Class numbers of ray class fields of imaginary quadratic fields

Let K be an imaginary quadratic field with class number one and let p ⊂ OK be a degree one prime ideal of norm p not dividing 6dK . In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields Kp heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura’s reciprocity law. We have discovered a very interesting phenomenon where p divides the class number of Kp. This is a counterexample to the elliptic analogue of Vandiver’s conjecture.