#### Vol. 14, No. 5, 2020

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$p$-adic distribution of CM points and Hecke orbits I: Convergence towards the Gauss point

### Sebastián Herrero, Ricardo Menares and Juan Rivera-Letelier

Vol. 14 (2020), No. 5, 1239–1290
##### Abstract

We study the asymptotic distribution of CM points on the moduli space of elliptic curves over ${ℂ}_{p}$, as the discriminant of the underlying endomorphism ring varies. In contrast with the complex case, we show that there is no uniform distribution. In this paper we characterize all the sequences of discriminants for which the corresponding CM points converge towards the Gauss point of the Berkovich affine line. We also give an analogous characterization for Hecke orbits. In the companion paper we characterize all the remaining limit measures of CM points and Hecke orbits.

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##### Keywords
equidistribution, elliptic curves, Hecke correspondences
##### Mathematical Subject Classification 2010
Primary: 11G15
Secondary: 11F32, 11S82
##### Milestones
Revised: 16 November 2019
Accepted: 6 February 2020
Published: 13 July 2020
##### Authors
 Sebastián Herrero Instituto de Matemáticas Pontificia Universidad Católica de Valparaíso Chile Ricardo Menares Facultad de Matemáticas Pontificia Universidad Católica de Chile Santiago Chile Juan Rivera-Letelier Mathematics Department University of Rochester NY United States