# zbMATH — the first resource for mathematics

Infinite global fields and the generalized Brauer-Siegel theorem. (English) Zbl 1004.11037
An infinite global field $$\mathcal K$$ is an infinite algebraic extension of a global field, namely it is either an infinite algebraic extension of the rational numbers, or an infinite algebraic extension of the rational function field in one variable over a finite field $${\mathbb{F}}_r$$ such that $$\mathcal K\cap\overline{\mathbb{F}}_r={\mathbb{F}}_r$$. In this paper, the authors study several invariants, and introduce and study a kind of zeta-function for infinite global fields. Among other things they obtain a generalization of the so called Odlyzko-Serre inequality on discriminants of number fields as well as a generalization of the Drinfeld-Vladut theorem for the number of points of degree one on algebraic curves over the finite field $$\mathbb{F}_r$$. Another interesting result is a generalization of the Brauer-Siegel theorem on certain sequences of global fields.

##### MSC:
 11G20 Curves over finite and local fields 11R42 Zeta functions and $$L$$-functions of number fields 11R37 Class field theory 14G05 Rational points 14G15 Finite ground fields in algebraic geometry 14H05 Algebraic functions and function fields in algebraic geometry
Full Text: