## 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
Full Text:

### References:

  Chabauty, C.: Sur les équations diophantiennes liées aux unités d’un corps de nombres algébriques fini. Annali di Math.17, 127-168 (1938) · Zbl 0019.00303  Evertse, J.H.: On sums ofS-units and linear recurrences. Compositio Math. (à paraître) · Zbl 0547.10008  Lang, S.: Fundamentals of diophantine geometry. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0528.14013  Liardet, P.: Sur une conjecture de S. Lang. Astérisque, no 24-25. Soc. Math. Fr 187-209 (1975) · Zbl 0315.14005  Mann, H.B.: On linear relations between roots of unity. Mathematika, London12, 107-117 (1965) · Zbl 0138.03102  Ono, T.: Arithmetic of algebraic tori. Annals of Math.74, 101-139 (1961) · Zbl 0119.27801  Samuel, P.: A propos du théorème des unités. Bull. Sci. Math.90, 89-96 (1966) · Zbl 0166.30701  Schlickewei, H.P.: Thep-adic Thue-Siegel-Roth-Schmidt theorem. Archiv der Math.29, 267-270 (1977) · Zbl 0365.10026  Schlickewei, H.P.: On norm-form equations. J. of Number Theory9, 370-380 (1977) · Zbl 0365.10016  Schlickewei, H.P., Van der Poorten, A.J.: The growth conditions for recurrences sequences, Report 82.0041, Macquarie University, N.S.W. Australia 1982  Schmidt, W.: Diophantine approximation. Lecture Notes in Mathematics vol. 785. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0421.10019  Waldschmidt, M.: Nombres transcendants. Lecture Notes in Mathematics vol. 402. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0302.10030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.