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arXiv · 1603.09734

Double integrals on a weighted projective plane and the Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$

Atsuhira Nagano
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Abstract

The aim of this paper is to give an explicit extension of the classical elliptic integrals to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We study a family of Kummer surfaces corresponding to the Humbert surface of invariant $5$ with two complex parameters. Our Kummer surface is given by a double covering of the weighted projective space $\mathbb{P}(1:1:2)$ branched along a parabola and a quintic curve. The period mapping for our family is given by double integrals of an algebraic function on chambers coming from an arrangement of a parabola and a quintic curve in $\mathbb{C}^2$.

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Atsuhira Nagano. 2016-03-26. Double integrals on a weighted projective plane and the Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$. https://doi.org/10.4064/aa167-4-2

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