Bad Reduction of Genus Three Curves with Complex Multiplication

  • Irene Bouw
  • Jenny Cooley
  • Kristin LauterEmail author
  • Elisa Lorenzo García
  • Michelle Manes
  • Rachel Newton
  • Ekin Ozman
Part of the Association for Women in Mathematics Series book series (AWMS, volume 2)


Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes \(\mathfrak{p}\) of M such that the stable reduction of C at \(\mathfrak{p}\) contains three irreducible components of genus 1.

MSC 2010:

11G15 14K22 15B33 



The authors would like to thank the Centre International de Rencontres Mathématiques in Luminy for sponsoring the Women in Numbers - Europe (Femmes en Nombre) workshop and for providing a productive and enjoyable environment for our initial work on this project. We would especially like to thank the organizers of WINE, Marie José Bertin, Alina Bucur, Brooke Feigon, and Leila Schneps for making the conference and this collaboration possible. We also thank the referee for the detailed and helpful report.

The work of MM was partially supported by NSF-DMS 1102858.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Irene Bouw
    • 1
  • Jenny Cooley
    • 2
  • Kristin Lauter
    • 3
    Email author
  • Elisa Lorenzo García
    • 4
  • Michelle Manes
    • 5
  • Rachel Newton
    • 6
  • Ekin Ozman
    • 7
    • 8
  1. 1.Institute of Pure MathematicsUniversity of UlmUlmGermany
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Microsoft ResearchRedmondUSA
  4. 4.Mathematisch InstituutUniversiteit LeidenLeidenThe Netherlands
  5. 5.Department of MathematicsUniversity of HawaiiHonoluluUSA
  6. 6.Max Planck Institute for MathematicsBonnGermany
  7. 7.Mathematics DepartmentUniversity of Texas at AustinAustinUSA
  8. 8.Department of MathematicsBogazici UniversityIstanbulTurkey

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