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 Q or of Fr(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 loghR/log√|D| valid without the standard assumption n/log√|D|→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.
Received:June 10, 2001; in revised form April 23, 2002
\Bibitem{TsfVla02}
\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
\mathnet{http://mi.mathnet.ru/mmj58}
\crossref{https://doi.org/10.17323/1609-4514-2002-2-2-329-402}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1944510}
\zmath{https://zbmath.org/?q=an:1004.11037}
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Linking options:
https://www.mathnet.ru/eng/mmj58
https://www.mathnet.ru/eng/mmj/v2/i2/p329
This publication is cited in the following 41 articles:
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SUSHANT KALA, “ON THE LOWEST ZERO OF THE DEDEKIND ZETA FUNCTION”, Bull. Aust. Math. Soc., 2024, 1
PENG-JIE WONG, “ON STARK'S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE”, Bull. Aust. Math. Soc., 106:2 (2022), 288
Dixit A.B., “On the Generalized Brauer-Siegel Theorem For Asymptotically Exact Families of Number Fields With Solvable Galois Closure”, Int. Math. Res. Notices, 2021:14 (2021), 10941–10956
Farshid Hajir, Christian Maire, Ravi Ramakrishna, “Cutting towers of number fields”, Ann. Math. Québec, 45:2 (2021), 321
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Anup B. Dixit, “A uniqueness property of general Dirichlet series”, Journal of Number Theory, 206 (2020), 123
Michael A. Tsfasman, “Serre's theorem and measures corresponding to abelian varieties over finite fields”, Mosc. Math. J., 19:4 (2019), 789–806
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S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Russian Math. Surveys, 73:2 (2018), 261–322
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Griffon R., “A Brauer-Siegel Theorem For Fermat Surfaces Over Finite Fields”, J. Lond. Math. Soc.-Second Ser., 97:3 (2018), 523–549
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
Philippe Lebacque, Alexey Zykin, “On $M$-functions associated with modular forms”, Mosc. Math. J., 18:3 (2018), 437–472
Luzzi L., Vehkalahti R., “Almost Universal Codes Achieving Ergodic Mimo Capacity Within a Constant Gap”, IEEE Trans. Inf. Theory, 63:5 (2017), 3224–3241
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