A note on $K3$ surfaces and sphere packings

Abstract

We study the Mordell-Weil lattices ([9]) of the elliptic $K3$ surfaces which have been introduced by Inose [4] and Kuwata [6]. The point is that the lattices (of rank up to 18) arising this way can be of some interest in terms of sphere packings. In this note, we treat the case of rank 16, 17, or 18, and show that these Morderll-Weil lattices are potentially comparable with the record lattices in these dimensions ([2]). The detailed accont is in preparation, which will include the corresponding results for other dimensions as well.