A theta expression of the Hilbert modular functions for 5 via the periods of K3 surfaces

Abstract

In this paper, we give an extension of the classical story of the elliptic modular function to the Hilbert modular case for $\mathbb{Q}\left(\sqrt{5}\right)$. We construct the period mapping for a family $\mathcal{F}=\left\{S\right(X,Y\left)\right\}$ 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 $\mathbb{Q}\left(\sqrt{5}\right)$. Moreover, we show an explicit expression of this inverse correspondence by theta constants.