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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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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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Additional Information
  • Omer Kucuksakalli
  • Affiliation: University of Massachusetts, Amherst, Department of Mathematics and Statistics, Amherst, Massachusetts 01003
  • Address at time of publication: Middle East Technical University, Department of Mathematics, 06531 Ankara, Turkey
  • Email: omerks@gmail.com
  • Received by editor(s): May 14, 2009
  • Received by editor(s) in revised form: January 1, 2010
  • Published electronically: September 2, 2010
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1099-1122
  • MSC (2010): Primary 11Y40
  • DOI: https://doi.org/10.1090/S0025-5718-2010-02413-5
  • MathSciNet review: 2772114