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Mathematics of Computation

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Class numbers of ray class fields of imaginary quadratic fields

Author: Omer Kucuksakalli
Journal: Math. Comp. 80 (2011), 1099-1122
MSC (2010): Primary 11Y40
Published electronically: September 2, 2010
MathSciNet review: 2772114
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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

Received by editor(s): May 14, 2009
Received by editor(s) in revised form: January 1, 2010
Published electronically: September 2, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.