Brauer-Siegel theorem for families of number fields over almost Sn fields

TL;DR

The paper develops a descent principle for Brauer-Siegel in number-field families by passing from base fields that are almost $S_n$-fields to quadratic extensions, enabling Siegel-type results to be extended to varying base fields. It proves BS for quadratic extensions over almost $S_n$-fields and, in the Tsfasman-Vladut framework, establishes GBS for asymptotically good towers built over such bases. It also shows that, conditional on Artin holomorphy, BS holds for families of $S_n$-fields, broadening the contexts in which BS is known. Collectively, the work broadens the reach of Brauer-Siegel results beyond fixed-base reductions and connects zero-free regions, L-functions, and group-theoretic decompositions in a unified descent approach.

Abstract

The classical Brauer-Siegel conjecture describes the asymptotic behaviour of the product of the class number and the regulator in families of number fields. All known cases of the conjecture rely on reducing the problem, via group theoretic methods, to Siegel's theorem for quadratic fields over Q or over a fixed base field. In this paper, we establish a new form of descent for the Brauer-Siegel conjecture. We show that if the conjecture holds for a family of almost Sn-fields, it necessarily holds for all quadratic extensions over that family, under mild conditions. This result may be viewed as an analogue of Siegel's theorem in which the base field is allowed to vary. In addition, we also establish the generalized Brauer-Siegel conjecture as formulated by Tsfasman-Vladut for asymptotically good towers of number fields over a family of almost Sn-fields.

Figures (3)

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Theorems & Definitions (21)

• Conjecture 1.1: BS
• Definition 1.3
• Theorem 1.4
• Conjecture 1.5: GBS
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5