Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Computation of the $ p$-part of the ideal class group of certain real abelian fields


Author: Hiroki Sumida-Takahashi
Journal: Math. Comp. 76 (2007), 1059-1071
MSC (2000): Primary 11R23, 11R70
DOI: https://doi.org/10.1090/S0025-5718-07-01926-6
Published electronically: January 5, 2007
MathSciNet review: 2291850
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Under Greenberg's conjecture, we give an efficient method to compute the $ p$-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the $ p$-part of the ideal class group of the maximal real subfield of $ \mathbf{Q}(\sqrt{-f},\zeta_{p^{n+1}})$ in the range $ 1 <f<200$ and $ 5 \le p <100000$. In order to explain our method, we show an example whose ideal class group is not cyclic.


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

  • 1. M. Aoki and T. Fukuda, An algorithm for computing $ p$-class groups of abelian number fields, Lecture Notes in Comput. Sci. 4076, Springer, Berlin, 2006, pp. 56-71.
  • 2. J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and A. M. Shokrollahi, Irregular primes and cyclotomic invariants to $ 12$ million, J. Symbolic Comput. 31 (2001), 89-96. MR 1806208 (2001m:11220)
  • 3. B. Ferrero and R. Greenberg, On the behavior of $ p$-adic $ L$-functions at $ s=0$, Invent. Math. 50 (1978), 91-102. MR 0516606 (80f:12016)
  • 4. B. Ferrero and L. Washington, The Iwasawa invariant $ \mu_p$ vanishes for abelian number fields, Ann. of Math. 109 (1979), 377-395. MR 0528968 (81a:12005)
  • 5. R. Gillard, Remarques sur les unités cyclotomiques et les unités elliptiques, J. Number Theory 11 (1979), 21-48. MR 0527759 (80j:12004)
  • 6. R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263-284. MR 0401702 (53:5529)
  • 7. H. Ichimura, Local units modulo Gauss sums, J. Number Theory 68 (1998), 36-56. MR 1492887 (98k:11154)
  • 8. H. Ichimura and H. Sumida, On the Iwasawa invariants of certain real abelian fields II, Internat. J. Math. 7 (1996), 721-744. MR 1417782 (98e:11128c)
  • 9. K. Iwasawa, Lectures on $ p$-adic L-functions, Ann. of Math. Stud., vol. 74, Princeton Univ. Press: Princeton, N.J., 1972. MR 0360526 (50:12974)
  • 10. -, On $ \mathbf{Z}_l$-extensions of algebraic number fields, Ann. of Math.,(2) 98 (1973), 246-326. MR 0349627 (50:2120)
  • 11. J. S. Kraft and R. Schoof, Computing Iwasawa modules of real quadratic number fields, Compositio Math. 97 (1995), 135-155. MR 1355121 (97b:11129)
  • 12. T. Kubota and H.W. Leopoldt, Eine $ p$-adische Theorie der Zetawerte, I. Einführung der $ p$-adischen Dirichletschen $ L$-Funktionen, J. Reine Angew. Math. 214/215 (1964), 328-339. MR 163900 (29:1199)
  • 13. B. Mazur and A. Wiles, Class fields of abelian extensions of $ \mathbf{Q}$, Invent. Math. 76 (1984), 179-330. MR 0742853 (85m:11069)
  • 14. M. Ozaki, On the cyclotomic unit group and the ideal class group of a real abelian number field. I, II, J. Number Theory 64 (1997), 211-222, 223-232. MR 1453211 (98c:11121)
  • 15. H. Sumida-Takahashi, Computation of Iwasawa invariants of certain real abelian fields, J. Number Theory 105 (2004), 235-250. MR 2040156 (2005a:11171)
  • 16. -, The Iwasawa invariants and the higher $ K$-groups associated to real quadratic fields, Exp. Math. 14 (2005), 307-316. MR 2172709 (2006j:11152)
  • 17. L. Washington, Introduction to cyclotomic fields. second edition, Graduate Texts in Math., vol. 83, Springer-Verlag: New York, 1997. MR 1421575 (97h:11130)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R23, 11R70

Retrieve articles in all journals with MSC (2000): 11R23, 11R70


Additional Information

Hiroki Sumida-Takahashi
Affiliation: Faculty and School of Engineering, The University of Tokushima, 2-1 Minamijosanjima-cho, Tokushima 770-8506, Japan
Email: hiroki@pm.tokushima-u.ac.jp

DOI: https://doi.org/10.1090/S0025-5718-07-01926-6
Keywords: Ideal class group, Iwasawa invariant, abelian field, Greenberg's conjecture
Received by editor(s): September 7, 2005
Received by editor(s) in revised form: January 20, 2006
Published electronically: January 5, 2007
Additional Notes: This work was partially supported by the Grants-in-Aid for Encouragement of Young Scientists (No. 16740019) from Japan Society for the Promotion of Science.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society