Nikulin's 2-elementary lattices

V. V. Nikulin, Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications, J. Soviet Math. 22 (1983), 1401–1475, DOI 10.1007/BF01094757, lists in Table 1 the even hyperbolic lattices \(L\) with \(2 A_L = 0\) that have a primitive embedding in the K3 lattice. By Remark 4.5.3 these are the invariant lattices of the nonsymplectic involutions of K3 surfaces, and by Theorem 4.3.2 Nikulin’s invariants \((r, a, \delta)\) determine each of them. The table has 75 rows, and the members of this collection are the 75 records, one for each row. Each record states \(\delta\), and its page shows \((r, a, \delta)\).

The 13 rows with \(a = r\) are the lattices \(M(2)\) for a unimodular lattice \(M\), since \(2 L^* \subseteq L\) and \(|A_L| = 2^r\) give \(L^* = \tfrac{1}{2} L\). The corpus records a lattice once up to twist, and these 13 twists are the only exception: the classification names \(M(2)\) and not \(M\). Each twist record and its \(M\) link to each other:

\(M(2)\) has \(\delta = 0\) exactly when \(M\) is even: the Gram matrix of \(M(2)\) is \(2G\), and the diagonal entries of \((2G)^{-1} = \tfrac{1}{2} G^{-1}\) are integers exactly when those of \(G^{-1}\) are even, which for a unimodular \(M\) holds exactly when \(M\) is even.