On the image of Galois representations and the Hodge conjecture for abelian varieties


First, I am going to discuss results of a joint work with Banaszak and Krason on the image of l-adic representation attached to Tate modules of abelian varieties of type I, II and III in the Albert classification. We computed the images explicitly and verified Mumford-Tate, Hodge and Tate conjectures for a large class of these varieties.

As another application of the computation of images, one obtains a criterion for detecting linear relations among given nontorsion points in the Mordell-Weil group of an abelian variety, by using reduction maps.

In the second part of the talk, I'll explain a recent result obtained in collaboration with Gornisiewicz on the criteria of this type for all abelian varieties.