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

Herrero, Sebastián; Menares, Ricardo; Rivera-Letelier, Juan

doi:10.2140/ant.2020.14.1239

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

$p$-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

DOI: 10.2140/ant.2020.14.1239

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.

Keywords

equidistribution, elliptic curves, Hecke correspondences

Mathematical Subject Classification 2010

Primary: 11G15

Secondary: 11F32, 11S82

Milestones

Received: 3 November 2018

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

Vol. 14, No. 5, 2020