Infinite global fields and the generalized Brauer--Siegel theorem

TL;DR

This paper develops a theory of infinite global fields, introduces invariants and zeta-functions for them, and generalizes the Brauer--Siegel theorem to include non-archimedean places and unramified towers, providing new asymptotic bounds.

Contribution

It introduces a new framework for infinite global fields, including invariants and zeta-functions, and extends classical bounds to broader cases, including unramified towers.

Findings

Generalized Odlyzko--Serre bounds for sequences of number fields.

Extended the Brauer--Siegel theorem without the standard assumptions.

Provided examples of class field towers demonstrating the necessity of assumptions.

Abstract

The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of ${\mathbb{Q}}$ or of ${\mathbb{F}}_r(t)$. We produce a series of invariants of such fields, and we introduce and study a kind of zeta-function for them. Second, for sequences of number fields with growing discriminant we prove generalizations of the Odlyzko--Serre bounds and of the Brauer--Siegel theorem, taking into account non-archimedean places. This leads to asymptotic bounds on the ratio ${{\log hR}/\log\sqrt{| D|}}$ valid without the standard assumption ${n/\log\sqrt{| D|}}\to 0,$ thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer--Siegel theorem to hold. As an easy consequence we ameliorate on existing bounds for regulators.