A theta expression of the Hilbert modular functions for $\sqrt{5}$ 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 $\mathbb{Q}(\sqrt{5})$. 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.

23 pages, 2 figures. A typo in the page 4 is corrected on 3 September 2016

References in corpus (1)

• Period differential equations for the families of $K3$ surfaces with $2$ parameters derived from the reflexive polytopes

Cited by in corpus (5)

• Geometric interpretation of Hermitian modular forms via Burkhardt invariants
• Inverse period mappings of $K3$ surfaces and a construction of modular forms for a lattice with the Kneser conditions
• Icosahedral invariants and Shimura curves
• Icosahedral invariants and a construction of class fields via periods of $K3$ surfaces
• Sequence of families of lattice polarized $K3$ surfaces, modular forms and degrees of complex reflection groups