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Effective bounds on differences of singular moduli that are S-units

Part of: Algebraic number theory: global fields Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  22 September 2022

FRANCESCO CAMPAGNA
FRANCESCO CAMPAGNA*
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany e-mail: [email protected]
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Abstract

Given a singular modulus $j_0$ and a set of rational primes S, we study the problem of effectively determining the set of singular moduli j such that $j-j_0$ is an S-unit. For every $j_0 \neq 0$ , we provide an effective way of finding this set for infinitely many choices of S. The same is true if $j_0=0$ and we assume the Generalised Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a “uniformity conjecture” for singular S-units.

MSC classification

Primary: 11G05: Elliptic curves over global fields 14K22: Complex multiplication 11G15: Complex multiplication and moduli of abelian varieties
Secondary: 11R52: Quaternion and other division algebras 11G50: Heights
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 2 , March 2023 , pp. 415 - 450
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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