RÉSUMÉ

The main object of our study is an "infinite" global field, i.e. an infinite algebraic extension either of $\Bbb Q$ or of $\Bbb F_r(t)$. We manage to produce a series of interesting invariants of such fields, provided the field is not "too big". These invariants give rise to certain zeta-functions of infinite global fields, and to some measures corresponding to limit distributions of zeroes of usual zeta functions. In order to understand such fields we study towers of usual global fields, both number and function, with growing discriminant (respectively genus). We generalize the Odlyzko-Serre bounds and the Brauer-Siegel theorem. This leads to asymptotic bounds on the ratio ${\log hR}\over {\log\sqrt D}$ valid without the standard assumption ${n}\over {\log\sqrt D}$ tends 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.

WWW page design par Thomas Papanikolaou & Christine Bachoc