7-Gons and genus three hyperelliptic curves
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 ℚ(ζ7 + ζ̄7). We study the algebraic correspondences on these curves which are lifts of algebraic correspondences on a conic in P2 associated with Poncelet 7-gons. These correspondences induce endomorphisms φ on the Jacobians which satisfy φ3+φ2-2φ-1=0. Moreover, we study Humbert's modular equations which characterize the curves of genus three having these real multiplications. © 2012 Springer-Verlag.
Publication Source (Journal or Book title)
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Hoffman, J., & Wang, H. (2013). 7-Gons and genus three hyperelliptic curves. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, 107 (1), 35-52. https://doi.org/10.1007/s13398-012-0079-1