Hodge classes on abelian varieties of low dimension

In this paper we study Hodge classes on complex abelian varieties. We prove some general results that allow us, in certain cases, to compute the Hodge group of a product abelian variety $X = X_1 \times X_2$ once we know the Hodge groups of the two factors. Using these results we can compute the Hodge groups of all abelian varieties of dimension $\leq 5$. We prove that the Hodge ring of any such abelian variety $X$ is generated by divisor classes together with the so-called Weil classes on (quotients of) $X$.