Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean

In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are “Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM.