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.


Zeta Function Prime Power Number Field Field Case Global Field 
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  1. M.A. Tsfasrnan, S.G. Vlăduţ. Asymptotic Properties of Zeta-Functions. J. Math. Sciences, 1997, v. 84, n. 5, pp. 1445–1467.CrossRefGoogle Scholar
  2. M.A. Tsfasman, S.G. Vlădut. Infinite Global Fields and the Generalized Brauer-Siegel Theorem.Moscow Math. J., 2002, v.2, n. 2.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • M. A. Tsfasman
    • 1
  1. 1.Institut de Mathématiques de LuminyIndependent University of Moscow, and Institute for Information Transmission ProblemsRussia

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