Adequate equivalence relations and Pontryagin products

Let A be an abelian variety over a field k. We consider CH₀(A) as a ring under Pontryagin product and relate powers of the ideal I⊆CH₀(A) of degree zero elements to powers of the algebraic equivalence relation. We also consider a filtration F⁰⊇F¹⊇… on the Chow groups of varieties of the form T×_kA (defined using Pontryagin products on A×_kA considered as an A-scheme via projection on the first factor) and prove that F^r coincides with the r-fold product (F¹)^*r as adequate equivalence relations on the category of all such varieties.