Goals: - To realize that in order to study Riemann surfaces with abelian fundamental group and having universal covering the upper half-plane, one needs to first classify Möbius transformations in general and in particular study among those that are automorphisms of the upper half-plane - To motivate how the classification of Möbius transformations can be done using two seemingly unrelated aspects: one of them being the set of fixed points in the extended complex plane and the other being the value of the square of the trace of the transformation. - To show that these two aspects, though one of them is geometric while the other numeric, are in fact precisely related to each other - To characterize Möbius transformations with exactly one fixed point in the extended complex plane as precisely those that are conjugate to a translation; to show that such transformations are also precisely the so-called parabolic transformations, where parabolicity is defined as the square of the trace being equal to four Topics: Upper half-plane, unit disc, abelian fundamental group, deck transformation group, Möbius transformation, universal covering, holomorphic automorphism, group isomorphism, linear fractional transformation, bilinear transformation, fixed point of a map, square of the trace of a Möbius transformation, parabolic Möbius transformations, translations, conjugation by a Möbius transformation, special linear group, projective special linear group, upper-triangular matrix
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.