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Arithmetic on a Family of Picard Curves

In: Finite Fields with Applications to Coding Theory, Cryptography and Related Areas

Author

Listed:
  • Rolf-Peter Holzapfel

    (Humboldt-Universität zu Berlin, Institut für Mathematik)

  • Florin Nicolae

    (Humboldt-Universität zu Berlin, Institut für Mathematik)

Abstract

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.

Suggested Citation

  • Rolf-Peter Holzapfel & Florin Nicolae, 2002. "Arithmetic on a Family of Picard Curves," Springer Books, in: Gary L. Mullen & Henning Stichtenoth & Horacio Tapia-Recillas (ed.), Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, pages 187-208, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-59435-9_14
    DOI: 10.1007/978-3-642-59435-9_14
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