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On quadratic fields with large 3-rank


Author: Karim Belabas
Translated by:
Journal: Math. Comp. 73 (2004), 2061-2074
MSC (2000): Primary 11R11, 11R16, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-04-01632-1
Published electronically: January 30, 2004
MathSciNet review: 2059751
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Abstract: Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the $O(X)$general cubic discriminants (real or imaginary) up to $X$ in time $O(X)$ and space $O(X^{3/4})$, or more generally in time $O(X + X^{7/4} / M)$ and space $O(M + X^{1/2})$ for a freely chosen positive $M$. A variant computes the $3$-ranks of all quadratic fields of discriminant up to $X$ with the same time complexity, but using only $M + O(1)$ units of storage. As an application we obtain the first $1618$ real quadratic fields with $r_3(\Delta) \geq 4$, and prove that $\mathbb{Q} (\sqrt{-5393946914743})$ is the smallest imaginary quadratic field with $3$-rank equal to $5$.


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  • 1. K. BELABAS, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), pp. 1213-1237. MR 97m:11159
  • 2. K. BELABAS, On the mean $3$-rank of quadratic fields, Compositio Mathematica 118 (1999), pp. 1-9. MR 2000g:11102
  • 3. M. BHARGAVA, A simple proof of the Davenport-Heilbronn theorem, 1999, preprint.
  • 4. M. BHARGAVA, Higher composition laws, Ph.D. thesis, Princeton University, 2001.
  • 5. H. COHEN, A course in computational algebraic number theory, Springer-Verlag, 1993. MR 94i:11105
  • 6. H. COHEN, Advanced topics in computational number theory, Springer-Verlag, 2000. MR 2000k:11144
  • 7. H. COHEN AND H. W. LENSTRA, JR., Heuristics on class groups of number fields, in Number theory, Noordwijkerhout 1983 (Berlin), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33-62. MR 85j:11144
  • 8. M. CRAIG, A construction for irregular discriminants, Osaka J. Math 14 (1977), pp. 365-402. MR 56:8522
  • 9. J. E. CREMONA, Reduction of binary cubic and quartic forms, LMS J. Comput. Math. 2 (1999), pp. 64-94 (electronic). MR 2000f:11040
  • 10. H. DAVENPORT, On the class number of binary cubic forms (i), J. Lond. Math. Soc. 26 (1951), pp. 183-192; errata ibid 27 (1951), p. 512. MR 13:323e
  • 11. H. DAVENPORT, On the class number of binary cubic forms (ii), J. Lond. Math. Soc. 26 (1951), pp. 192-198. MR 13:323f
  • 12. H. DAVENPORT AND H. HEILBRONN, On the density of discriminants of cubic fields (ii), Proc. Roy. Soc. Lond. A 322 (1971), pp. 405-420. MR 58:10816
  • 13. B. N. DELONE AND D. K. FADDEEV, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, vol. 10, American Mathematical Society, 1964. MR 28:3955
  • 14. F. DIAZ Y DIAZ, On some families of imaginary quadratic fields, Math. Comp. 32 (1978), no. 142, pp. 637-650. MR 58:5582
  • 15. F. DIAZ Y DIAZ, Sur le $3$-rang des corps quadratiques réels, Prépublications de la faculté d'Orsay, 1978. MR 80i:12005
  • 16. P. DUTARTE, Compatibilité avec le Spiegelungssatz de probabilités conjecturales sur le $p$-rang du groupe des classes, in Number theory, 1983-1984 (Besançon), Univ. Franche-Comté, Besançon, 1984, pp. Exp. No. 4, 11. MR 86m:11103
  • 17. V. ENNOLA AND R. TURUNEN, On totally real cubic fields, Math. Comp. 44 (1985), no. 170, pp. 495-518. MR 86e:11100
  • 18. H. HASSE, Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage, Math. Zeitschrift. 31 (1930), pp. 565-582.
  • 19. C. S. HERZ, Seminar on Complex Multiplication. VII. Construction of class fields, Lect. Notes in Math., vol. 21, Springer-Verlag, Berlin, 1966. MR 34:1278
  • 20. D. E. KNUTH, The art of computer programming. vol. 2: Seminumerical algorithms, Addison-Wesley, 1969. MR 44:3531
  • 21. F. LEPRÉVOST, Courbes modulaires et $11$-rang de corps quadratiques, Experiment. Math. 2 (1993), no. 2, pp. 137-146. MR 94m:11073
  • 22. P. LLORENTE AND J. QUER, On the $3$-Sylow subgroup of the class group of quadratic fields, Math. Comp. 50 (1988), no. 181, pp. 321-333. MR 89b:11083
  • 23. D. C. MAYER, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), no. 198, pp. 831-847, S55-S58. MR 92f:11154
  • 24. J.-F. MESTRE, Corps quadratiques dont le $5$-rang du groupe des classes est $\geq 3$, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 4, pp. 371-374. MR 93f:11076
  • 25. PARI/GP, version 2.1.5, Bordeaux, 2003, http://www.parigp-home.de.
  • 26. J. QUER, Corps quadratiques de $3$-rang $6$ et courbes elliptiques de rang $12$, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), pp. 215-218. MR 88j:11074
  • 27. D. P. ROBERTS, Density of cubic field discriminants, Math. Comp. 70 (2001), no. 236, pp. 1699-1705 (electronic). MR 2002e:11142
  • 28. R. J. SCHOOF, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, pp. 295-302. MR 84h:12005
  • 29. T. SHINTANI, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, Sec. Ia 22 (1975), pp. 25-66. MR 52:5590
  • 30. J. J. SOLDERITSCH, Quadratic fields with special class groups, Ph.D. thesis, Lehigh University, 1977.
  • 31. Y. YAMAMOTO, On unramified Galois extensions of quadratic number fields, Osaka J. Math. 7 (1970), pp. 57-76. MR 42:1800

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Additional Information

Karim Belabas
Affiliation: Université Paris-Sud, Département de Mathématiques (bât. 425), F-91405 Orsay, France
Email: Karim.Belabas@math.u-psud.fr

DOI: https://doi.org/10.1090/S0025-5718-04-01632-1
Keywords: Cubic fields, quadratic fields, 3-rank
Received by editor(s): April 8, 2002
Received by editor(s) in revised form: May 3, 2003
Published electronically: January 30, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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