Isogeny formulas for the Picard modular form and a three terms AGM

In this paper we study the theta contants appeared in [S] those induced the modular function for the family of Picard curves C(ξ) given by (1). Our theta constants θk(u, v) (k = 0, 1, 2) , given by (3), are ”Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball B, given by (2), with respect to a index finite subgroup Γθ of the Picard modular group Γ = PGL(M,Z[exp(2πi/3)]). We define a simultaneous isogeny for the family of Jacobian varieties of C(ξ). Our main result is stated in Theorem (3.1). There we show the explicit relations between theta constants θk(u, v) and θk( √ −3u, 3v) which are corresponding to isogenous Jacobian varieties. In the theory of elliptic theta functions we have the relation { θ00(2τ) = 1 2 (θ 2 00(τ) + θ 2 01(τ)) θ01(2τ) = θ00(τ)θ01(τ).