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

TL;DR

This paper introduces a generalized arithmetic-geometric mean for hyperelliptic curves and Calabi-Yau varieties, linking theta constants, period integrals, and complex algebraic geometry.

Contribution

It defines a new generalized mean $rac{rac{g}$ for hyperelliptic curves and Calabi-Yau varieties, connecting theta constants with period integrals in algebraic geometry.

Findings

Expressed rac{rac{g}$ in terms of hyperelliptic curve periods.

Derived a period integral expression for Calabi-Yau g-folds.

Established a link between theta constants and algebraic geometric structures.

Abstract

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 $\mathbf{P}^g$.