Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein's icosahedral invariants $\mathfrak{ A},\mathfrak{ B}$ and $\mathfrak{ C}$ give the Hilbert modular forms for $\sqrt{ 5}$ via the period mapping for a family of $K3$ surfaces. Using the period mappings for several families of $K3$ surfaces, we obtain explicit models of Shimura curves with small discriminant in the weighted projective space ${ \rm Proj} (\mathbb{ C}[\mathfrak{ A},\mathfrak{ B},\mathfrak{ C}])$.
Categories: math.NT