On the image of the 𝑙-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field

Abstract

Let Y Y be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic > 2 >2 . It is shown that the Tate conjecture implies the surjectivity of the l l -adic Abel-Jacobi map, a Y , l r : C H h o m r ( Y ) → H 2 r − 1 ( Y , Z l ( r ) ) ⊗ Q l / Z l \mathbf {a}^{r}_{Y,l}:CH^{r}_{hom}(Y)\to H^{2r-1}(Y,\mathbb Z_l (r))\otimes \mathbb Q_l /\mathbb Z_l , for all r r and almost all l l . For a special class of threefolds the surjectivity of a Y , l 2 \mathbf {a}^{2}_{Y,l} is proved without assuming any conjectures.