There are at most finitely many singular moduli that are S-units
Number Theory
2023-09-07 v3 Algebraic Geometry
Dynamical Systems
Abstract
We show that for every finite set of prime numbers S, there are at most finitely many singular moduli that are S-units. The key new ingredient is that for every prime number p, singular moduli are p-adically disperse. We prove analogous results for the Weber modular functions, the lambda invariants and the McKay-Thompson series associated to the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.
Keywords
Cite
@article{arxiv.2102.05041,
title = {There are at most finitely many singular moduli that are S-units},
author = {Sebastián Herrero and Ricardo Menares and Juan Rivera-Letelier},
journal= {arXiv preprint arXiv:2102.05041},
year = {2023}
}
Comments
Improved version of Theorem A. Corrected Lemma 2.8(ii), which was false as stated in the previous version (this does not affect the main results). Accepted for publication in Compositio Mathematica