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.C/ and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification Xpan of XQp . We partition the set of CM points of sufficiently high conductor in XQp into finitely many explicit basins BV , indexed by the irreducible components V of the mod-p reduction of the canonical model of X. We prove that a sequence zn of local Galois orbits of CM points with p-conductor going to infinity has a limit in Xpan if and only if it is eventually supported in a single basin BV . If so, the limit is the unique point of Xpan 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.
- arithmetic dynamics
- Berkovich spaces
- CM points
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