Hodge classes and Tate classes on simple abelian fourfolds

Families of Motives and the Mumford–Tate Conjecture

“…There is a rich literature on the topic, and wonderful results have been obtained based on such methods. See for instance [58], [59], [63], [64], [70], [88], [89], [90]. The method has its limitations, though.…”

“…The method has its limitations, though. Already for abelian fourfolds, there are cases where the Hodge ring is not generated by divisor classes (see for instance [58], [107]), and apart from a couple of exceptions (see [72], [74], [99]) the Hodge conjecture is not known to be true in these cases. Also, in general it is not possible to determine MT(X) based only on information about End 0 (X).…”

“…Still with X simple of dimension 4, the HC and TC are known for X for some types of endomorphism algebras, but not in general. See [58].…”

Hodge groups of abelian varieties with purely multiplicative reduction

“…The following result follows easily from a result in [8], and characterizes the endomorphism algebras of abelian varieties whose Hodge groups are not semisimple. Proof.…”

Isogeny classes of abelian varieties over function fields

“…Recall that S is arcwise connected with respect to the complex topology. If t is another point of S then every path γ in S from s to t defines an isomorphism γ * : Q)) be the Hodge group of X s [22] (see also [18,19,37]). Recall that Hdg(X s ) is a connected reductive algebraic Q-group and its centralizer End Hdg(Xs) (H 1 (X s , Q)) in End Q (H 1 (X s , Q)) coincides with End 0 (X s ).…”