Finiteness results for multiplicatively dependent points on complex curves. (English) Zbl 1048.11056

Denote by \({\mathbb G}_m^n\) the \(n\)-dimensional linear torus and for \(r<n\), denote by \({\mathcal H}_r\) the union of all algebraic subgroups of \({\mathbb G}_m^n\) of dimension at most \(r\). Further, let \(K\) be a field of characteristic \(0\), and \(\overline{K}\) the algebraic closure of \(K\). In the paper being reviewed here, the authors prove the following result.
Theorem. Let \(C\) be an irreducible curve in \({\mathbb G}_m^n\) defined over \(\overline{K}\) which is not contained in any translate of an irreducible algebraic subgroup of \({\mathbb G}_m^n\) of dimension \(n-1\). Then \(C\cap {\mathcal H}_{n-2}\) has at most finitely many \(\overline{K}\)-rational points.
In a previous paper [Int. Math. Res. Not. 1999, 1119–1140 (1999; Zbl 0938.11031)] the authors proved this result but only for \(K={\mathbb Q}\).
One of the ingredients in the proof of the theorem is the following result which is of independent interest: Suppose that the field \(K\) introduced above is finitely generated over \({\mathbb Q}\). Let \(C\) be as in the theorem. Let \(D\) be a positive integer. Then \(C\cap {\mathcal H}_{n-1}\) has at most finitely many \(\overline{K}\)-rational points \(P\) which are defined over an extension of \(K\) of degree at most \(D\).
Further, the authors observe that \(C\cap {\mathcal H}_{n-1}\) always has infinitely many \(\overline{K}\)-rational points. Therefore, in the last mentioned result, the bound on the degree of the field defined by \(P\) is necessary.


11G99 Arithmetic algebraic geometry (Diophantine geometry)
14H99 Curves in algebraic geometry


Zbl 0938.11031
Full Text: DOI


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