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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.

References [Enhancements On Off] (What's this?)

  • 1. Á. Lozano-Robledo, Bernoulli numbers, Hurwitz numbers, $ p$-adic $ L$-functions and Kummer's criterion. RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 101 (2007), no. 1, 1-32. MR 2324575 (2008d:11124)
  • 2. G. Robert, Nombres de Hurwitz et Unités Elliptiques. Ann. Scient. Éc. Norm. Sup. (1978), $ 4^e$ série, 11, 297-389. MR 521636 (80k:12010)
  • 3. R. Schoof, Class numbers of real cyclotomic fields of prime conductor. Math. Comp. 72 (2003), no. 242, 913-937 MR 1954975 (2004f:11116)
  • 4. H. M. Stark, $ L$-Functions at s=1. IV. First Derivatives at s=0. Adv. in Math. 35 (1980), no. 3, 197-235. MR 563924 (81f:10054)
  • 5. H. M. Stark, A complete determination of the complex quadratic fields of class-number one. Michigan Math. J. 14 1967 1-27. MR 0222050 (36:5102)
  • 6. L. C. Washington, Introduction to Cyclotomic Fields; second edition. Graduate Texts in Math. 83, Springer-Verlag, Berlin, Heidelberg, New York, 1997. MR 1421575 (97h:11130)
  • 7. PARI/GP, version 2.3.2,, Bordeaux, 2006.

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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.

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