Goals: - To see how the quotient of the region of discontinuity by a Kleinian subgroup of Möbius transformations is a union of Riemann surfaces - To see how the quotients above are ramified (or) branched coverings of Riemann surfaces, with ramifications at the points with nontrivial isotropies (stabilizers) - To see in detail how to get a complex coordinate chart at the image point of a point of ramification Topics: Upper half-plane, unimodular group, fixed point, projective special linear group, quotient by a subgroup of Möbius transformations, holomorphic automorphisms, extended plane, properly discontinuous action, stabilizer (or) isotropy subgroup, region of discontinuity of a subgroup of Möbius transformations, limit set of a subgroup of Möbius transformations, elliptic Möbius transformations, isolated point, discrete subset, Kleinian subgroup of Möbius transformations, quotient topology, ramification (or) branch points, ramified (or) branched covering, unramified (or) unbranched covering, branch cut, slit disc
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.