Exponential diophantine equations. (Équations diophantiennes exponentielles.) (French) Zbl 0554.10009

Laurent, Michel

Invent. Math. 78, 299-327 (1984).

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)\(\}\).

Reviewer: Frits Beukers (Utrecht)

MSC:

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

Keywords:

Lang conjecture; commutative algebraic group; quantitative version; solutions of systems of exponential equations; algebraic numbers; polynomials with algebraic coefficients

Citations:

Zbl 0533.10011

References:

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