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Diophantine approximation on semi-abelian varieties. (Approximation diophantienne sur les variétés semi-abéliennes.) (French) Zbl 1081.11053

An algebraic group \(A\) for which there exists an exact sequence of algebraic groups \(0\to {\mathbb G}_m^n\to A\to A_0\to 0\) where \(A_0\) is an algebraic variety and \({\mathbb G}_m^n\) the \(n\)-dimensional linear torus is called a semi-abelian variety. Let \(A\) be a semi-abelian variety defined over an algebraic closure \(\overline{{\mathbb Q}}\) of \({\mathbb Q}\), with group operation \(+\). It is possible to define a canonical height \(h_{\text{can}}\) on \(A(\overline{{\mathbb Q}})\). This canonical height can be decomposed as \(h_{\text{can}}=h_{\text{lin}}+h_{\text{quad}}\), where \(h_{\text{lin}}\) satisfies the triangle inequality on \(A(\overline{{\mathbb Q}})\), while \(h_{\text{quad}}\) is a positive definite quadratic form, basically a Néron-Tate height on \(A_0\).
Let \(\Gamma\) be a subgroup of \(A(\overline{{\mathbb Q}})\) of finite rank. For \(\varepsilon >0\) define the ‘cylinder’ \(\Gamma_{\varepsilon}:= \{ x+y:\, x\in\Gamma,\, y\in A(\overline{{\mathbb Q}}),\, h_{\text{can}}(y)\leq \varepsilon\}\) and the ‘cone’ \(C(\Gamma ,\varepsilon ):= \{ x+y:\, x\in\Gamma,\, y\in A(\overline{{\mathbb Q}}),\, h_{\text{can}}(y)\leq \varepsilon (1+h_{\text{can}}(x))\}\). Let \(X\) be a subvariety of \(A\), defined over \(\overline{{\mathbb Q}}\) such that \(X\) is not a translate of a semi-abelian subvariety of \(A\). It follows from work of Vojta and McQuillan that \(X(\overline{{\mathbb Q}})\cap \Gamma\) is not Zariski-dense in \(X\). In 1999, B. Poonen [Invent. Math. 137, 413–425 (1999; Zbl 0995.11040)] posed the following ‘Lang-Bogomolov’ conjecture, combining the above with a generalization of a conjecture of Bogomolov: There is \(\varepsilon >0\) such that \(X(\overline{{\mathbb Q}})\cap \Gamma_{{\varepsilon}}\) is not Zariski-dense in \(X\). This conjecture was proved independently by Poonen himself in the above paper and by S.-W. Zhang [Duke Math. J. 103, 39–46 (2000; Zbl 0972.11053)] in the case that \(A\) is isogenous to a product of a linear torus and an abelian variety. In this paper, the author proves the conjecture in full generality. More precisely, he proves the following result concerning the cone \(C(\Gamma ,\varepsilon )\) mentioned above. Let \(Z_X\) denote the union of all positive-dimensional semi-abelian subvarieties contained in \(X\). Then there is \(\varepsilon >0\) such that \((X\setminus Z_X)(\overline{{\mathbb Q}})\cap C(\Gamma ,\varepsilon )\) is finite.
The author’s proof has two main ingredients. The first is a result by S. David and P. Philippon, implying the Bogomolov conjecture for semi-abelian varieties, i.e., the above conjecture with \(\Gamma =\{ 0\}\) [see C. R. Acad. Sci., Paris, Sér. I Math. 331, 587–592 (2000; Zbl 0972.11059)]. The second is ‘the generalized Vojta inequality’, proved by the author in [Prépublication de l’Institut Fourier 584 (2003)], which is a generalization of an inequality proved by Vojta in his proof of the Mordell conjecture. In the special case considered here, the generalized Vojta inequality implies that if \(x_1,\ldots , x_m\) \((m=\dim X +1)\) are points in \(X(\overline{{\mathbb Q}})\) such that \(h_{\text{can}}(x_1)\) and the quotients \(h_{\text{can}}(x_{i+1})/h_{\text{can}}(x_i)\) \((i=1,\ldots m-1)\) are sufficiently large and such that \(x_1,\ldots , x_m\) ‘point almost in the same direction’, then at least one of \(x_1,\ldots ,x_m\) lies in \(Z_X(\overline{{\mathbb Q}})\).

MSC:

11J95 Results involving abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
11G50 Heights
14K15 Arithmetic ground fields for abelian varieties
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