$p$-adic Dynamics of Hecke Operators on Modular Curves

TL;DR

This paper investigates the $p$-adic dynamics of Hecke operators on modular curves, focusing on ordinary and supersingular reductions, and analyzing their effects on CM points using advanced deformation and period maps.

Contribution

It introduces a detailed analysis of Hecke operator dynamics on modular curves in $p$-adic settings, employing Serre-Tate coordinates and Gross-Hopkins period maps for the first time in this context.

Findings

Characterization of dynamics on CM points

Application of Serre-Tate coordinates in ordinary reduction

Use of Gross-Hopkins period map in supersingular reduction

Abstract

In this paper we study the $p$-adic dynamics of prime-to-$p$ Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre-Tate coordinates, while in the case of supersingular reduction, we use a parameter on the deformation space of the unique formal group of height $2$ over $\overline{\mathbb{F}}_p$, and take advantage of the Gross-Hopkins period map.