An explicit André-Oort type result for P¹(C) x G_m(C) based on logarithmic forms

2014/03/12 by Roland Paulin, Paulin, Roland
Computer Science · Mathematics · #11G18 (Primary) 11J86 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11G18 #msc:11J86

paper · pdf · doi:10.48550/arxiv.1403.2949

9 pages

arxiv created 2014/03/12 · openalex publication_date 2014/03/12 · arxiv updated 2014/03/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Using linear forms in logarithms we prove an explicit result of André-Oort type for ℙ¹(ℂ) × \mathbbG_m(ℂ). In this variation the special points of ℙ¹(ℂ) × \mathbbG_m(ℂ) are of the form (α, λ), with α a singular modulus and λ a root of unity. The qualitative version of our result states that if C is a closed algebraic curve in ℙ¹(ℂ) × \mathbbG_m(ℂ), defined over a number field, not containing a horizontal or vertical line, then C contains only finitely many special points. The proof is based on linear forms in logarithms. This differs completely from the method used by the author recently in the proof of the same kind of statement, where class field theory was applied.

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