## Abstract

Let

We partition the set of CM points of sufficiently high conductor in XQp into finitely many explicit \emph{basins}

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.

*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^{an}_{p}of XQp.We partition the set of CM points of sufficiently high conductor in XQp into finitely many explicit \emph{basins}

*B*, indexed by the irreducible components_{V}*V*of the mod-p reduction of the canonical model of*X*. We prove that a sequence*z*of local Galois orbits of CM points with p-conductor going to infinity has a limit in X_{n}^{an}p if and only if it is eventually supported in a single basin BV. If so, the limit is the unique point of Xanp 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.

Original language | English |
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State | Published - 2019 |

## Keywords

- math.NT
- math.AG
- 11G15, 14K22