Infinite global fields and the generalized Brauer--Siegel theorem

Details Infinite global fields and the generalized Brauer--Siegel theorem Infinite global fields and the generalized Brauer--Siegel theorem

2002-05-13

arxiv.org/abs/math/0205129v1

Mathematics

Number Theory

81 pages, to appear in Moscow Mathematical Journal, v. 2, No. 2

Scientific paper

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.

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