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

TL;DR

This paper extends classical elliptic integrals to the Hilbert modular setting for b(\u221a{5}), using double integrals on Kummer surfaces related to specific algebraic curves, providing explicit period mappings.

Contribution

It introduces an explicit extension of elliptic integrals to the Hilbert modular case for b({5}), involving double integrals on Kummer surfaces associated with particular algebraic curves.

Findings

Explicit description of Kummer surfaces related to b({5})

Construction of double integrals representing period mappings

Connection between algebraic functions and Hilbert modular forms

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 $\mathbb{C}^2$.