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 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 80 Issue: 274
  • Publication Date: 2011
  • Doi Number: 10.1090/s0025-5718-2010-02413-5
  • Journal Name: MATHEMATICS OF COMPUTATION
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1099-1122
  • Middle East Technical University Affiliated: Yes

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.