Drinfeld modules and torsion in the Chow groups of certain threefolds

Let E → B be an elliptic surface defined over the algebraic closure of a finite field of characteristic greater than 5. Let W be a resolution of singularities of E ×B E. We show that the l-adic Abel–Jacobi map from the l-power-torsion in the second Chow group of W to H3(W,Zl(2)) ⊗ Ql/Zl is an isomorphism for almost all primes l. A main tool in the proof is the assertion that certain CM-cycles in fibres of W → B are torsion, which is proven using results from the theory of Drinfeld modular curves