Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-17T13:21:49.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Redei reciprocity, governing fields and negative Pell

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  19 April 2021

PETER STEVENHAGEN
PETER STEVENHAGEN*
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands e-mail: psh@math.leidenuniv.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .

MSC classification

Primary: 11R11: Quadratic extensions 11R37: Class field theory
Secondary: 11R16: Cubic and quartic extensions
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 3 , May 2022 , pp. 627 - 654
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

References

Boneh, D. and Silverberg, A.. Applications of multilinear forms to cryptography. Topics in algebraic and noncommutative geometry (Luminy/Annapolis, MD, 2001). Contemp. Math. vol. 324 (Amer. Math. Soc., Providence, RI, 2003), pp. 71–90.CrossRefGoogle Scholar
Bosma, W. and Stevenhagen, P.. Density computations for real quadratic units. Math. Comp. 65 (1996), no. 215, 1327–1337.CrossRefGoogle Scholar
Chan, S., Koymans, P., Milovic, D. and Pagano, C.. On the negative Pell equation, arXiv:1908.01752v1 (2019).Google Scholar
Cohn, H. and Lagarias, J. C.. On the existence of fields governing the 2-invariants of the classgroup of ${\bf Q}(\sqrt{dp})$ as p varies, Math. Comp. 41 (1983), no. 164, 711–730.Google Scholar
Corsman, J.. Rédei symbols and governing fields. ProQuest LLC, Ann Arbor, MI, 2007. PhD thesis McMaster University (Canada).Google Scholar
Fouvry, É. and Klüners, J.. On the negative Pell equation. Ann. of Math. (2) 172 (2010). no. 3, 2035–2104.CrossRefGoogle Scholar
Fouvry, É. and Klüners, J.. The parity of the period of the continued fraction of $\sqrt{d}$ , Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 337–391.CrossRefGoogle Scholar
Gärtner, J.. Rédei symbols and arithmetical mild pro-2-groups. Ann. Math. Qué. 38 (2014), no. 1, 13–36 (English, with English and French summaries).CrossRefGoogle Scholar
H. W. Lenstra Jr. Solving the Pell equation. Algorithmic number theory: lattices, number fields, curves and cryptography. Math. Sci. Res. Inst. Publ., vol. 44 (Cambridge Univerity Press, Cambridge, 2008) pp. 1–23.Google Scholar
Milovic, D.. On the 16-rank of class groups of ${\mathbf{Q}}(\sqrt{-8p})$ for $p \equiv -1$ mod 4, Geom. Funct. Anal. 27 (2017), no. 4, 973–1016.CrossRefGoogle Scholar
Koymans, P. and Milovic, D. Z.. Spins of prime ideals and the negative Pell equation $x^{2} - 2py^{2}=-1$ . Composition Math. 155 (2019), no. 1, 100–125.CrossRefGoogle Scholar
Mináć, J. and Duy Tân, N.. Construction of unipotent Galois extensions and Massey products. Adv. Math. 304 (2017), 1021–1054.CrossRefGoogle Scholar
Morishita, M.. Knots and Primes (Universitext, Springer, London, 2012). An introduction to arithmetic topology.CrossRefGoogle Scholar
Rédei, L. and Reichardt, H.. Die Anzahl der durch vier teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers J. Reine Angew. Math. 170 (1934), 69–74.Google Scholar
Rédei, L.. Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper. J. Reine Angew. Math. 171 (1934), 55–60.CrossRefGoogle Scholar
Rédei, L.. Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper. I. J. Reine Angew. Math. 180 (1939), 1–43.Google Scholar
Rieger, G. J.. Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke. J. Reine Angew. Math. 217 (1965), 200–216.CrossRefGoogle Scholar
Smith, A.. Governing fields and statistics for 4-Selmer groups and 8-class groups, ArXiv:1607.07860 (2016), 1–29.Google Scholar
Smith, A.. 2 $\infty$ -Selmer groups, 2 $\infty$ -class groups, and Goldfeld’s conjecture, ArXiv:1702.02325 (2017), 1–72.Google Scholar
Stevenhagen, P.. Ray class groups and governing fields, Théorie des nombres, Année 1988/89, Fasc. 1, Publ. Math. Fac. Sci. (Besançon, Univ. Franche-Comté, Besançon, 1989), pp. 1–93.CrossRefGoogle Scholar
Stevenhagen, P.. The number of real quadratic fields having units of negative norm. Experiment. Math. 2 (1993), no. 2, 121–136. MR1259426CrossRefGoogle Scholar
Stevenhagen, P.. On a problem of Eisenstein. Acta Arith. 74 (1996), no. 3, 259–268.CrossRefGoogle Scholar