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The goal of this paper is to study a $p$-adic analog of the joint of theconjectures of Andr\'e--Oort and Andr\'e--Pink. More precisely, on a product ofordinary Siegel formal moduli schemes, we study the distribution of pointswhose components are either CM points or points in Hecke orbits. We uselinearity of formal subschemes of the product as the $p$-adic analog ofgeodesicness over complex numbers. Moreover, we relax the usual incidencerelations by using $p$-adic distance. We also study a $p$-adic formal schemetheoretic analog of the Ax--Lindemann theorem.

Linearity on ordinary Siegel moduli schemes and joint unlikely almost intersections