Commas, points extrémaux et arêtes des corps possédant une formule du produit. (Commas, extremal points and edges of fields with product formula). (French) Zbl 0578.12001

Hellegouarch, Y.; Paysant-le Roux, R.

C. R. Math. Acad. Sci., Soc. R. Can. 7, 291-296 (1985).

The authors define and investigate the notions of ”commas”, ”extremal points” and ”arrows” in a field E possessing a set of absolute values (”places”) with a product formula, as defined by E. Artin. Such points are elements of E with certain minimality properties with respect to a finite set S of places. For global fields E, upper bounds are given for the number of orbits of such sets under the action of the group of S- units, depending on
a) Riemann-Roch’s theorem (if E is a global function field);
b) Minkowski’s lemma (if E is a number field).

Reviewer: E.-U.Gekeler

MSC:

11R04	Algebraic numbers; rings of algebraic integers
11R58	Arithmetic theory of algebraic function fields
11H99	Geometry of numbers

Keywords:

commas; extremal points; arrows; absolute values; places; product formula; minimality properties; Riemann-Roch’s theorem; global function field; Minkowski’s lemma; number field