Notions relatives de régulateurs et de hauteurs. (Relative notions of regulators and of heights). (French) Zbl 0642.12011

Bergé, A.-M.; Martinet, J.

Acta Arith. 54, No. 2, 155-170 (1989).

In a previous paper [Théorie des Nombres, Sémin. Paris, exposé du 12 oct. 1987 (to appear)], we defined the relative regulator for an extension L/K of number fields by the formula \(R_{L/K}=Q_{L/K}R_ L/R_ K\), where \(Q_{L/K}\) is a “Hasse index”. We first prove in this paper a generalization to relative regulators of a theorem of J. H. Silverman [J. Number Theory 19, 437-442 (1984; Zbl 0552.12003)] which gives general lower bounds in terms of the norm of the relative discriminant of L/K (only primitive extensions were dealt with in our previous paper).
The proof involves an argument of diophantine geometry via the notion of height of an algebraic number. We took this opportunity to introduce, for a given number field K, the more suitable notion of a K-height for an algebraic number \(\theta\), namely \(H(K;\theta)=Inf H(\epsilon \theta\) \(n)^{1/n}\), n a natural integer, \(\epsilon\) a unit in K(H: the usual height; obvious generalizations to projective spaces).
The second part of our paper is devoted to the explicit calculation of this height, its properties (analogous to the usual ones, roots of unity being replaced by units with a power in K) and eventually to extensions of notions and problems classically attached to heights (P. V. and Salem numbers, Lehmer problem,...).

Reviewer: A.-M.Bergé

MSC:

11R99	Algebraic number theory: global fields
11R23	Iwasawa theory
14H05	Algebraic functions and function fields in algebraic geometry

Keywords:

relative regulator; Hasse index; relative discriminant; height; Salem numbers; Lehmer problem

Citations:

Zbl 0552.12003