Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields


Author: Stéphane R. Louboutin
Journal: Math. Comp. 76 (2007), 455-473
MSC (2000): Primary 11R16, 11R20, 11R29, 11R42, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-06-01888-6
Published electronically: September 11, 2006
MathSciNet review: 2261031
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{K_m\}$ be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials $ P_m(x) =x^3 -mx^2-(m+3)x+1$ and $ P_m(x) =x^4 -mx^3-6x^2+mx+1$. We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields $ {\bf Q}(\zeta_p)^+$ of prime conductors $ p\ge 3$ and class numbers $ h_p^+$ greater than or equal to $ p$. However, in accordance with Vandiver's conjecture, we found no example of $ p$ for which $ p$ divides $ h_p^+$.


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

  • [BE71] B. C. Berndt and R. J. Evans.
    Sums of Jacobi, Gauss, and Jacobsthal.
    J. Number Theory 11 (1979), 349-398. MR 0544263 (81j:10054)
  • [BE82] -,
    The determination of Gauss sums.
    Bull. Amer. Math. Soc. 5 (2) (1981), 107-129. Corrigendum in 7 (2) (1982), 441. MR 0621882 (82h:10051); MR 0663795 (83i:10049)
  • [BEW] B. C. Berndt, R. J. Evans and K. S. Williams.
    Gauss and Jacobi sums.
    Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1998. MR 1625181 (99d:11092)
  • [Bye] D. Byeon.
    Class number $ 3$ problem for the simplest cubic fields.
    Proc. Amer. Math. Soc. 128 (2000), 1319-1323. MR 1664337 (2000j:11158)
  • [CW] G. Cornell and L. C. Washington.
    Class numbers of cyclotomic fields.
    J. Number Theory 21 (1985), 260-274. MR 0814005 (87d:11079)
  • [Dav] H. Davenport.
    Multiplicative Number Theory.
    Springer-Verlag, Grad. Texts Math., 74, Third Edition, 2000. MR 1790423 (2001f:11001)
  • [Gra1] M. N. Gras.
    Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré $ 4$ de $ {\bf Q}$.
    Publ. Math. Besancon, 1977/78, fasc. 2, pp1-26&1-53.
  • [Gra2] -,
    Special units in real cyclic sextic fields.
    Math. Comp. 48 (1988), 543-556. MR 0866107 (88m:11092)
  • [IR] K. Ireland and M. Rosen.
    A classical introduction to modern number theory. Second edition.
    Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1990. MR 1070716 (92e:11001)
  • [Jean] S. Jeannin.
    Nombre de classes et unités des corps de nombres cycliques quintiques d'E. Lehmer.
    J. Théor. Nombres Bordeaux 8 (1996), no. 1, 75-92. MR 1399947 (97k:11154)
  • [Lan] S. Lang.
    Algebraic Number Theory. Second edition.
    Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)
  • [Laz1] A. J. Lazarus.
    Class numbers of simplest quartic fields.
    Number theory (Banff, AB, 1988), 313-323, Walter de Gruyter, Berlin, 1990. MR 1106670 (92d:11119)
  • [Laz2] -,
    On the class number and unit index of simplest quartic fields.
    Nagoya Math. J. 121 (1991), 1-13. MR 1096465 (92a:11129)
  • [Laz3] -,
    Gaussian periods and units in certain cyclic fields.
    Proc. Amer. Math. Soc. 115 (1992), 961-968. MR 1093600 (92j:11118)
  • [Laz4] -,
    The sextic period polynomial.
    Bull. Austral. Math. Soc. 49 (1994), 293-304. MR 1265365 (95e:11118)
  • [Leh] E. Lehmer.
    Connection between Gaussian periods and cyclic fields.
    Math. Comp. 50 (1988), 535-541. MR 0929551 (89h:11067a)
  • [Lon] Robert L. Long.
    Algebraic Number Theory.
    Monographs and Textbooks in Pure and Applied Mathematics, Vol. 41. Marcel Decker, Inc., New York-Basel, 1977. MR 0469888 (57:9668)
  • [Lou1] S. Louboutin.
    Minoration au point $ 1$ des fonctions $ L$ et détermination des corps sextiques abéliens totalement imaginaires principaux.
    Acta Arith. 62 (1992), 109-124. MR 1183984 (93h:11100)
  • [Lou2] -,
    Computation of relative class numbers of CM-fields by using Hecke $ L$-functions.
    Math. Comp. 69 (2000), 371-393. MR 1648395 (2000i:11172)
  • [Lou3] -,
    The exponent three class group problem for some real cyclic cubic number fields.
    Proc. Amer. Math. Soc. 130 (2002), 353-361. MR 1862112 (2002h:11106)
  • [Lou4] -,
    Efficient computation of class numbers of real abelian number fields.
    Lect. Notes in Comp. Sci. 2369 (2002), 134-147. MR 2041079 (2005d:11182)
  • [Lou5] -,
    Computation of class numbers of quadratic number fields.
    Math. Comp. 71 (2002), 1735-1743. MR 1933052 (2003i:11163)
  • [Lou6] -,
    The simplest quartic fields with ideal class groups of exponents less than or equal to $ 2$.
    J. Math. Soc. Japan 56 (2004), 717-727. MR 2071669 (2005e:11137)
  • [Lou7] -,
    Class numbers of real cyclotomic fields.
    Publ. Math. Debrecen 64 (2004), 451-461. MR 2058916 (2005b:11172)
  • [LP] F. Lemmermeyer and A. Pethö.
    Simplest cubic fields.
    Manuscripta Math. 88 (1995), 53-58. MR 1348789 (96g:11131)
  • [Sha] D. Shanks.
    The simplest cubic fields.
    Math. Comp. 28 (1974), 1137-1152. MR 0352049 (50:4537)
  • [Sta] H. M. Stark.
    Dirichlet's class-number formula revisited.
    Contemp. Math. 143 (1993), 571-577. MR 1210543 (94a:11133)
  • [SW] R. Schoof and L. C. Washington.
    Quintic polynomials and real cyclotomic fields with large class numbers.
    Math. Comp. 50 (1988), 543-556. MR 0929552 (89h:11067b)
  • [SWW] E. Seah, L. C. Washington and H. C. Williams.
    The calculation of a large cubic class number with an application to real cyclotomic fields.
    Math. Comp. 41 (1983), 303-305. MR 0701641 (84m:12008)
  • [Tha] F. Thaine.
    Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers.
    Math. Comp. 69 (2000), 1653-1666. MR 1653998 (2001a:11179)
  • [Wa1] L. C. Washington.
    Class numbers of the simplest cubic fields.
    Math. Comp. 48 (1987), 371-384. MR 0866122 (88a:11107)
  • [Wa2] -,
    A family of cyclic quartic fields arising from modular curves.
    Math. Comp. 57 (1991), no. 196, 763-775. MR 1094964 (92a:11120)
  • [Wa3] -,
    Introduction to Cyclotomic Fields. Second edition.
    Graduate Texts in Mathematics, 83, Springer-Verlag, 1997. MR 1421575 (97h:11130)
  • [WB] H. C. Williams and J. Broere.
    A computational technique for evaluating $ L(1,\chi )$ and the class number of a real quadratic field.
    Math. Comp. 30 (1976), 887-893. MR0414522 (54:2623)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R16, 11R20, 11R29, 11R42, 11Y40

Retrieve articles in all journals with MSC (2000): 11R16, 11R20, 11R29, 11R42, 11Y40


Additional Information

Stéphane R. Louboutin
Affiliation: Institut de Mathématiques de Luminy, UMR 6206, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email: loubouti@iml.univ-mrs.fr

DOI: https://doi.org/10.1090/S0025-5718-06-01888-6
Keywords: Real abelian number field, class number, Gauss sums, simplest cubic field, simplest quartic field, simplest quintic field, simplest sextic field.
Received by editor(s): July 8, 2005
Received by editor(s) in revised form: October 14, 2005
Published electronically: September 11, 2006
Dedicated: Dedicated to Danièle B.
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society