Almost 40 years ago, Nikulin and Vinberg completed the classification of K3 surfaces with finite automorphism group. Since for Picard rank at least 3 there are only finitely many Picard lattices of such K3 surfaces, one may wonder how complicated the automorphism group becomes in the other cases. I will introduce a special class of K3 surfaces, called of zero entropy, that exhibit particularly simple dynamics, and we will see that the automorphism group of these surfaces is infinite, but abelian up to finite index. I will report on a joint project with Simon Brandhorst, in which we completely classify the Picard lattices of K3 surfaces of zero entropy. I will then show some possible applications to arithmetic problems, such as the study of the set of rational points of K3 surfaces over number fields.

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