In this elegant paper the authors study the moduli space of Picard curves, i.e. curve of genus \(3\) which have an equation of the form \(y^3=x(x-1)(x-\lambda_1)(x-\lambda_2)\). The Jacobian variety of these curves has complex multiplication by \(\mathbb Z[\sqrt{-3}]\). The inverse of the period map in terms of theta functions has been known for some time. The authors construct an isogeny with kernel \((\mathbb Z/3\mathbb Z)^3\) on the set of Picard curves and determine explicitly the action on theta functions. This is analogous to the relations \(\vartheta_{00}^2(2\tau)=\frac 12(\vartheta_{00}(\tau)^2 +\vartheta_{01}(\tau)^2)\) and \(\vartheta_{01}^2(2\tau)= \vartheta_{00}(\tau) \vartheta_{01}(\tau)\) which appear in Gauss’ theory of the AGM. Correspondingly the authors find a map \(\Psi\) defined on a cone in \(\mathbb R^3\) which represents this isogeny. The iterates converge and the inverse of the limit can be represented as a hypergeometric function in two variables. This represents a three-variable AGM. On a two-dimensional subset it reduces to the cubic AGM of J. M. Borwein and P. B. Borwein [Trans. Am. Math. Soc. 323, No. 2, 691–701 (1991; Zbl 0275.33014)].