A note on \(K3\) surfaces and sphere packings
The main object of the paper under review is the Mordell-Weil lattice of an elliptic \(K3\) surface defined over an algebraically closed field \(k\) contained in the field of complex numbers. To be more precise, the focus is on a special class of such surfaces studied earlier by Inose and Kuwata; they are related to the Kummer surfaces \(\text{Km}(C_1\times C_2)\) where \(C_1\) and \(C_2\) are elliptic curves over \(k\). The reason is that some of such surfaces have high Mordell-Weil rank given by \(r=16+h\) where \(h=\text{rk Hom}(C_1,C_2)\). For the Mordell--Weil lattice of such a surface, assuming that \(C_1\) and \(C_2\) are not isomorphic, the author's main results give the precise value of the minimal norm (\(\mu =4\)) and expressions for the determinant and the center density depending on some parameters. This leads to lattices denser than the recorded ones in dimensions 16, 17, and 18 under the condition that surfaces with required parameters really exist.
scientific article; zbMATH DE number 1538525