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Given two varieties V,W in the n-fold product of modular curves, we answeraffirmatively a question (formulated by Shou-Wu Zhang's AIM group) on whetherthe set of points in V that are Hecke translations of some point on W is densein V. We need to make some (necessary) assumptions on the dimensions of V,W butfor instance, when V is a divisor and W is a curve, no further assumptions areneeded. We also examine the necessity of our assumptions in the case ofunlikely intersections and show that, contrary to exceptions, two curves in ahigh dimensional space over a finite field can intersect infinitely often up toHecke translations.

Unlikely and just likely intersections for high dimensional families of elliptic curves