manuscripta mathematica

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

CM fields of Dihedral type and the Colmez conjecture

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dodson, B.: The structure of Galois groups of \({\rm CM}\)-fields. Trans. Am. Math. Soc. 283(1), 1–32 (1984)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Obus, A.: On Colmez’s product formula for periods of CM-abelian varieties. Math. Ann. 356(2), 401–418 (2013)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Yang, T.: The Chowla–Selberg formula and the Colmez conjecture. Can. J. Math. 62(2), 456–472 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Yang, T.: Arithmetic intersection on a Hilbert modular surface and the Faltings height. Asian J. Math. 17(2), 335–381 (2013)MathSciNetCrossRefMATHGoogle 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

Personalised recommendations