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**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.

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.

Reviewer: Jan-Hendrik Evertse (Leiden)

### Citations:

Zbl 0938.11031
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\textit{E. Bombieri} et al., Mich. Math. J. 51, No. 3, 451--466 (2003; Zbl 1048.11056)

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