Icosahedral invariants and Shimura curves

Abstract

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 $\mathrm{Proj}(\mathbb{C}[\mathfrak{A},\mathfrak{B},\mathfrak{C}])$.

24 pages, 5 figures. This paper is corrected and shortened on 4 January 2016. Minor corrections on 7 August 2016

References in corpus (6)

• K3 surfaces and equations for Hilbert modular surfaces
• Period differential equations for the families of $K3$ surfaces with $2$ parameters derived from the reflexive polytopes
• A theta expression of the Hilbert modular functions for $\sqrt{5}$ via the periods of $K3$ surfaces
• Double integrals on a weighted projective plane and the Hilbert modular functions for $\mathbb{Q}(\sqrt{5})$
• Modular surfaces associated with toric K3 hypersurfaces
• Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19

Cited by in corpus (1)

• Icosahedral invariants and a construction of class fields via periods of $K3$ surfaces