Formal Brauer groups arising from certain weighted K3 surfaces (Q1124977)

The formal Brauer groups arising from certain K3 surfaces in weighted projective \(3\)-spaces are studied. Here the varieties are defined over base rings \(k,\) where \(k\) is either an algebraically closed field of characteristic \(p>0\) or a Noetherian ring which is flat and of finite type over \(\mathbb Z\). In the case where \(k\) is an algebraically closed field of positive characteristic, the height of a formal Brauer group is an integer between 1 and 10, provided the formal group is of finite height. The authors provide K3 surfaces with formal Brauer groups of heights 1, 2, 3, 4, 6, and 10. Many examples are provided, in terms of the field characteristic and the weightings. When \(k\) is a Noetherian ring that is flat and of finite type over \(\mathbb Z\), the author discusses the issue of representing a formal Brauer group arising from a weighted diagonal (or quasi-diagonal) \(K3\) surface by a formal group law. The formal group law is described in terms of its logarithm. Finally, an analogue of the Atkin and Swinnerton-Dyer congruence for elliptic curves is proved for these K3 surfaces.