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Lower bounds for relative class numbers of CM-fields

Author: Stéphane Louboutin
Journal: Proc. Amer. Math. Soc. 120 (1994), 425-434
MSC: Primary 11R42; Secondary 11R29
MathSciNet review: 1169041
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Abstract: Let $ {\mathbf{K}}$ be a CM-field that is a quadratic extension of a totally real number field $ {\mathbf{k}}$. Under a technical assumption, we show that the relative class number of $ {\mathbf{K}}$ is large compared with the absolute value of the discriminant of $ {\mathbf{K}}$, provided that the Dedekind zeta function of $ {\mathbf{k}}$ has a real zero $ s$ such that $ 0 < s < 1$. This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.

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Keywords: Class number, CM-fields, zeta function
Article copyright: © Copyright 1994 American Mathematical Society