manuscripta mathematica

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

CM fields of Dihedral type and the Colmez conjecture

  • Tonghai YangEmail author
  • Hongbo Yin


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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  1. 1.
    Andreatta, F., Goren, E., Howard, B., Pera, K.M.: Faltings heights of abelian varieties with complex multiplication, p. 129 (Preprint, 2016)Google Scholar
  2. 2.
    Bruinier, J., Howard, B., Kudla, S., Rapoport, M., Yang, T.H.: Modularity of generating series of divisors on unitary shimura varieties ii: arithmetic applications (Preprint, 2017)Google Scholar
  3. 3.
    Barquero-Sanchez, A., Masri, R.: On the Colmez conjecture for non-abelian CM fields, p. 35 (Preprint, 2016)Google Scholar
  4. 4.
    Colmez, P.: Périodes des variétés abéliennes à multiplication complexe. Ann. of Math. (2) 138(3), 625–683 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dodson, B.: The structure of Galois groups of \({\rm CM}\)-fields. Trans. Am. Math. Soc. 283(1), 1–32 (1984)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kudla, S.S., Rapoport, M., Yang, T.: On the derivative of an Eisenstein series of weight one. Int. Math. Res. Notices 7, 347–385 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Obus, A.: On Colmez’s product formula for periods of CM-abelian varieties. Math. Ann. 356(2), 401–418 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Tate, J.: Les conjectures de Stark sur les fonctions L d’Artin en s = 0, volume 47 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1984). Lecture notes edited by Dominique Bernardi and Norbert SchappacherGoogle Scholar
  9. 9.
    Tsimerman, J.: A proof of the Andre–Oort conjecture for \({A}_g\) (Preprint)Google Scholar
  10. 10.
    Yang, T.: An arithmetic intersection formula on Hilbert modular surfaces. Am. J. Math. 132(5), 1275–1309 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yang, T.: The Chowla–Selberg formula and the Colmez conjecture. Can. J. Math. 62(2), 456–472 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yang, T.: Arithmetic intersection on a Hilbert modular surface and the Faltings height. Asian J. Math. 17(2), 335–381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yuan, X., Zhang, S.-W.: On the average Colmez conjecture (Preprint)Google Scholar

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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