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Intersecting a curve with algebraic subgroups of multiplicative groups. (English) Zbl 0938.11031
Let \(C\) be a closed absolutely irreducible curve in \({\mathbb G}_m^{n}\) with \(n\geq 2\), defined over the field \(\overline{\mathbb Q}\) of algebraic numbers, and not contained in a translate of a proper algebraic subtorus of \({\mathbb G}_m^{n}\). Then the union of all sets \(C\cap H(\overline{\mathbb Q})\), for \(H\) ranging over all proper algebraic subgroups of \({\mathbb G}_m^{n}\), is a set of bounded height.
A concrete example has been worked out by P. Cohen and U. Zannier, namely with \(n=2\) and the curve \(x+y=1\): the height of any algebraic number \(x\not\in\{0,1\}\) for which \(x\) and \(1-x\) are multiplicatively dependent is at most \(\log 2\) [ Multiplicative dependence and bounded height, an example. Halter-Koch, Franz (ed.) et al., Algebraic number theory and diophantine analysis. Proceedings of the international conference, Graz, Austria, 1998. Berlin: Walter de Gruyter, 93–101 (2000; Zbl 0959.11032)].
The authors also show that under the same assumptions the points in \(C\cap H(\mathbb C)\) for \(H\) ranging over all algebraic subgroups of \({\mathbb G}_m^{n}\) with \(\dim H\leq n-2\) is a finite set.
The proofs combine properties of the height with geometry of numbers. They also use a generalization, due to F. Amoroso and S. David, of Dobrowolski’s theorem to multiplicatively independent numbers [J. Reine Angew. Math. 513, 145–179 (1999; Zbl 1011.11045); see also C. R. Acad. Sci., Paris, Sér. I, Math. 326, 1163–1166 (1998; Zbl 0927.11042)].

MSC:
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G50 Heights
14G05 Rational points
11D61 Exponential Diophantine equations
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