The theory of complex multiplication provides us with a means of computing a generating polynomial for the Hilbert class field of a given imaginary quadratic number field. The classical approach of using the modular j-function yields polynomials with huge coefficients, and as was disovered by Weber already, we can do better by using `smaller' functions.

In this talk we focus on new p-adic algorithms to compute such generating polynomials. For the j-function this is based on a paper of Couveignes and Henocq, and we explain how to generalize their approach to cope with smaller functions over p-adic fields by using modular curves.