Abstract
Recently Yu. Bilu, P. Habegger and L. Kühne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-unit. Here we prove that if S is the set of all primes p congruent to 1 modulo 3, no singular modulus is an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.
Similar content being viewed by others
References
Bilu, Yu., Habegger, P., Kühne, L.: No singular modulus is a unit. Int. Math. Res. Not. IMRN (2018) rny274, https://doi.org/10.1093/imrn/rny274
Bilu, Yu., Masser, D., Zannier, U.: An effective “Theorem of André” for CM points on a plane curve. Math. Proc. Camb. Philos. Soc. 154, 145–152 (2013)
Cox, D.: Primes of the Form \(x^2+ny^2\), 2nd edn. Wiley, Hoboken (2013)
Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörpen. M. Abh. Math. Semin. Univ. Hambg. 14, 197–272 (1941)
Gross, B., Zagier, D.: On singular moduli. J. Reine Angew. Math 355, 191–220 (1984)
Habegger, P.: Singular moduli that are algebraic units. Algebra Number Theory 9(7), 1515–1524 (2015)
Herrero, S., Menares, R., Rivera-Letelier, J.: p-Adic distribution of CM points and Hecke orbits. I. Convergence towards the Gauss point (2020). arXiv:2002.03232
Kühne, L.: An effective result of André–Oort type. Ann. Math. 176, 651–671 (2012)
Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2. B. G. Teubner, Leipzig und Berlin (1909)
Lang, S.: Elliptic Functions. Graduate Texts in Mathematics, vol. 112, 2nd edn. Springer, New York (1987)
Lauter, K., Viray, B.: On singular moduli for arbitrary discriminants. Int. Math. Res. Not. IMRN 2015(19), 9206–9250 (2014)
Li, Y.: Singular units and isogenies between CM elliptic curves (2019). arXiv:1810.13214
Montgomery, H., Vaughan, R.: Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2006)
SageMath: The Sage Mathematics Software System (Version 0.4.2), The Sage Developers (2019). https://wiki.sagemath.org/Publications_using_SageMath
Serre, J.P., Tate, J.: Good reduction of abelian varieties. Ann. Math. 88, 492–517 (1968)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106, 2nd edn. Springer, New York (2009)
Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, New York (1994)
Schwartz, W.: Über die Summe \(\sum _{n\le x} \varphi (f(n))\) und verwandte Probleme. Monaths. Math. 66, 43–54 (1962)
Acknowledgements
The author would like to thank his supervisor Fabien Pazuki, for his guidance and advice, and Philipp Habegger for the helpful suggestions and comments. He would also like to thank Riccardo Pengo and Peter Stevenhagen for the useful discussions, and the anonymous referee for the careful reading and the many insightful comments.
Funding
This project has received funding from the European Union Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 801199
.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Some numerical computations
Appendix: Some numerical computations
In this appendix we collect in a table some numerical computations, obtained using SAGE [14], concerning the norm factorizations for singular moduli of discriminant \(-3f^2\). In the first column of the table we list the conductors f of different orders of complex multiplication inside \(\mathbb {Q}(\sqrt{-3})\); in the second column we compute, up to a sign, the norm factorizations of the corresponding singular moduli (since singular moduli relative to the same order form a Galois orbit in \(\overline{\mathbb {Q}}\), they all have the same norm). The factorizations are obtained simply by factoring the constant term in the Hilbert class polynomial of discriminant \(-3f^2\).
Rights and permissions
About this article
Cite this article
Campagna, F. On singular moduli that are S-units. manuscripta math. 166, 73–90 (2021). https://doi.org/10.1007/s00229-020-01230-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-020-01230-1