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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Q}} (\zeta_7 + \overline{\zeta}_7)}$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi}$$\end{document} on the Jacobians which satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi^3+\phi^2-2\phi-1=0}$$\end{document}. Moreover, we study Humbert’s modular equations which characterize the curves of genus three having these real multiplications.