Goals: - To ask for a description of the set of holomorphic isomorphism classes of complex tori - To state the Theorem on the Moduli of Elliptic Curves that not only answers the question above but also shows that the set above has a beautiful God-given geometry - To see how the upper half-plane and the unimodular group (integral projective special linear group) enter into the discussion - To use the theory of covering spaces to prove a part of the Theorem on the Moduli of Elliptic Curves, namely that the set of holomorphic isomorphism classes of complex 1-dimensional tori is in a natural bijective correspondence with the set of orbits of the unimodular group in the upper half-plane Topics: Real torus, complex torus, Möbius transformation, translation, abelian group, holomorphic universal covering, admissible neighborhood, fundamental group, deck transformation group, biholomorphism class (or) holomorphic isomorphism class, locally biholomorphic map, upper half-plane, projective special linear group, unimodular group, orbits of a group action, action of a subgroup, underlying fixed geometric structure, superimposed (or) overlying (or) extra geometric structure, variation of extra structure for a fixed underlying structure (or) moduli problem, quotient by a group, equivalence relation induced by a group action, universal property of the universal covering, unique lifting property, moduli of elliptic curves, forming the fundamental group is functorial
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.