## 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.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}^{an} of X_{Q}_{p} . We partition the set of CM points of sufficiently high conductor in X_{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}^{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}^{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.

Original language | English |
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Pages (from-to) | 635-668 |

Number of pages | 34 |

Journal | Commentarii Mathematici Helvetici |

Volume | 97 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 2022 |

## Keywords

- arithmetic dynamics
- Berkovich spaces
- CM points
- equidistribution

## All Science Journal Classification (ASJC) codes

- Mathematics(all)