Families of abelian varieties with many isogenous fibres

Abstract

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 André 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--Wüstholz 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.

Gap in Lemma 3.3 found and corrected by Gabriel Dill

References in corpus (1)

• Ax-Lindemann for \mathcal{A}_g

Cited by in corpus (6)

• Height bounds and the Siegel property
• Quantitative reduction theory and unlikely intersections
• On compatibility between isogenies and polarisations of abelian varieties
• Torsion points on isogenous abelian varieties
• Lattices with skew-Hermitian forms over division algebras and unlikely intersections
• Unlikely intersections with isogeny orbits in a product of elliptic schemes