Goals of Lecture 42: - To complete the proof of the fact that a suitable region in the upper half-plane, described in the previous lecture and shown there to be a fundamental region for the unimodular group, is also a fundamental region for the elliptic modular j-invariant function - In view of the above, we complete the proof of the theorem on the Moduli of Elliptic Curves: the natural Riemann surface structure, on the set of holomorphic isomorphism classes of complex 1-dimensional tori (complex algebraic elliptic curves) identified with the set of orbits of the unimodular group in the upper half-plane, is holomorphically isomorphic via the j-invariant to the complex plane Topics: Upper half-plane, quotient by the unimodular group, orbits of the unimodular group, representative of an orbit, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) weight two modular form, full modular function (or) full modular form, period two modular form, congruence-mod-2 normal subgroup of the unimodular group, projective special linear group with mod-2 coefficients, finite group, kernel of a group homomorphism, zeros of the derivative of the Weierstrass phe-function, pole of order two (or) double pole with residue zero, universal cover, neighborhood of infinity, lower half-plane, rational function, kernel of a group homomorphism, functional equations satisfied by the weight two modular form, j-invariant of a complex torus (or) j-invariant of an algebraic elliptic curve, Fundamental theorem of Algebra, complex field is algebraically closed, fundamental region for the full modular form, fundamental region for the unimodular group, ramified (or) branched covering, group-invariant holomorphic maps, fundamental region for a group-invariant holomorphic map, fundamental parallelogram associated to a lattice in the plane, Galois theory, Galois group, Galois extension of function fields of meromorphic functions on Riemann surfaces, symmetric group, Galois covering.
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.