$p$-adic equidistribution of CM points

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