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Exponential diophantine equations. (Équations diophantiennes exponentielles.) (French) Zbl 0554.10009
Let \(\mathbb G\) be a commutative algebraic group over \(\mathbb C\), not containing any algebraic subgroup isomorphic to the additive group \(\mathbb G_ a\). Let \(\Gamma\) be a subgroup of \(\mathbb G(\mathbb C)\) of finite rank, that is, there is a finitely generated subgroup \(\Gamma'\) of \(\Gamma\) such that all elements of \(\Gamma/\Gamma'\) have finite order. Let \(V\) be a subvariety of \(\mathbb G\). Lang conjectured: The set \(V\cap \Gamma\) is a finite union of subsets of the form \(\gamma (H\cap \Gamma)\), where \(\gamma\) denotes an element of \(\Gamma\) and \(H\) is an algebraic subgroup of \(\mathbb G\), such that \(\gamma H\subseteq V\).
The author proves this conjecture for the case \(\mathbb G=(\mathbb C^*)^ n\), where \(\mathbb C^*\) is the multiplicative group of non-zero complex numbers. The proof consists of an ingenious use of arguments from Kummer theory and a result of Evertse on sums of \(S\)-units. The author also gives a quantitative version, of which we mention one interesting application. It characterizes solutions of systems of exponential equations of the form \(\sum_{\mu}Q_ k(\mu)a^{m_ 1}_{1,k}\cdots a^{m_ r}_{r,k}=0\), indexed by \(k\), and where \(a_{i,k}\) are non-zero algebraic numbers, \(Q_ k\) polynomials with algebraic coefficients, and \(\mu =(m_ 1,m_ 2,\ldots,m_ r)\) are the unknown rational integers. Unfortunately, the precise result is too long to quote here. \(\{\) A short version was published in C. R. Acad. Sci., Paris, Sér. I 296, 945–947 (1983; Zbl 0533.10011)\(\}\).

MSC:
11G35 Varieties over global fields
11D61 Exponential Diophantine equations
14L40 Other algebraic groups (geometric aspects)
14G25 Global ground fields in algebraic geometry
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References:
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