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We use lattice theory to study the isogeny class of a K3 surface. Startingfrom isotropic Brauer classes, we construct isogenies via Kneser method ofneighboring lattices. We also determine the fields of definition of isogenousK3 surfaces, and study Kneser construction over number fields. We then applyour results to relate conjectures about the finiteness of Brauer groups andN\'{e}ron-Severi lattices of K3 surfaces.

Neighbors and arithmetic of isogenous K3 surfaces