K3 surfaces and equations for Hilbert modular surfaces
Let \(D\) be the discriminant of the ring of integers \(\mathcal O_D\) of the real quadratic field \(\mathbb Q(\sqrt D)\). The Hilbert modular surface \(Y_-(D)\) is obtained as compactification of the space parametrizing abelian surfaces with an action of \(\mathcal O_D\). The aim of the present paper is to give explicit equations for rational models of these surfaces for fundamental discriminant \(D\), with \(1<D<100\). The first step in order to obtain equations is to parametrize K3 surfaces which are related to abelian surfaces with real multiplication by \(\mathcal O_D\) via a Shioda-Inose structure. Since these K3 surfaces admit a particular sublattice \(L_D\) of the Néron-Severi lattice, this allows to obtain a moduli space \(\mathcal M_D\) of \(L_D\)-polarized K3 surfaces as a family of elliptic surfaces. Thanks to the previous description and using properties on the special fibers of these elliptic surfaces and their Mordell-Weil group, the authors show \(Y_- (D)\) as double cover of \(\mathbb P^2\) branched over a curve of small degree and thus give equations. Calculations are carried out for each value of \(D\) between 5 (the smallest fundamental discriminant for real multiplication) and 97. The authors suggest that the method can possibly be extended to higher \(D\).
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