Lower bounds for normalized heights of subvarieties of tori. (Minorations des hauteurs normalisées des sous-variétés des tores.) (French) Zbl 1002.11055

David, Sinnou; Philippon, Patrice

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, No. 3, 489-543 (1999).

In a previous paper [Minorations des hauteurs normalisées des sous-variétés de variétés abéliennes, Contemp. Math. 210, 333-364 (1998; Zbl 0899.11027)] the authors investigated algebraic subvarieties \(V\) of an abelian variety \(A\) over the field \({\overline{\mathbb Q}}\) of algebraic numbers. They showed that if \(V\) has a sufficiently small height (in terms of its degree), then \(V\) is a torsion subvariety of \(A\), namely a translate by a torsion point of an abelian subvariety of \(A\). As a consequence the set of points on \(V({\overline{\mathbb Q}})\) of small height is not Zariski dense in \(V\).
Here they consider a similar question for tori \(\mathbb{G}_{m}^{g}\). They produce explicit lower bounds for the height of an algebraic subvariety defined over \({\overline{\mathbb Q}}\) which is not a translate of an algebraic subgroup of \(\mathbb{G}_{m}^{g}\). The bounds depend only on the degree of \(V\). Next they prove that for an algebraic subvariety \(V\) of \(\mathbb{G}_{m}^{g}\) over \({\overline{\mathbb Q}}\), the set of points \(x\) in \(V({\overline{\mathbb Q}})\), which belong to no translate of an algebraic subgroup of dimension \(\geq 1\) contained in \(V\), and which have sufficiently small height \(\widehat{h}(x)\), is finite. The upper bounds, for \(\widehat{h}(x)\) in the hypothesis and for the number of points in the conclusion, are explicitly given; they depend only on \(g\) and on the degree of \(V\). Also they give a lower bound for the height \(\widehat{h}(x)\) of a point \(x\in V({\overline{\mathbb Q}})\) which does not belong to a torsion subvariety included in \(V\), assuming \(V\) is defined over \(\mathbb Q\) and is \(\mathbb Q\)-irreducible. This refines earlier estimates due to W. M. Schmidt [Heights of points on subvarieties of \(\mathbb G_{m}^{n}\), Contemp. Math. 210, 97-99 (1998; Zbl 0897.11021)]. In particular these new estimates do not depend on the height of \(V\).
For the proof they first reduce the question to planar curves in \(\mathbb{G}_{m}^{2}\). Next they prove a new explicit estimate for the arithmetic Hilbert function, where the remainder term is explicit; the proof of this estimate requires a new absolute Siegel lemma.

Reviewer: Michel Waldschmidt (Paris)

MSC:

11G35	Varieties over global fields
11G50	Heights
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
11J81	Transcendence (general theory)
11J95	Results involving abelian varieties

Keywords:

torsion subvariety; explicit lower bounds; height; algebraic subvariety; heights of points; arithmetic Hilbert function; absolute Siegel lemma

Citations:

Zbl 0899.11027; Zbl 0897.11021

References:

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