Arithmetic of K3 Surfaces

Being surfaces of intermediate type, i.e., neither geometrically rational or ruled, nor of general type, K3 surfaces have a rich yet accessible arithmetic theory, which has started to come into focus over the last fifteen years or so. These notes, written to accompany a 4-hour lecture series at the 2015 Arizona Winter School, survey some of these developments, with an emphasis on explicit methods and examples. They are mostly expository, though I have included at the end two admittedly optimistic conjectures on uniform boundedness of Brauer groups (modulo constants) for lattice polarized K3 surfaces over number fields, which to my knowledge have not appeared in print before (Conjectures 4.5 and 4.6). The topics treated in these notes are as follows.