Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

Abstract

We give explicit upper and lower bounds for relative class numbers of imaginary abelian number fields and of nonabelian CM-fields K. We then use them to obtain Brauer–Siegel type results for relative class numbers of CM-fields. The main feature of this paper is a new method which enables us to easily deal with the case where K contains an imaginary quadratic subfield. We will make use of the ideas introduced in [Lou04] to obtain in Theorem 1 upper bounds on |L(1, χ)| for even and odd primitive Dirichlet characters χ, whereas our previous ideas (developed in [Lou98b] and [Lou01]) failed to embrace the case of odd characters. These bounds depend on whether L(s, χ) has or does not have a real zero β in the range 0 < β < 1. They will then enable us to obtain in Theorems 18, 28 and 31 bounds for relative class numbers hK of CM-fields K, especially in the case that K contains an imaginary quadratic subfield L. Apart from the proof of Lemma 17, which can be found in [Lou03], this paper provides the reader with a self-contained exposition of how one can obtain (as in Corollaries 20, 21, 23 and 25 where several footnotes clearly show that our approach is more efficient than the ones formerly developed by various authors) good enough explicit lower bounds for relative class numbers to enable him to solve various class number problems for CM-fields or to simplify the existing proofs (e.g., see [CK98], [CK00a], [CK00b], [Lou95], [Lou97],[Lou98a], [Lou99], [MM] and [Yam]). Whereas almost all the papers in the literature dealing with explicit lower bounds for relative class numbers of CM-fields (or values at s = 1 of L-functions)