Singular Moduli of Shimura Curves

TL;DR

This paper demonstrates that coordinate maps for certain Shimura curves are Borcherds lifts of modular forms, enabling explicit computation of singular moduli norms and verification of conjectural CM point values.

Contribution

It establishes that Shimura curve coordinate maps are Borcherds lifts, allowing algebraic calculation of CM point norms on these curves.

Findings

Explicit formulas for singular moduli norms on Shimura curves with discriminants 6 and 10.

Verification of conjectural values for rational CM points.

Method for calculating CM point norms with large negative discriminant.

Abstract

The $j$-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over $\mathbb{Q}$. This paper shows that the coordinate maps for the Shimura curves associated to the quaternion algebras with discriminants 6 and 10 are Borcherds lifts of vector-valued modular forms. This property is then used to explicitly compute the rational norms of singular moduli on these curves. This method not only verifies the conjectural values for the rational CM points, but also provides a way of algebraically calculating the norms of CM points on these Shimura curves with arbitrarily large negative discriminant.