A theta expression of the Hilbert modular functions for \(\sqrt{5}\) via the periods of \(K3\) surfaces (Q394149)

It is well-known that the elliptic modular function \(\lambda(z)\) on the moduli space \({\mathbb{H}}/\Gamma(2)\) is given by the the inverse of the period map for a family of elliptic curves (in the Legendre form). The modular function \(\lambda(z)\) has an explicit expression in terms of the Jacobi theta functions.NEWLINENEWLINEIn this paper, the above classical facts are generalized to the Hilbert modular functions for \({\mathbb{Q}}(\sqrt{5})\) by using a family of \(K3\) surfaces. The family \({\mathcal{F}}:=\{S(X,Y)\}\) of \(K3\) surfaces with two complex parameters \(X\) and \(Y\) used for this purpose is given by an affine equation in \((x,y,z)\)-space: NEWLINE\[NEWLINES(X,Y): z^2=x^3-4y^2(4y-5)x^2+20Xy^3x+Yy^4.NEWLINE\]NEWLINE It is shown that the inverse of the period map for this family of \(K3\) surfaces gives a system of generators of Hilbert modular functions for \({\mathbb{Q}}(\sqrt{5})\). Moreover, the two parameters \(X\) and \(Y\) may be regarded as Hilbert modular functions for \({\mathbb{Q}}(\sqrt{5})\), and it is shown that they can be expressed as quotients of theta constants (i.e., Müller's modular forms).