Cycles on Abelian varieties and exceptional numbers

The article considers a technique for proving the Hodge, Tate, and Mumford–Tate conjectures for a simple complex Abelian variety $J$ of non-exceptional dimension under the condition that $\operatorname{End}(J)\otimes \mathbb R\in\bigl\{\mathbb R,M_2(\mathbb R), \mathbb K,\mathbb C\bigr\}$, where $\mathbb K$ is the skew field of classical quaternions. The simple $2p$-dimensional Abelian varieties over a number field ($p$ is a prime, $p\geqslant 17$) are studied in detail. An application is given of Minkowski's theorem on unramified extensions of the field $\mathbb Q$ to the arithmetic and geometry of certain Abelian varieties over the field of rational numbers.