Class numbers of ray class fields of imaginary quadratic fields

Kucuksakalli, Omer

doi:10.1090/S0025-5718-2010-02413-5

Class numbers of ray class fields of imaginary quadratic fields
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by Omer Kucuksakalli PDF

Math. Comp. 80 (2011), 1099-1122 Request permission

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.

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