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Mosc. Math. J., 2002, Volume 2, Number 2, Pages 329–402 (Mi mmj58)  

This article is cited in 23 scientific papers (total in 23 papers)

Infinite global fields and the generalized Brauer–Siegel theorem

M. A. Tsfasmanabc, S. G. Vlăduţac

a Institute for Information Transmission Problems, Russian Academy of Sciences
b Independent University of Moscow
c Institut de Mathématiques de Luminy

Abstract: The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of $\mathbb{Q}$ or of $\mathbb{F}_r(t)$. We produce a series of invariants of such fields, and we introduce and study a kind of zeta-function for them. Second, for sequences of number fields with growing discriminant, we prove generalizations of the Odlyzko–Serre bounds and of the Brauer–Siegel theorem, taking into account non-archimedean places. This leads to asymptotic bounds on the ratio ${\log hR}/\log\sqrt{|D|}$ valid without the standard assumption $n/\log\sqrt{|D|}\to 0$, 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. As an easy consequence we ameliorate existing bounds for regulators.

Key words and phrases: Global field, number field, curve over a finite field, class number, regulator, discriminant bound, explicit formulae, infinite global field, Brauer–Siegel theorem.


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MSC: 11G20, 11R37, 11R42, 14G05, 14G15, 14H05
Received: June 10, 2001; in revised form April 23, 2002

Citation: M. A. Tsfasman, S. G. Vlăduţ, “Infinite global fields and the generalized Brauer–Siegel theorem”, Mosc. Math. J., 2:2 (2002), 329–402

Citation in format AMSBIB
\by M.~A.~Tsfasman, S.~G.~Vl{\u a}du\c t
\paper Infinite global fields and the generalized Brauer--Siegel theorem
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 2
\pages 329--402

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    This publication is cited in the following articles:
    1. Tsfasman M.A., “Asymptotic properties of global fields”, Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, 2002, 328–334  crossref  mathscinet  zmath  isi
    2. A. I. Zykin, “The Brauer-Siegel and Tsfasman–Vlǎdut̨ theorems for almost normal extensions of number fields”, Mosc. Math. J., 5:4 (2005), 961–968  mathnet  mathscinet  zmath
    3. Lebacque Ph., “Generalised Mertens and Brauer-Siegel theorems”, Acta Arithmetica, 130:4 (2007), 333–350  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Kunyavskii B.E., Tsfasman M.A., “Brauer-Siegel Theorem for Elliptic Surfaces”, International Mathematics Research Notices, 2008, rnn009  mathscinet  isi  elib
    5. A. I. Zykin, “Brauer–Siegel Theorem for Families of Elliptic Surfaces over Finite Fields”, Math. Notes, 86:1 (2009), 140–142  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. I. Zykin, “Asymptotic properties of the Dedekind zeta-function in families of number fields”, Russian Math. Surveys, 64:6 (2009), 1145–1147  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Zykin A., “On the generalizations of the Brauer-Siegel theorem”, Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Mathematics, 487, 2009, 195–206  crossref  mathscinet  zmath  isi
    8. Zykin A.I., Lebacque P., “On logarithmic derivatives of zeta functions in families of global fields”, Doklady Mathematics, 81:2 (2010), 201–203  crossref  mathscinet  zmath  isi  elib
    9. Schmidt A., “Regarding Pro-Fundamental group markers of arithmetic curves”, Journal fur Die Reine und Angewandte Mathematik, 640 (2010), 203–235  crossref  mathscinet  zmath  isi
    10. Lebacque Ph., “On Tsfasman-Vladut Invariants of Infinite Global Fields”, Int J Number Theory, 6:6 (2010), 1419–1448  crossref  mathscinet  zmath  isi
    11. Lebacque Ph., Zykin A., “On Logarithmic Derivatives of Zeta Functions in Families of Global Fields”, Int J Number Theory, 7:8 (2011), 2139–2156  crossref  mathscinet  zmath  isi
    12. Emmanuel Hallouin, Marc Perret, “Recursive towers of curves over finite fields using graph theory”, Mosc. Math. J., 14:4 (2014), 773–806  mathnet  mathscinet
    13. Ngo Thi Ngoan, Nguyen Quoc Thang, “on Some Hasse Principles For Algebraic Groups Over Global Fields. II”, Proc. Jpn. Acad. Ser. A-Math. Sci., 90:8 (2014), 107–112  crossref  mathscinet  zmath  isi
    14. Zykin A., “Asymptotic Properties of Zeta Functions Over Finite Fields”, Finite Fields their Appl., 35 (2015), 247–283  crossref  mathscinet  zmath  isi  elib
    15. Lebacque Ph., “Some Effective Results on the Tsfasman-Vladut Invariants”, Ann. Inst. Fourier, 65:1 (2015), 63–99  crossref  mathscinet  zmath  isi
    16. Zykin A., “Uniform Distribution of Zeroes of l-Functions of Modular Forms”, Algorithmic Arithmetic, Geometry, and Coding Theory, Contemporary Mathematics, 637, eds. Ballet S., Perret M., Zaytsev A., Amer Mathematical Soc, 2015, 295–299  crossref  mathscinet  zmath  isi
    17. Marc Hindry, Amílcar Pacheco, “An analogue of the Brauer–Siegel theorem for abelian varieties in positive characteristic”, Mosc. Math. J., 16:1 (2016), 45–93  mathnet  mathscinet
    18. Luzzi L., Vehkalahti R., “Almost Universal Codes Achieving Ergodic Mimo Capacity Within a Constant Gap”, IEEE Trans. Inf. Theory, 63:5 (2017), 3224–3241  crossref  mathscinet  zmath  isi
    19. S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Russian Math. Surveys, 73:2 (2018), 261–322  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. A. L. Smirnov, “Kummerova bashnya i bolshie dzeta-funktsii”, Algebra i teoriya chisel. 1, Posvyaschaetsya pamyati Olega Mstislavovicha FOMENKO, Zap. nauchn. sem. POMI, 469, POMI, SPb., 2018, 151–159  mathnet
    21. Maire Ch., Oggier F., “Maximal Order Codes Over Number Fields”, J. Pure Appl. Algebr., 222:7 (2018), 1827–1858  crossref  mathscinet  zmath  isi  scopus
    22. Griffon R., “A Brauer-Siegel Theorem For Fermat Surfaces Over Finite Fields”, J. Lond. Math. Soc.-Second Ser., 97:3 (2018), 523–549  crossref  mathscinet  zmath  isi  scopus
    23. Hajir F., Maire Ch., “On the Invariant Factors of Class Groups in Towers of Number Fields”, Can. J. Math.-J. Can. Math., 70:1 (2018), 142–172  crossref  mathscinet  zmath  isi
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