Chow Groups of Abelian Varieties and Beilinson's Conjecture
  In the present paper we introduce the property AA of a subsemigroup of the endomorphism semigroup of an abelian variety, which holds for semigroup of endomorphisms of an abelian variety defined over a number field, and show that the orbit of any cycle under a semigroup with property AA in the Chow group $\otimes\Q$ has finite dimensional span. 
Bogdan Zavyalov
math.AG math.NT