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Arithmetic on a Family of Picard Curves

  • Rolf-Peter Holzapfel
  • Florin Nicolae

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.

Keywords

Zeta Function Finite Field Abelian Variety Endomorphism Ring Exceptional Point 
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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

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