An extended Gauss AGM and corresponding Picard modular forms

We construct a new system of three terms arithmetic geometric mean (we say AGM). Our system is an extension of the classical Gauss AGM. It is induced from an isogeny formula of the modular forms defined on the complex 2-dimensional hyperball with respect to a principal congruence subgroup of the Picard modular group . Our AGM function is expressed via the Appell hypergeometric function. These results are the extension of the properties discovered by Gauss himself. Our result is based on the work of K. Matsumoto in 1989 [K. Matsumoto, On modular functions in 2 variables attached to a family of hyperelliptic curves of genus 3, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 16 (4) (1989) 557–578]. And our present paper is a continuation of the previous work [K. Koike, H. Shiga, Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean, J. Number Theory 124 (2007) 123–141. [Ko-Shi]].