Algebraic cycles on a simple \(4P\)-dimensional abelian variety over a number field
Let \(J\) be an abelian variety over a number field \(k\subset\mathbb C\) and set \(J_{\mathbb C}=J\otimes_k\mathbb C\). There are a number of conjectures on algebraic cycles of \(J_{\mathbb C}\) (which can be stated even in a more general setting) such as: the Hodge conjecture, the general Hodge-Grothendieck conjecture, the Tate conjecture and the Mumford-Tate conjecture. For example, the Hodge conjecture asserts that the \(\mathbb Q\)-space \(H^{2r}(J_{\mathbb C},\mathbb Q)\cap H^{r,r}\) is generated by the cohomology classes of algebraic cycles of dimension \(r\) on \(J_{\mathbb C}\), and the Tate conjecture asserts that for every prime \(l\), the \(\mathbb Q_l\)-space \(H^{2r}_{et}(J\otimes_k\overline{k},\mathbb Q_l(r))^ {\text{Gal}(\overline{k}|k)}\) is generated by the cohomology classes of algebraic cycles of codimension \(r\) on \(J\). In the paper under review the author proves four interesting theorems which basically assert that (under certain additional hypotheses) these conjectures hold true for a simple \(4p\)-dimensional abelian variety \(J\) over \(k\) (where \(p\) is a prime natural number).
scientific article; zbMATH DE number 1213212
scientific article