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

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.