On singular moduli that are S-units

Francesco Campagna

doi:10.1007/s00229-020-01230-1

On singular moduli that are S-units

Francesco Campagna

Department of Mathematical Sciences

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Abstract

Recently Yu. Bilu, P. Habegger and L. K\"uhne 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 $S$-units. Here we prove that when the set $S$ contains only primes congruent to 1 modulo 3 then no singular modulus can be 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.

Original language	English
Journal	Manuscripta Mathematica
Volume	166
Pages (from-to)	73–90
ISSN	0025-2611
DOIs	https://doi.org/10.1007/s00229-020-01230-1
Publication status	Published - 2021

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10.1007/s00229-020-01230-1

On singular moduli that are S-unitsAccepted author manuscript, 335 KBLicence: Other

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AB - Recently Yu. Bilu, P. Habegger and L. K\"uhne 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 $S$-units. Here we prove that when the set $S$ contains only primes congruent to 1 modulo 3 then no singular modulus can be 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.

U2 - 10.1007/s00229-020-01230-1

DO - 10.1007/s00229-020-01230-1

M3 - Journal article

SN - 0025-2611

VL - 166

SP - 73

EP - 90

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

ER -