Icosahedral invariants and Shimura curves (Q2397950)

Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication, but it is a non-trivial problem to find explicit models of Shimura curves. This paper obtains explicit models of Shimura curves for quaternion algebra with small discriminant. Explicit defining equations of the Shimura curves are obtained in the weighted projective space \({\mathbb{P}}(1:3:5)=\text{{Proj}}({\mathbb{C}}[A,B,C])\) where \(A,B,C\) are the Klein icosahedral invariants of weight \(1,3\) and \(5\), respectively. The Klein invariants \(A, B\) and \(C\) give the Hilbert modular forms associated to \(\sqrt{5}\) via the period map for a family of \(K3\) surfaces. Using the period map for several families of \(K3\) surfaces, explicit models of Shimura curves with small discriminants are produced.