Title: The ball and some Hilbert problems
First published: 1995
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
LC classification: QA565 .H65 1995