Arithmetic-geometric means for hyperelliptic curves and Calabi-Yau varieties

In this paper, we define a generalized arithmetic-geometric mean ${\mu }_g$ among $2^g$ terms motivated by $2\tau$-formulas of theta constants. By using Thomae’s formula, we give two expressions of ${\mu }_g$ when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve $C$ of genus $g$. The other is by a period integral of a certain Calabi-Yau $g$-fold given as a double cover of the $g$-dimensional projective space ${\mathbf{P}}^g$.