or) elliptic functions, the stereotype of which is given by the famous Weierstrass phe-function Topics: Upper half-plane, unimodular group, projective special linear group, set of orbits, quotient Riemann surface, lattice (or) grid in the plane, fundamental parallelogram associated to a lattice, complex torus associated to a lattice, translation, j-invariant for a complex torus, Felix Klein, group-invariant function, bounded entire function, Liouville's theorem, singularity of an analytic function, poles, meromorphic function, doubly-periodic meromorphic function (or) elliptic function, Karl Weierstrass, algebraic curve, elliptic curve, Weierstrass phe-function, Residue theorem, double pole
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.