The Ball and Some Hilbert Problems 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 Torellis theorem Picard modular forms Theta functions class fields and transcen dental values in an effective manner