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

Zbl 0533.10011
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References:

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