Double integrals on a weighted projective plane and Hilbert modular functions for $\mathbb {Q}(\sqrt {5})$
Volume 167 / 2015
Acta Arithmetica 167 (2015), 327-345
MSC: Primary 11F46; Secondary 14J28, 33E05.
DOI: 10.4064/aa167-4-2
Abstract
The aim of this paper is to give an explicit extension of 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 $\mathbb {C}^2$.
