Torsion points and height jumping in higher-dimensional families of abelian varieties

Abstract: In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replacefinite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.