Class numbers of ray class fields of imaginary quadratic fields

Abstract:

Let $K$ be an imaginary quadratic field with class number one and let $\mathfrak {p} \subset \mathcal {O}_K$ be a degree one prime ideal of norm $p$ not dividing $6d_K$. In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields $K_{\mathfrak {p}}$ 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 $K_{\mathfrak {p}}$. This is a counterexample to the elliptic analogue of Vandiver’s conjecture.