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Explicit Lower bounds for residues at $s=1$ of Dedekind zeta functions and relative class numbers of CM-fields


Author: Stéphane Louboutin
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
Published electronically: April 25, 2003
MathSciNet review: 1974676
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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

DOI: https://doi.org/10.1090/S0002-9947-03-03313-0
Keywords: Dedekind zeta functions, CM-field, relative class number
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.
Article copyright: © Copyright 2003 American Mathematical Society

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