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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Explicit Lower bounds for residues at $s=1$ of Dedekind zeta functions and relative class numbers of CM-fields
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by Stéphane Louboutin PDF
Trans. Amer. Math. Soc. 355 (2003), 3079-3098 Request permission

Abstract:

Let $S$ be a given set of positive rational primes. Assume that the value of the Dedekind zeta function $\zeta _K$ of a number field $K$ is less than or equal to zero at some real point $\beta$ in the range ${1\over 2} <\beta <1$. We give explicit lower bounds on the residue at $s=1$ of this Dedekind zeta function which depend on $\beta$, the absolute value $d_K$ of the discriminant of $K$ and the behavior in $K$ of the rational primes $p\in S$. Now, let $k$ be a real abelian number field and let $\beta$ be any real zero of the zeta function of $k$. We give an upper bound on the residue at $s=1$ of $\zeta _k$ which depends on $\beta$, $d_k$ and the behavior in $k$ of the rational primes $p\in S$. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields $K$ which depend on the behavior in $K$ of the rational primes $p\in S$. We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.
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Additional Information
  • Stéphane Louboutin
  • Affiliation: Institut de Mathématiques de Luminy, UPR 9016, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
  • Email: loubouti@iml.univ-mrs.fr
  • Received by editor(s): April 23, 2002
  • Received by editor(s) in revised form: January 6, 2003
  • Published electronically: April 25, 2003

  • Dedicated: Dedicated to Jacqueline G.
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 3079-3098
  • MSC (2000): Primary 11R42; Secondary 11R29
  • DOI: https://doi.org/10.1090/S0002-9947-03-03313-0
  • MathSciNet review: 1974676