Families of abelian varieties with many isogenous fibres

Let $V$ be a subvariety of the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. Suppose that a Zariski dense set of points of $V$ lie in a single Hecke orbit; in other words they correspond to abelian varieties from a single polarised isogeny class. The Zilber-Pink conjecture predicts that $V$ is weakly special. We will prove this when $\dim V=1$ using the Pila-Zannier method and the Masser-Wustholz 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.