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manuscripta mathematica

, Volume 156, Issue 1–2, pp 1–22 | Cite as

CM fields of Dihedral type and the Colmez conjecture

  • Tonghai Yang
  • Hongbo Yin
Article

Abstract

In this paper, we consider some CM fields which we call of dihedral type and compute the Artin L-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the ‘abelian case’ of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of \(\mathbb {Q}\) (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for ‘unitary CM types of signature \((n-1, 1)\)’ and holds on average for ‘unitary CM types of a fixed CM number field of signature \((n-r, r)\)’.

Mathematics Subject Classification

11G15 11F41 14K22 

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin MadisonMadisonUSA
  2. 2.Academy of Mathematics and Systems Science, Morningside center of MathematicsChinese Academy of SciencesBeijingChina

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