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We study Gushel-Mukai (GM) varieties of dimension 4 or 6 in characteristic$p$. Our main result is the Tate conjecture for all such varieties overfinitely generated fields of characteristic $p\geq 5$. In the case of GMsixfolds, we follow the method used by Madapusi Pera in his proof of the Tateconjecture for K3 surfaces. As input for this, we prove a number of basicresults about GM sixfolds, such as the fact that there are no nonzero globalvector fields. For GM fourfolds, we prove the Tate conjecture by reducing it tothe case of GM sixfolds by making use of the notion of generalised partnersplus the fact that generalised partners in characteristic 0 have isomorphicChow motives in middle degree. Several steps in the proofs rely on results incharacteristic 0 that are proven in a companion paper.

The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$