Dual lattices and discriminant groups

Let \(L\) be an integral lattice that is nondegenerate.

Dual lattice

The dual lattice \(L^*\) is \(\{x \in L \otimes_{\mathbb{Z}} \mathbb{Q} : b(x, y) \in \mathbb{Z} \text{ for all } y \in L\}\), with the \(\mathbb{Q}\)-bilinear extension of \(b\).

\(L^*\) contains \(L\), and its chosen basis is dual to that of \(L\).

Discriminant group

The discriminant group \(A_L\) of \(L\) is the cokernel of the correlation \(L \to \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{Z})\), \(x \mapsto b(x, -)\).

The isomorphism \(L^* \to \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{Z})\), \(y \mapsto b(y, -)\), gives \(A_L \cong L^*/L\), a finite group of order \(|\det L|\).

The page states \(A_L\) by its invariant factors \(d_1 \mid d_2 \mid \dots\), each greater than 1; the empty list is the trivial group.

The discriminant form is \(b_{A_L}(x + L, y + L) = b(x, y) + \mathbb{Z}\), with values in \(\mathbb{Q}/\mathbb{Z}\).

Unimodular lattices

\(L\) is unimodular when the correlation \(L \to \operatorname{Hom}_{\mathbb{Z}}(L, \mathbb{Z})\) is an isomorphism; equivalently \(A_L = 0\), and equivalently \(\det L = \pm 1\).

2-elementary lattices

\(L\) is 2-elementary when \(A_L \cong (\mathbb{Z}/2)^a\) with \(a \geq 1\).

An even lattice \(L\) with \(2 A_L = 0\) has Nikulin’s invariants \((r, a, \delta)\): \(r\) is the rank, \(A_L \cong (\mathbb{Z}/2)^a\), and \(\delta = 0\) exactly when \(b(x, x)\) is an integer for every \(x\) in the dual lattice, else \(\delta = 1\). The even unimodular lattices are the case \(a = 0\), with \(\delta = 0\).

\(\delta = 0\) exactly when every diagonal entry of \(G^{-1}\) is an integer, for the Gram matrix \(G\) of \(L\).

The basis of \(L^*\) dual to the chosen basis of \(L\) has Gram matrix \(G^{-1}\). Since \(2 L^* \subseteq L\), every \(2 b(x, y)\) with \(x, y \in L^*\) is an integer, so for \(x = \sum_i c_i f_i\) the value \(b(x, x) = \sum_i c_i^2 (G^{-1})_{ii} + \sum_{i < j} 2 c_i c_j (G^{-1})_{ij}\) is an integer for every \(x\) exactly when every \((G^{-1})_{ii}\) is.

An indefinite even 2-elementary lattice is determined by its signature and \((r, a, \delta)\) (Nikulin 1980: V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR-Izv. 14 (1980), 103–167).

An even hyperbolic lattice \(S\) with \(2 A_S = 0\) is determined by \((r, a, \delta)\). The isomorphism classes of involutions of the K3 lattice whose invariant lattice is hyperbolic correspond to the isomorphism classes of their invariant lattices, which are the even hyperbolic lattices \(S\) with \(2 A_S = 0\) that have a primitive embedding in the K3 lattice; Table 1 of the source lists them, and they are the invariant lattices of the nonsymplectic involutions of K3 surfaces (Nikulin 1983: 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, Theorem 4.3.2, Table 1 and Remark 4.5.3).