$p$-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 $X(\mathbf{C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. We study the equidistribution problem in the Berkovich analytification $X\_{p}^{\operatorname{an}}$ of $X\_{\operatorname{Q}\_{p}}$. We partition the set of CM points of sufficiently high conductor in $X\_{\operatorname{Q}{p}}$ into finitely many explicit basins $B{V}$, indexed by the irreducible components $V$ of the $\bmod\text{-}p$ 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 $X\_{p}^{\operatorname{an}}$ if and only if it is eventually supported in a single basin $B\_{V}$. If so, the limit is the unique point of $X\_{p}^{\operatorname{an}}$ 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 quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin–Tate space.