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)\(\}\).


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: DOI EuDML


[1] 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 · doi:10.1007/BF02410698
[2] Evertse, J.H.: On sums ofS-units and linear recurrences. Compositio Math. (à paraître) · Zbl 0547.10008
[3] Lang, S.: Fundamentals of diophantine geometry. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0528.14013
[4] Liardet, P.: Sur une conjecture de S. Lang. Astérisque, no 24-25. Soc. Math. Fr 187-209 (1975) · Zbl 0315.14005
[5] Mann, H.B.: On linear relations between roots of unity. Mathematika, London12, 107-117 (1965) · Zbl 0138.03102 · doi:10.1112/S0025579300005210
[6] Ono, T.: Arithmetic of algebraic tori. Annals of Math.74, 101-139 (1961) · Zbl 0119.27801 · doi:10.2307/1970307
[7] Samuel, P.: A propos du théorème des unités. Bull. Sci. Math.90, 89-96 (1966) · Zbl 0166.30701
[8] Schlickewei, H.P.: Thep-adic Thue-Siegel-Roth-Schmidt theorem. Archiv der Math.29, 267-270 (1977) · Zbl 0365.10026 · doi:10.1007/BF01220404
[9] Schlickewei, H.P.: On norm-form equations. J. of Number Theory9, 370-380 (1977) · Zbl 0365.10016 · doi:10.1016/0022-314X(77)90072-5
[10] Schlickewei, H.P., Van der Poorten, A.J.: The growth conditions for recurrences sequences, Report 82.0041, Macquarie University, N.S.W. Australia 1982
[11] Schmidt, W.: Diophantine approximation. Lecture Notes in Mathematics vol. 785. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0421.10019
[12] Waldschmidt, M.: Nombres transcendants. Lecture Notes in Mathematics vol. 402. Berlin-Heidelberg-New York: Springer 1974 · Zbl 0302.10030
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