Bilu, Yuri; Masser, David; Zannier, Umberto An effective “Theorem of André” for \(CM\)-points on a plane curve. (English) Zbl 1263.14028 Math. Proc. Camb. Philos. Soc. 154, No. 1, 145-152 (2013). Yves André proved that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve \(Y_0(n)\) for some \(n\). His result is partially ineffective due to the use of Siegel’s class-number estimates.The purpose of the paper under review is to observe that in fact a modification of André leads to a completely effective result, and to work out an example. The example is \(XY=1\), i.e., there are no imaginary quadratic numbers \(\tau_1\) and \(\tau_2\) with \(j(\tau_1)\,j(\tau_2)=1\), where \(j\) is the classical modular function.Lars Kühne had independently obtained an effective version of André’s Theorem a bit earlier than the authors of this paper, but the concise exposition of effectiveness is the value of their work. The authors add at the end of paper whether there are any imaginary quadratic number \(\tau\) such that \(j(\tau)\) is a unit, and report that Habegger recently showed that there are at most finitely many. Reviewer: Sungkon Chang (Savannah) Cited in 2 ReviewsCited in 25 Documents MSC: 14G05 Rational points 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 11G15 Complex multiplication and moduli of abelian varieties 11G18 Arithmetic aspects of modular and Shimura varieties Keywords:theorem of André; CM points PDFBibTeX XMLCite \textit{Y. Bilu} et al., Math. Proc. Camb. Philos. Soc. 154, No. 1, 145--152 (2013; Zbl 1263.14028) Full Text: DOI References: [1] Husemöller, Elliptic Curves (1987) · doi:10.1007/978-1-4757-5119-2 [2] Gross, J. Reine Angew. Math. 355 pp 191– (1985) [3] DOI: 10.4007/annals.2012.176.1.13 · Zbl 1341.11035 · doi:10.4007/annals.2012.176.1.13 [4] Masser, Elliptic functions and transcendence (1975) · doi:10.1007/BFb0069432 [5] Lang, Elliptic Functions (1973) [6] DOI: 10.1515/crll.1998.505.203 · Zbl 0918.14010 · doi:10.1515/crll.1998.505.203 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.