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.


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


Zbl 1011.11045
Full Text: EuDML


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