Sur le th\'eor\`eme de Brauer-Siegel g\'en\'eralis\'e

TL;DR

This paper extends the Brauer-Siegel theorem to broader families of number fields by introducing Galois complexity, unifying previous results, and proving a new principle related to zeta functions and exceptional zeroes.

Contribution

It introduces Galois complexity for number field extensions and proves the generalized Brauer-Siegel theorem under controlled complexity growth, unifying prior results.

Findings

Proves the generalized Brauer-Siegel theorem for new families of number fields.

Introduces Galois complexity as a key concept in the theorem's extension.

Establishes a new version of Stark's principle for exceptional zeroes.

Abstract

We extend the Brauer-Siegel theorem to new families of number fields, both in the classical setting of asymptotically bad families and in the more general framework due to Tsfasman and Vl\u{a}du\c{t} of asymptotically exact families. We introduce a notion of Galois complexity for extensions of number fields, and show that the generalized Brauer-Siegel theorem, as conjectured by Tsfasman and Vl\u{a}du\c{t}, holds for families in which the complexity does not grow too fast. This allows to unify and extend all previously known results. The crucial step in our work is the proof of a new version -- stated in terms of our Galois complexity -- of a fundamental principle due to Stark descending exceptional zeroes of zeta functions down to quadratic number fields. Among the hitherto unknown cases we are able to treat are the families of number fields contained in the solvable Galois closure of a given number field.