Linearity on ordinary Siegel moduli schemes and joint unlikely almost intersections

TL;DR

This paper investigates a $p$-adic analog of joint conjectures related to special points on Siegel moduli schemes, focusing on distributions of CM points and Hecke orbits using linearity and $p$-adic distances.

Contribution

It introduces a $p$-adic framework for joint unlikely intersections on Siegel moduli schemes, extending classical conjectures with new $p$-adic techniques.

Findings

Establishes $p$-adic linearity properties of formal subschemes.

Develops a $p$-adic analog of the Ax--Lindemann theorem.

Analyzes distributions of CM points and Hecke orbits in the $p$-adic setting.

Abstract

The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of Andr\'e--Oort and Andr\'e--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.