$p$-adic equidistribution of CM points

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 \emph{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.