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An effective “Theorem of André” for \(CM\)-points on a plane curve. (English) Zbl 1263.14028

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.

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
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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
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