Tag 0002
\(\langle 2 \rangle\)
<2>·(r, a, delta) = (1, 1, 1)
Invariants
| Rank | 1 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 0) |
| Determinant | \(2\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}\).
| \(e_{1}\) | |
|---|---|
| \(e_{1}\) | 2 |
[[2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \(\mathbb{Z}/2\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 1 |
| Nikulin invariants \((r, a, \delta)\) | \((1, 1, 1)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s)\, L^\Sigma(s, \chi_{2n})\) |
| Genus symbol | \(\mathrm{II}_{1,0}\ \bigl(2\colon [2^{1}]_{1}\bigr)\) |
| Class number of the genus | 1 |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 2 |
| Order of the isometry group | \(2 = 2\) |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 2q^{2} + 2q^{8} + O(q^{13}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{2\}\) |
| Number of roots | 2 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \langle 2 \rangle\) |
| Rank of \(R(L)\) | 1 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(A_{1}\) | \(\{2\}\) | 2 | \(\cong \langle 2 \rangle\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{1}\) | \(2\) |
Notes
This is the even 2-elementary lattice of signature \((1, 0)\) with invariants \((r, a, \delta) = (1, 1, 1)\).
Since \(a = r\), \(L^* = \tfrac{1}{2} L\), and \(L\) is the twist \(M(2)\) of the unimodular lattice \(M = \langle 1 \rangle\), the record 0001. The corpus records a twist \(M(n)\), \(|n| \geq 2\), only for the 13 rows \(a = r\) of Nikulin’s table, because the classification names the twist and not \(M\).
Nikulin lists it in Table 1 of Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections (1983), the table of the even hyperbolic lattices \(L\) with \(2 A_L = 0\) that have a primitive embedding in the K3 lattice. These are the invariant lattices of the nonsymplectic involutions of K3 surfaces (Remark 4.5.3), and Theorem 4.3.2 states that \((r, a, \delta)\) determines each of them.
\(\langle a \rangle\) is the lattice of rank 1 with basis \(e\) and \(b(e, e) = a\), and \(A_1(-1) = \langle -2 \rangle\).
Related lattices
Morphisms
References
- 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, §4, Table 1 and Theorem 4.3.2. https://doi.org/10.1007/BF01094757