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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