CLASS NUMBERS OF RAY CLASS FIELDS OF IMAGINARY QUADRATIC FIELDS


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Kucuksakalli O.

MATHEMATICS OF COMPUTATION, vol.80, no.274, pp.1099-1122, 2011 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 80 Issue: 274
  • Publication Date: 2011
  • Doi Number: 10.1090/s0025-5718-2010-02413-5
  • Title of Journal : MATHEMATICS OF COMPUTATION
  • Page Numbers: pp.1099-1122

Abstract

Let K be an imaginary quadratic field with class number one and let p subset of 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(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(p). This is a counterexample to the elliptic analogue of Vandiver's conjecture.