# p-adic equidistribution of CM points

## TL;DR

This paper investigates the distribution of CM points on modular and Shimura curves in the p-adic Berkovich setting, establishing conditions for their convergence to specific points based on their Galois orbits and local properties.

## Contribution

It introduces a novel partition of CM points into explicit basins in the p-adic analytification and characterizes the limits of Galois orbits in this setting, extending previous equidistribution results.

## Key findings

- Sequences supported in a single basin converge to a unique limit point.
- The limit point corresponds to the generic point of the associated irreducible component.
- The results apply to Shimura curves over totally real fields.

## Abstract

Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. We study the equidistribution problem in the Berkovich analytification $X_{p}^{\rm an}$ of $X_{{\bf Q}_{p}}$. We partition the set of CM points of sufficiently high conductor in $X_{{\bf Q}_{p}}$ into finitely many explicit `basins' $B_{V}$, indexed by the irreducible components $V $ of the mod-$p$ reduction of the canonical model of $X$. We prove that a sequence $z_{n}$ of local Galois orbits of CM points with $p$-conductor going to infinity has a limit in $X_{p}^{\rm an}$ if and only if it is eventually supported in a single basin $B_{V}$. If so, the limit is the unique point of $X_{p}^{\rm an}$ whose mod-$p$ reduction is the generic point of $V$. The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasicanonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin--Tate space.