Gisbert Wuestholz

Hilbert's seventh problem asks for transcendence of algebraic powers of algebraic numbers and the twelfth Hilbert Problem asks whether certain algebraic number fields can be generated by the values of transcendental functions. It turns out that there is a close relation between the two questions which we shall discuss in the talk. We shall show how the classical theory of hypergeometric functions, modular forms and Shimura varieties come into the picture. Finally we shall briefly discuss how all this is related to a conjecture of Oort on subvarieties of Shimura varieties.