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Cohen, Henri; Thorne, Frank (2013)
Publisher: University of Michigan, Department of Mathematics
Languages: English
Types: Article
Subjects: 11R29, 11R37, 11R16, Mathematics - Number Theory, [MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT], 11Y40
International audience; Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. In the present paper we carry the analysis further and prove explicit formulas for these Dirichlet series over Q. In a companion paper we do the same for quartic fields having a given cubic resolvent. As an application (not present in the initial version), we compute tables of the number of S_3-sextic fields E with |Disc(E)| < X, for X ranging up to 10^23. An accompanying PARI/GP implementation is available from the second author's website.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] J. Armitage and A. Fr¨ohlich, Class numbers and unit signatures, Mathematika 14 (1967), 94-98.
    • [2] K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), no. 219, 1213-1237; accompanying software available at http://www.math.u-bordeaux1.fr/~belabas/research/software/cubic-1.2.tgz.
    • [3] M. Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), no. 2, 1031-1063.
    • [4] M. Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. (2007), no. 17, Art. ID rnm052, 20 pp.
    • [5] M. Bhargava, Higher composition laws and applications, Proceedings of the International Congress of Mathematicians, Vol. II, 271294, Eur. Math. Soc., Zu¨rich, 2006.
    • [6] M. Bhargava, The density of discriminants of quintic rings and fields, Ann. Math. (2) 172 (2010), no. 3, 1559- 1591.
    • [7] M. Bhargava, A. Shankar, and J. Tsimerman, On the Davenport-Heilbronn theorem and second order terms, preprint; available at http://arxiv.org/abs/1005.0672
    • [8] H. Cohen, F. Diaz y Diaz, and M. Olivier, Counting discriminants of number fields, J. Th´eor. Nombres Bordeaux 18 (2006), no. 3, 573-593.
    • [9] H. Cohen and A. Morra, Counting cubic extensions with given quadratic resolvent, J. Algebra 325 (2011), 461- 478. (Theorem numbers refer to the published version, which are different than in the arXiv version.)
    • [10] H. Cohn, The density of abelian cubic fields, Proc. Amer. Math. Soc. 5 (1954), 476-477.
    • [11] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405-420.
    • [12] H. Hasse, Arithmetische Theorie der kubischen Zahlk¨orper auf klassenk¨orpertheoretischer Grundlage, Math. Zeit. 31 (1930), 565-582, and Math. Abhandlungen, Walter de Gruyter (1975), 423-440.
    • [13] A. Morra, Comptage asymptotique et algorithmique d'extensions cubiques relatives (in English), thesis, Universit´e Bordeaux I, 2009. Available online at http://perso.univ-rennes1.fr/anna.morra/these.pdf.
    • [14] J. Nakagawa, On the relations among the class numbers of binary cubic forms, Invent. Math. 134 (1998), no. 1, 101-138.
    • [15] Y. Ohno, A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms, Amer. J. Math. 119 (1997), no. 5, 1083-1094.
    • [16] D. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), no. 236, 1699-1705.
    • [17] T. Taniguchi and F. Thorne, Secondary terms in counting functions for cubic fields, submitted; available at http://arxiv.org/abs/1102.2914
    • [18] F. Thorne, Shintani's zeta function is not a finite sum of Euler products, submitted; preprint available at http://arxiv.org/pdf/1112.1397.
    • [19] F. Thorne, Four perspectives on a curious secondary term, submitted; preprint available at http://arxiv.org/abs/1202.3965.
    • [20] L. Washington, Introduction to cyclotomic fields (second edition), Springer-Verlag, New York, 1997. Universit´e Bordeaux I, Institut de Math´ematiques, U.M.R. 5251 du C.N.R.S, 351 Cours de la Lib´eration, 33405 TALENCE Cedex, FRANCE E-mail address: Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC 29208, USA E-mail address:
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