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On singular moduli that are S-units

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.

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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

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Correspondence to Francesco Campagna.

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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 \(-3f^2\).

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Campagna, F. On singular moduli that are S-units. manuscripta math. 166, 73–90 (2021). https://doi.org/10.1007/s00229-020-01230-1

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Mathematics Subject Classification

  • 11G15