Number Theory Seminar

Speaker: Richard Griffon (Université Clermont-Auvergne)

Title: New cases of the generalised Brauer—Siegel theorem

Abstract: Given an infinite sequence S of numbers fields, one may try to describe, as K runs through S, the asymptotic behaviour of the product of the class number of K by the regulator of units of K, in terms of the discriminant of K. The classical Brauer—Siegel theorem completely solves this problem in case the number fields in S have bounded degree. Much more recently, Tsfasman and Vladuts formulated a conjectural answer to this question for more general sequences S. If true, their conjecture, call it BS-TV, would provide a vast generalisation of the Brauer—Siegel theorem. BS-TV is known to hold conditionally to GRH, but also unconditionally in a limited number of cases (for instance, Lebacque and Zykin have shown that it holds for asymptotically exact families of stepwise Galois number fields). In this talk, I will report on a recent joint work with Philippe Lebacque, where we exhibit new families of number fields where BS-TV is true. In particular, we prove that any infinite global field contained in a p-class field tower satisfies the generalised form of the Brauer—Siegel theorem.