志村曲线上CM点之间距离的一个下界
A lower bound for the distance between CM points on Shimura curves
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中文总结 AI 辅助
本文针对志村曲线上的CM点建立定量丢番图逼近结果，通过富克斯一致化等方法，依据CM点自同态环判别式，证明了收敛到固定CM点的一列CM点与该固定点距离的下界，给出哈贝格尔结果的志村曲线类似物。
中文摘要 AI 辅助
在本文中，我们为志村曲线上的复乘（CM）点建立了一个定量丢番图逼近结果。具体而言，我们根据一列收敛到志村曲线\(X(D,1)\)上固定CM点\(P\)的CM点\(P_n\)的自同态环的判别式，证明了\(P_n\)与\(P\)之间距离的一个下界。证明利用了富克斯一致化的复几何、基础四元数代数的显式矩阵表示以及刘维尔不等式。我们表明上半平面中相应固定点\(\tau_n\)和\(\tau\)之间的距离由判别式的负幂乘以一个正常数下界界定。该结果提供了哈贝格尔关于奇异模和模曲线结果的志村曲线类似物。
英文摘要
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.
