On the image of the \(l\)-adic Abel-Jacobi map for a variety over the algebraic closure of a finite field. (English) Zbl 0917.14004
Summary: Let \(Y\) be a smooth projective variety of dimension at most 4 defined over the algebraic closure of a finite field of characteristic \(>2\). It is shown that the Tate conjecture implies the surjectivity of the \(l\)-adic Abel-Jacobi map, \(\mathbf{a}^{r}_{Y,l}: \text{CH}^{r}_{\text{hom}}(Y)\to H^{2r-1}(Y,\mathbb{Z}_l (r))\otimes \mathbb{Q}_l /\mathbb{Z}_l\), for all \(r\) and almost all \(l\). For a special class of threefolds the surjectivity of \(\mathbf{a}^{2}_{Y,l}\) is proved without assuming any conjectures.
Keywords:
algebraic cycles; \(l\)-adic Abel-Jacobi map; finite ground field; Tate conjecture; threefoldsReferences:
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