Asymptotic Properties of Global Fields

Abstract

The main object of our study is an “infinite” global field, i.e., an infinite algebraic extension either of ℚ or of F _r (t). In order to understand such fields we study sequences of usual global fields, both number and function, with growing discriminant (respectively, genus). We manage to generalize the Odlyzko—Serre bounds and the Brauer—Siegel theorem. This leads to asymptotic bounds on the ratio $\log {\text{ }}hR/\log \sqrt {\left| D \right|}$ valid without the standard assumption $n/\log \sqrt {\left| D \right|} \to 0$, 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. To understand what is going on, we introduce zeta-functions of infinite global fields, and study measures corresponding to limit distributions of zeroes of usual zeta functions.

Supported in part by RBRF 99-01-01204. This lecture is based on my joint work with Serge Vlăduţ.