ACFA and the Manin-Mumford Conjecture
We give Hrushovski’s proof of the (extended) Manin-Mumford conjecture over number fields. Recall that the Manin-Mumford conjecture states that if X is a nonsingular projective curve of genus > 1 over an algebraically closed field of characteristic 0, and A is the Jacobian variety of X then X ∩ Tor (A) is finite (where Tor (A) denotes the group of torsion points of A). This is of course a special case of the Mordell-Lang conjecture. Raynaud  proved that if A is an abelian variety (in characteristic 0) and X is a subvariety of A then X ∩ Tor(A) is a finite union of cosets (proving in particular Manin-Mumford). Hindry  proved the same thing, but with A an arbitrary commutative algebraic group.
KeywordsIrreducible Component Abelian Variety Number Field Finite Union Residue Field
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