A lower bound for the distance between CM points on Shimura curves

2026/07/25 by Daniel Rodriguez
Mathematics · #math.NT

paper · pdf

Abstract

In this paper, we establish a quantitative Diophantine approximation result for complex multiplication (CM) points on Shimura curves. Specifically, we prove a lower bound for the distance between a sequence of CM points P_n converging to a fixed CM point P on a Shimura curve X(D,1) in terms of the discriminant of the endomorphism rings of P_n. The proof exploits the complex geometry of the Fuchsian uniformization, the explicit matrix representation of the underlying quaternion algebra, and Liouville's inequality. We show that the distance between the corresponding fixed points τ_n and τ in the upper half-plane is bounded below by a positive constant times a negative power of the discriminant. This result provides a Shimura curve analogue of a result of Habegger on singular moduli and modular curves.

Citations

• There are at most finitely many singular moduli that are S-units
• The Manin-Mumford conjecture and the Tate-Voloch conjecture for a product of Siegel moduli spaces
• The Tate-Voloch Conjecture in a Power of a Modular Curve
• A finiteness property of torsion points

Related

• Equidistribution of CM points on Shimura Curves and ternary theta series
• Shimura curve computations
• Singular Moduli of Shimura Curves
• CM-points and Lattice counting on arithmetic compact Riemann surfaces
• Counting intersection numbers of closed geodesics on Shimura curves