The higher-dimensional Lehmer problem. (Le problème de Lehmer en dimension supérieure.) (French) Zbl 1011.11045

Amoroso, Francesco; David, Sinnou

J. Reine Angew. Math. 513, 145-179 (1999).

Summary: We study a higher-dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if \(\alpha_1,\dots, \alpha_n\) are multiplicatively independent algebraic numbers, we provide a lower bound for the product of the heights of the \(\alpha_i\)’s in terms of the degree \(D\) of the number field generated by the \(\alpha_i\)’s. This enables us to study the successive minima for the height function in a given number field. Our bound is a generalization of an earlier result of E. Dobrowolski [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)] and is best possible up to a power of \(\log(D)\). This, in particular, shows that the Lehmer problem is true for number fields having a ‘small’ Galois group.
The main result bases on two theorems, first an analogue of Dobrowolski’s key lemma and second a version of Philippon’s zero estimates with multiplication.

MSC:

11G50	Heights
11J95	Results involving abelian varieties
14G40	Arithmetic varieties and schemes; Arakelov theory; heights

Keywords:

Siegel lemma; extrapolation; rank estimate; higher-dimensional Lehmer problem; power of the multiplicative group; lower bound; heights; successive minima for the height function

Citations:

Zbl 0416.12001