## 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

### Keywords:

semi-abelian varieties; Bogomolov-Lang conjecture

### Citations:

Zbl 0995.11040; Zbl 0972.11053; Zbl 0972.11059
Full Text:

### References:

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