Torsion points on isogenous abelian varieties

Investigating a conjecture of Zannier, we study irreducible subvarieties abelian schemes that dominate the base and contain Zariski dense set torsion points lie on pairwise isogenous fibers. If everything is defined over algebraic numbers scheme has maximal variation, prove geometric generic fiber such subvariety union cosets. We go to fully or partially explicit versions this result in fibered powers Legendre family elliptic curves. Finally, apply recent Galateau-Mart\'inez obtain uniform bounds number cosets Manin-Mumford problem across given isogeny class. For proofs, adapt strategy, due Lang, Serre, Tate, Hindry, using Galois automorphisms act as homotheties setting.