## Distribution of points of small height in multiplicative groups. (Distribution des points de petite hauteur dans les groupes multiplicatifs.)(French)Zbl 1150.11021

The authors enhance a main result of their earlier paper [J. Reine Angew. Math. 513, 145–179 (1999; Zbl 1011.11045)]. Suppose $$n$$ is a positive integer and $$V\subset{\mathbb G}_m^n$$ is an algebraic subvariety of $${\mathbb G}_m^n$$ defined over a cyclotomic extension $$K$$ of $${\mathbb Q}$$. Denote by $$V^*$$ the complement of the torsion subvarieties, i.e.of translates of proper subtori of $${\mathbb G}_m^n$$ by torsion points. Then there is a universal effectively computable positive real constant $$c(n)$$ such that if $$V$$ is the intersection of hypersurfaces of degree at most $$\delta$$, then for all $$\alpha\in V^*$$ $h(\alpha)\geq c(n)^{-1}\delta^{-1} \left[\log\big(3[K:{\mathbb Q}]\delta\big)\right]^{-\kappa(n)},$ where $$\kappa(n)=2n(n+1)!^n-1$$ (this value corrects a minor mistake in the paper cited above) and $$h$$ is the height, given by some fixed projective embedding of $${\mathbb G}_m^n$$. For $$K={\mathbb Q}$$ this is close to being best possible.
The proofs closely follow arguments from (loc. cit.), making the changes needed to replace $${\mathbb Q}$$ by $$K$$ and some other refinements.

### MSC:

 11G10 Abelian varieties of dimension $$> 1$$ 11J81 Transcendence (general theory) 14G40 Arithmetic varieties and schemes; Arakelov theory; heights

Zbl 1011.11045
Full Text:

### References:

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