Families of abelian varieties with many isogenous fibres

Let Z be a subvariety of the moduli space of principally polarised abelian varieties of dimension g over the complex numbers. Suppose that Z contains a Zariski dense set of points which correspond to abelian varieties from a single isogeny class. A generalisation of a conjecture of Andr'e and Pink predicts that Z is a weakly special subvariety. We prove this when dim Z = 1 using the Pila–Zannier method and the Masser–W"ustholz isogeny theorem. This generalises results of Edixhoven and Yafaev when the Hecke orbit consists of CM points and of Pink when it consists of Galois generic points.