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7-Gons and genus three hyperelliptic curves

  • J. William Hoffman
  • Haohao Wang
Original Paper

Abstract

In this paper, we will give a general but completely elementary description for hyperelliptic curves of genus three whose Jacobian varieties have endomorphisms by the real cyclotomic field \({{\mathbb{Q}} (\zeta_7 + \overline{\zeta}_7)}\). We study the algebraic correspondences on these curves which are lifts of algebraic correspondences on a conic in P 2 associated with Poncelet 7-gons. These correspondences induce endomorphisms \({\phi}\) on the Jacobians which satisfy \({\phi^3+\phi^2-2\phi-1=0}\). Moreover, we study Humbert’s modular equations which characterize the curves of genus three having these real multiplications.

Keywords

Curves of genus three Real multiplication Abelian variety 

Mathematics Subject Classification (2000)

Primary 11G10 11G15 14H45 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Department of MathematicsSoutheast Missouri State UniversityCape GirardeauUSA

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