Goals: - To show that the Weierstrass phe-function associated to a lattice satisfies a first order degree two cubic ODE - The ODE mentioned above is the key to studying the geometry of the complex torus associated to the lattice and eventually leads to the classification (moduli) theory of complex tori Topics: Upper half-plane, invariants for complex tori, lattice (or) grid in the plane, fundamental parallelogram (or) period parallelogram associated to a lattice, complex torus associated to a lattice, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, isolated double pole, singular part of the Laurent expansion, analytic part of the Laurent expansion, antiderivative for the Weierstrass phe-function, Identity theorem for power series or Laurent series, differentiating term-by-term and integrating term-by-term under uniform convergence, even function, odd function, entire elliptic functions are constants, algebraic elliptic cubic 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.