Icosahedral invariants and Shimura curves

TL;DR

This paper constructs explicit models of Shimura curves with small discriminant using Klein's icosahedral invariants and period mappings of K3 surfaces, advancing the understanding of their moduli spaces in number theory.

Contribution

It provides new explicit models of Shimura curves in weighted projective space derived from Klein's invariants and K3 surface period mappings.

Findings

Explicit models of Shimura curves with small discriminant obtained.

Connection established between Klein's invariants and Shimura curves.

Enhanced understanding of moduli spaces of abelian surfaces with quaternion multiplication.

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