\(K3\) surfaces and sphere packings (Q998934)

It is known that the Kummer surface \(\text{Km}(C_{1}\times C_{2})\) of the product of two elliptic curves \(C_{1}\) and \(C_{2}\) is isomorphic to the elliptic surface given by the Weierstrass equation \(F_{\alpha,\beta}^{2}:y^{2}=x^{3}-3\alpha x + (t^{2}+1/t^{2}-2\beta)\) for some \(\alpha\) and \(\beta\) [\textit{H. Inose}, in: Proc. int. Symp. on algebraic geometry, Kyoto 1977, 495--502 (1977; Zbl 0411.14009)]. Let \(F_{\alpha,\beta}^{n}:y^{2}=x^{3}-3\alpha x + (t^{n}+1/t^{n}-2\beta)\) be the elliptic surfaces obtained from the base change. When \(n\leq 6\), \(F_{\alpha,\beta}^{n}\) is a \(K3\) surface, and over the field of complex numbers, a transcendental argument due to Inose shows that the rank of the transcendental lattice is independent of \(n\). We thus know the rank of the Mordell-Weil lattice \(\text{MW}(F_{\alpha,\beta}^{n})\) [\textit{M. Kuwata}, Comment. Math. Univ. St. Pauli 49, No. 1, 91--100 (2000; Zbl 1018.14013)]. For example, if \(C_{1}\) and \(C_{2}\) are not isomorphic, we have \(16\leq \text{rank}\text{MW}(F_{\alpha,\beta}^{n})\leq 18\) for \(n=5,6\). However, the lattice structure of \(\text{MW}(F_{\alpha,\beta}^{n})\) had not been known, not to mention its generators. In the paper under review the author determines the structure of \(\text{MW}(F_{\alpha,\beta}^{n})\) explicitly, and showed a method to obtain an explicit set of generators. Key ingredients of the proof include the upper bound of the center density of the lattice sphere packing, and the structure of the Mordell-Weil lattice of the rational elliptic surface obtained by letting \(s=t+1/t\). The results have some applications to singular \(K3\) surfaces (i.e., \(K3\) surfaces with maximal possible Picard number).