CM points have everywhere good reduction

TL;DR

This paper proves that all CM points on any Shimura variety can be extended to integral models with good reduction, ensuring their stability across various primes.

Contribution

It establishes the existence of integral models for Shimura varieties where all CM points have good reduction, a significant advancement in understanding their arithmetic properties.

Findings

All CM points have good reduction in suitable integral models.

Characterization of points with potentially-good reduction.

Extension of CM points to integral points of the model.

Abstract

We prove that for every Shimura variety $S$, there is an integral model $\mathcal{S}$ such that all CM points of $S$ have good reduction with respect to $\mathcal{S}$. In other words, every CM point is contained in $\mathcal{S}(\overline{\mathbb{Z}})$. This follows from a stronger local result wherein we characterize the points of $S$ with potentially-good reduction (with respect to some auxiliary prime $\ell$) as being those that extend to integral points of $\mathcal{S}$.