Abstract
Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; we regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Artin, E., Hasse, H.: Die beiden Ergänzungssätze zum reziprozitätsgesetz der \(l^n\)-ten potenzreste im körper der \(l^n\)-ten Einheitswurzeln. Abh. Math. Sem. Univ. Hamburg 6(1), 146–162 (1928)
Brückner, H.: Eine explizite Formel zum Reziprozitätsgesetz für Primzahlexponenten \(p\), Algebraische Zahlentheorie (Ber. Tagung Math. Forschungsinst. Oberwolfach,: Bibliographisches Inst. Mannheim. 1967, 31–39 (1964)
Dainobu, N.: On an explicit reciprocity law in local class field theory via \((\varphi , {\Gamma })\)-modules, in preparation, (2022)
Hirano, H.: On mod 2 arithmetic Dijkgraaf-Witten invariants for certain real quadratic number fields. Osaka J. Math. (2022) (To appear)
Kim, J., Morishita, M., Noda, T., Terashima, Y.: On 3-dimensional foliated dynamical systems and hilbert type reciprocity law. Münster J. Math. 13, 323–348 (2021)
Kummer, E.E.: Über die allgemeinen Reziprozitätsgesetze der Potenzreste. J. Reine Angew. Math. 56, 270–279 (1858)
Mazur, B.: Primes, Knots and Po, Lecture Notes for the Conference “Geometry, Topology and Group Theory” . In: Hhonor of the 80th Birthday of Valentin Poenaru, (2012)
McMullen, C.T.: Knots which behave like the prime numbers. Compos. Math. 149(8), 1235–1244 (2013)
Mihara, T.: Cohomological approach to class field theory in arithmetic topology. Canad. J. Math. 71(4), 891–935 (2019)
Morishita, M.: Knots and primes: an introduction to arithmetic topology. In Universitext. Springer, London, (2012)
Neukirch, J.: Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. In: Harder, G. (ed.) Translated from the 1992 German Original and With a Note by Norbert Schappacher, vol. 322. Springer-Verlag, Berlin (1999)
Niibo, H.: Idèlic class field theory for 3-manifolds. Kyushu J. Math. 68(2), 421–436 (2014)
Niibo, H., Ueki, J.: Idèlic class field theory for 3-manifolds and very admissible links. Trans. Am. Math. Soc. 371(12), 8467–8488 (2019)
Spanier, E.H.: Algebraic Topology. McGraw-Hill Book Co., New York-Toronto (1966)
Ueki, J.: On the homology of branched coverings of 3-manifolds. Nagoya Math. J. 213, 21–39 (2014)
Ueki, J.: Chebotarev links are stably generic. Bull. Lond. Math. Soc. 53(1), 82–91 (2021)
Ueki, J.: Modular knots obey the chebotarev law, preprint. ar**v:2105.10745 (2021)
Acknowledgements
We are grateful to Masanori Morishita for raising an interesting problem, Tomoki Mihara for fruitful discussion, the organizers of the conference “Kyushu Algebraic Number Theory 2021 Spring –hybrid–” for their great hospitality, and the anonymous referees for essential comments. The second author has been partially supported by JSPS KAKENHI Grant Number JP19K14538.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Niibo, H., Ueki, J. A Hilbert reciprocity law on 3-manifolds. Res Math Sci 10, 3 (2023). https://doi.org/10.1007/s40687-022-00364-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40687-022-00364-w