Arithmetic on a Family of Picard Curves

  • Rolf-Peter Holzapfel
  • Florin Nicolae


The L-function of the curve C a : Y 3 = X 4 - aX over an algebraic number field k which contains \({\zeta _9}: = \exp \left({ \frac{{2\pi i}}{9}} \right)\)is the inverse of a product of six Hecke L-functions with Grössencharakter. The Euler factors at primes of good reduction are determined by means of Jacobi sums associated to certain powers of the 9-th power residue character. The number of points of C a over a finite field is given in terms of such sums. The jacobian variety of C a over the field of complex numbers has complex multiplication by the ring ℤζ9.


Zeta Function Finite Field Abelian Variety Endomorphism Ring Exceptional Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [CER]
    Cherdieu. J.-P., Estrada-Sarlabous, J., Reinaldo-Barreiro, E., Efficient Reduction on the Jacobian Variety of Picard Curves, in: Coding Theory, Cryptography and Related Areas, Proceedings of the ICCC-98, J. Buchmann, T. Hohold, H. Stichtenoth, H. Tapia-Recillas (eds.), pp. 13–28, Springer-Verlag, 2000.Google Scholar
  2. [Da-Ha]
    Davenport, H., Hasse, H., Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J.Reine Angew. Math. 172 (1934), 151–182.Google Scholar
  3. [De]
    Deuring, M., Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, Nachr. Akad. Wiss. Göttingen, 1953, 85–94.Google Scholar
  4. Feu]Feustel, J.M.,Kompaktifizierung und Singularitäten des Faktorraumes einer arithmetischen Gruppe, die in der zweidimensionalen Einheitskugel wirkt, Diplomarbeit, Humboldt-Univ. Berlin, 1976 (unpublished)Google Scholar
  5. [Hal]
    Hasse, H., Zetafunktion und L-Funktionen zu einem arithmetischen Funktionenkörper vorn Fermatschen Typus, Abhandlungen der Deutschen Akademie der Wissenschaften Berlin, Math.-Nat. Kl. 1954, Nr. 4, 5–70Google Scholar
  6. [Ha2]
    Hasse, H. Zahlentheorie. Akademie Verlag, Berlin, 1963zbMATHGoogle Scholar
  7. [He]
    Hecke, E., Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Zeitschr. 1(1918), 357–376, 6 (1920), 1151.Google Scholar
  8. [Hol]
    Holzapfel, R.-P., Geometry and Arithmetic around Euler partial differential equations, Dt. Verlag d. Wiss., Berlin/Reidel Publ. Comp., Dordrecht, 1986Google Scholar
  9. [Ho2]
    Holzapfel, R.-P., Hierarchies of endomorphism algebras of abelian varieties corresponding to Picard modular surfaces, Schriftenreihe Komplexe Mannigfaltigkeiten 190, Univ. Erlangen, 1994Google Scholar
  10. [Ho3]
    Holzapfel, R.-P., The ball and some Hilbert problems. Lect. in Math. ETH Zürich, Birkhäuser, Basel-Boston-Berlin, 1995Google Scholar
  11. [Lac]
    Lachaud, G., Courbes diagonales et courbes de Picard, Prétirage No. 97–30, Institut de Mathematiqués de Luminy, 1997Google Scholar
  12. [La]
    Lang, S., Complex multiplication, Grundl. Math. Wiss. 255, Springer, 1983Google Scholar
  13. [Neu]
    Neukirch. J., Algebraische Zahlentheorie, Springer, Berlin-Heidelberg, 1992zbMATHGoogle Scholar
  14. [Ta]
    Taniyama, Y., L-functions of number fields and zeta functions of abelian varieties. J. Math. Soc. Japan 9 (1957), 330–366.MathSciNetzbMATHCrossRefGoogle Scholar
  15. We] Weil, A., On Jacobi sums as “Grössencharaktere”, Transact. Amer. Math.Google Scholar
  16. Soc. 73(1952). 487–495.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Rolf-Peter Holzapfel
    • 1
  • Florin Nicolae
    • 1
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations