The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian subvariety translated by a torsion element.


Both Raynaud's proof and the equidistribution proof prove this by reduction to the case of number fields. The hint as to why this works is "by a specialization argument". Let me now present my thoughts on this. I will now assume Manin-Mumford for number fields.


The abelian variety and its subscheme are certainly defined over some finite type Q-algebra S lying inside F. Denote the models of our given data by $\mathcal{A}$ and $\mathcal{Z}$. It suffices to show that the torsion points in $\mathcal{Z}$ aren't dense. Their specializations over any closed point are not dense by assumption. I don't know how to conclude from here. I want to study the failure of commutativity of intersecting the defining ideals and tensoring, ideally obtaining generic commutativity of tensoring with certain limits.



Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?
What you want to do is, by induction on the theorem in the number field $K$ case, prove that all torsion points in $\mathcal Z\_K$ lie in a finite union of torsion translates of abelian varieties (contained in $Z\_K$).


It follows that all torsion points in $\mathcal Z\_{\eta}$ specialize to points in a finite union of torsion translates of abelian varieties.


But since the specialization map is injective on torsion points, all torsion points in $\mathcal Z\_\eta$ lie in torsion translates of lifts of those abelian varieties. (We can ensure the abelian varieties list by choosing $K$ appropriately using Masser's specialization theorem, though this can't be quite what Raynaud did for timeline reasons.)


So all torsion points in $Z$ lie in finitely many torsion translates of abelian subvarieties, but these subvarieties may not be contained in $Z$. However, they have dimension at most the dimension of $Z$, so the only way the torsion points of $Z$ can be Zariski dense is if $Z$ is equal to one of these torsion translates, because otherwise their intersections have smaller dimension and are not dense.


Probably one can avoid Masser's theorem by thinking more carefully about what the lifts of torsion points on an abelian subvariety that doesn't lift look like.


