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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.

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
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