Chow Groups of Abelian Varieties

Mumford in his paper [Mum68] has proven that if a smooth projective surface over complex numbers has a global regular 2-form then its Chow group of zero cycles is not finitely generated. However, there are conjectures that predict that the situation over number fields is completely different. Namely, the Beilinson conjecture implies that for any smooth projective variety over number field its Chow groups are finitely generated. This conjecture is not known in any particular case, when a variety has a global regular 2-form. The most straightforward way to try to produce a counterexample is to take some variety, a cycle on it and act on it by the endomorphism ring of this variety. We prove that actually the orbit of any cycle α ∈ CHp(A)Q under the action of zero-preserving endomorphisms of abelian variety spans finite-dimensional Q-vector space. Actually, it turns out that the specifics of a number field is not important in this case, and this statement holds over any field.