A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces

TL;DR

This paper extends classical elliptic modular functions to the Hilbert modular case for , constructing a period mapping for K3 surfaces and expressing Hilbert modular functions explicitly via theta constants.

Contribution

It introduces a novel period mapping for K3 surfaces related to  and provides explicit theta constant expressions for Hilbert modular functions.

Findings

Constructed a period mapping for K3 surface family .

Derived generators of Hilbert modular functions from the inverse period map.

Expressed Hilbert modular functions explicitly using theta constants.

Abstract

In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We construct the period mapping for a family $\mathcal{F}=\{S(X,Y)\}$ of $K3 $ surfaces with $2$ complex parameters $X$ and $Y$. The inverse correspondence of the period mapping gives a system of generators of Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$. Moreover, we show an explicit expression of this inverse correspondence by theta constants.