Equations in three singular moduli: The equal exponent case

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Authors

  • Guy Fowler
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Details

Original languageEnglish
Pages (from-to)256-297
Number of pages42
JournalJournal of number theory
Volume243
Early online date20 Oct 2022
Publication statusPublished - Feb 2023

Abstract

Let a∈Z>0 and ϵ123∈{±1}. We classify explicitly all singular moduli x1,x2,x3 satisfying either ϵ1x1a2x2a3x3a∈Q or (x1ϵ1x2ϵ2x3ϵ3)a∈Q×. In particular, we show that all the solutions in singular moduli x1,x2,x3 to the Fermat equations x1a+x2a+x3a=0 and x1a+x2a−x3a=0 satisfy x1x2x3=0. Our proofs use a generalisation of a result of Faye and Riffaut on the fields generated by sums and products of two singular moduli, which we also establish.

Keywords

    Andre–Oort conjecture, Singular moduli

ASJC Scopus subject areas

Cite this

Equations in three singular moduli: The equal exponent case. / Fowler, Guy.
In: Journal of number theory, Vol. 243, 02.2023, p. 256-297.

Research output: Contribution to journalArticleResearchpeer review

Fowler G. Equations in three singular moduli: The equal exponent case. Journal of number theory. 2023 Feb;243:256-297. Epub 2022 Oct 20. doi: 10.48550/arXiv.2105.12696, 10.1016/j.jnt.2022.09.012
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