The ball and some Hilbert problems

Description:

The famous twelfth Hilbert problem calls for holomorphic functions in several variables with properties analogous to the exponential function and the elliptic modular function with a view to the explicit construction of (Hilbert) class fields by means of special values. The lecture notes present those functions living on the two-dimensional complex unit ball. In the course of their construction, the reader is introduced to work with complex multiplication, moduli fields, moduli space of curves, surface uniformizations, Gauss-Manin connection, Jacobian varieties, Torelli's theorem, Picard modular forms, Theta functions, class fields and transcendental values in an effective manner.

subjects: Curves, Elliptic,  Elliptic Curves,  Picard schemes,  Unit ball,  Curves,  Geometry, algebraic,  Functions of several complex variables,  Algebraic Geometry,  Number theory,  Geometry,  Mathematics,  Global analysis (Mathematics),  Analysis