Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-27T03:24:20.125Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE $4$-RANK OF CLASS GROUPS OF DIRICHLET BIQUADRATIC FIELDS

Part of: Algebraic number theory: global fields Multiplicative number theory

Published online by Cambridge University Press:  22 December 2020

Étienne Fouvry ,
Peter Koymans Open the ORCID record for Peter Koymans [Opens in a new window]  and
Carlo Pagano
Étienne Fouvry
Affiliation:
Université Paris–Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405Orsay, France (Etienne.Fouvry@universite-paris-saclay.fr)
Peter Koymans
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111Bonn, Germany (koymans@mpim-bonn.mpg.de)
Carlo Pagano
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111Bonn, Germany (carlein90@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for $100\%$ of the odd, square free integers $n> 0$ , the $4$ -rank of $\text {Cl}(\mathbb{Q} (i, \sqrt {n}))$ is equal to $\omega _3(n) - 1$ , where $\omega _3$ is the number of prime divisors of n that are $3$ modulo $4$ .

Keywords

class groupsarithmetic statisticsbiquadratic fields

MSC classification

Primary: 11N45: Asymptotic results on counting functions for algebraic and topological structures
Secondary: 11R29: Class numbers, class groups, discriminants
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1543 - 1570
Check for updates
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Azizi, A., Zekhnini, A. and Taous, M., On the 2–class field tower of $\mathbb{Q}\left(\sqrt{p_1{p}_2},i\right)$ and the Galois group of its second Hilbert 2–class field, Collect. Math. 65(1) (2014), 131–141.Google Scholar
Azizi, A., Zekhnini, A. and Taous, M., On the unramified quadratic and biquadratic extensions of the field $\mathbb{Q}\left(\sqrt{d},i\right)$ , Int. J. Algebra 6(21–24) (2012), 1169–1173.Google Scholar
Azizi, A., Zekhnini, A. and Taous, M., Structure of $\mathsf{Gal}\left({k}_2^{(2)}/ k\right)$ for some fields $k=\mathbb{Q}\left(\sqrt{2{p}_1{p}_2},i\right)$ with ${\mathsf{Cl}}_2(k)\cong \left(2,2,2\right)$ , Abh. Math. Semin. Univ. Hambg. 84(2) (2014), 203–231.CrossRefGoogle Scholar
Azizi, A., Zekhnini, A. and Taous, M., Coclass of $\mathsf{Gal}\left({k}_2^{(2)}/ k\right)$ for some fields $k=\mathbb{Q}\left(\sqrt{p_1{p}_2q},\sqrt{-1}\right)$ with 2–class groups of types $\left(2,2,2\right)$ , J. Algebra Appl. 15(2) (2016), 1650027.Google Scholar
Azizi, A., Zekhnini, A. and Taous, M., On the strongly ambiguous classes of some biquadratic number fields, Math. Bohem. 141(3) (2016), 363–384.Google Scholar
Azizi, A., Zekhnini, A. and Taous, M., On the capitulation of the $2$ –ideal classes of the field $\mathbb{Q}\left(\sqrt{p_1{p}_2q},i\right)$ of type $\left(2,2,2\right)$ , Bol. Soc. Parana. Mat. (3) 38(4) (2020), 127–135.Google Scholar
Azizi, A., Zekhnini, A., Taous, M. and Mayer, D. C., Principalization of 2–class groups of type $\left(2,2,2\right)$ of biquadratic fields $\mathbb{Q}\left(\sqrt{p_1{p}_2q},\sqrt{-1}\right)$ , Int. J. Number Theory 11(4) (2015), 1177–1215.CrossRefGoogle Scholar
Bartel, A. and Lenstra, H. W. Jr., On class groups of random number fields. Proceedings of the London Mathematical Society, Preprint, 2018, https://arxiv.org/abs/1803.06903 Google Scholar
Bhargava, M., The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162(2) (2005), 1031–1063.Google Scholar
Bhargava, M., Shankar, A. and Tsimerman, J., On the Davenport–Heilbronn theorems and second order terms, Invent. Math. 193(2) (2013), 439–499.CrossRefGoogle Scholar
Bhargava, M. and Varma, I., On the mean number of $2$ -torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields, Duke Math. J. 164(10) (2015), 1911–1933.Google Scholar
Cohen, H. and Lenstra, H. W. Jr., Heuristics on Class Groups of number fields, Number Theory, Noordwijkerhout 1983, 33–62, Lecture Notes in Math., 1068 (Springer, Berlin, 1984).Google Scholar
Cohen, H. and Martinet, J., Class groups of number fields: numerical heuristics, Math. Comp. 48(177) (1987), 123–137.Google Scholar
Davenport, H. and Heilbronn, H., On the density of discriminants of cubic fields: II, Proc. A 322(1551) (1971), 405–420.Google Scholar
Dirichlet, G.L., Recherches sur les formes quadratiques à coëfficients et à indéterminées complexes: première partie, J. Reine Angew. Math. 24 (1842), 291–371.Google Scholar
Fouvry, É. and Klüners, J., On the 4-rank of class groups of quadratic number fields, Invent. Math. 167(3) (2007), 455–513.Google Scholar
Fouvry, É. and Klüners, J., On the negative Pell equation, Ann. of Math. (2) 172(3) (2010), 2035–2104.Google Scholar
Fouvry, É. and Klüners, J., The parity of the period of the continued fraction of $\sqrt{d}$ , Proc. Lond. Math. Soc. (3) 101(2) (2010), 337–391.CrossRefGoogle Scholar
Fouvry, É. and Klüners, J., Weighted distribution of the 4-rank of class groups and applications, Int. Math. Res. Not. 2011(16) (2011), 3618–3656.CrossRefGoogle Scholar
Fouvry, É. and Koymans, P., On Dirichlet biquadratic fields, Preprint, 2020, https://arxiv.org/abs/2001.05350 Google Scholar
Fröhlich, A., Central Extensions, Galois Groups and Ideal Class Groups of Number Fields , Contemp. Math., 24 (American Mathematical Society, Providence, 1983).Google Scholar
Ho, W., Shankar, A. and Varma, I., Odd degree number fields with odd class number, Duke Math. J. 167(5) (2018), 995–1047.Google Scholar
Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory , 2nd ed., Grad. Texts in Math., 84 (Springer-Verlag, New York, 1990).Google Scholar
Klys, J., The distribution of $p$ –torsion in degree $p$ cyclic fields. Algebra and Number Theory, Preprint. 2016, https://arxiv.org/abs/1610.00226 Google Scholar
Koymans, P. and Pagano, C., On the distribution of $\mathsf{Cl}(K)\left[{l}^{\infty}\right]$ for degree $l$ cyclic fields, Preprint, 2018, https://arxiv.org/abs/1812.06884 Google Scholar
Koymans, P. and Pagano, C., Higher genus theory, Preprint, 2019, https://arxiv.org/abs/1909.13871 Google Scholar
Lemmermeyer, F., Reciprocity Laws: From Euler to Eisenstein, Monogr. Math. (Springer-Verlag, Berlin, 2000).CrossRefGoogle Scholar
Liu, Y., Wood, M. M. and Zureick-Brown, D., A predicted distribution for Galois groups of maximal unramified extensions, Preprint, 2019, https://arxiv.org/abs/1907.05002 Google Scholar
Montgomery, H.L. and Vaughan, R.C., Multiplicative Number Theory: I. Classical Theory, Cambridge Stud. Adv. Math., 97 (Cambridge University Press, Cambridge, 2007).Google Scholar
Pagano, C. and Sofos, E., 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields, Preprint, 2017, https://arxiv.org/abs/1710.07587 Google Scholar
Smith, A., ${2}^{\infty }$ -Selmer groups, ${2}^{\infty }$ -class groups, and Goldfeld’s conjecture, Preprint, 2017, https://arxiv.org/abs/1702.02325 Google Scholar
Taniguchi, T. and Thorne, F., Secondary terms in counting functions for cubic fields, Duke Math. J. 162(13) (2013), 2451–2508.Google Scholar
Varma, I., The mean number of 3-torsion elements in ray class groups of quadratic fields, forthcoming in Israel J. Math. Google Scholar
Wang, W. and Wood, M. M., Moments and interpretations of the Cohen-Lenstra-Martinet heuristics, Preprint, 2019, https://arxiv.org/abs/1907.11201 Google Scholar