Non-divisible cycles on products of very general Abelian varieties

In this paper, we give a recipe for producing infinitely many non-divisible codimension $2$ cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of "field of definition" for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension $2$ cycles we call "primitive cycles," the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori's isogeny method \cite{N} (modified by Totaro \cite{T}) to produce infinitely many non-divisible cycles. As an application, we prove the Chow group modulo a prime of a product of $3$ or more very general elliptic curves is infinite, generalizing work of Schoen.