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

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

2010-01-27

arxiv.org/abs/1001.4868v1

Mathematics

Algebraic Geometry

latex 10 pages

Scientific paper

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 $\mathbf{P}^g$.

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