The Relative Manin-Mumford Conjecture

We prove the Relative Manin-Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila-Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier in the case of curves. The key new ingredients compared to previous applications of this approach are a height inequality proved by both authors of the current paper and Dimitrov, and the first-named author's study of certain degeneracy loci in subvarieties of abelian schemes. We also strengthen this result and prove a criterion for torsion points to be dense in a subvariety of an abelian scheme over $\mathbb{C}$. The Uniform Manin-Mumford Conjecture for curves embedded in their Jacobians was first proved by K\"{u}hne. We give a new proof, as a corollary to our main theorem, that does not use equidistribution.