Tag 029H
\(\mathrm{Nik}(-1)\)
Nik(-1)·Nikulin lattice with the sign reversed
Invariants
| Rank | 8 |
|---|---|
| Signature \((n_+, n_-)\) | (8, 0) |
| Determinant | \(64\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{8}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | |
|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| \(e_{2}\) | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 1 |
| \(e_{3}\) | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 |
| \(e_{4}\) | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 1 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 |
| \(e_{8}\) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 |
[[2, 0, 0, 0, 0, 0, 0, 1], [0, 2, 0, 0, 0, 0, 0, 1], [0, 0, 2, 0, 0, 0, 0, 1], [0, 0, 0, 2, 0, 0, 0, 1], [0, 0, 0, 0, 2, 0, 0, 1], [0, 0, 0, 0, 0, 2, 0, 1], [0, 0, 0, 0, 0, 0, 2, 1], [1, 1, 1, 1, 1, 1, 1, 4]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{6}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 514 |
| Nikulin invariants \((r, a, \delta)\) | \((8, 6, 0)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{4} \det L\) | trivial |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 7)\, \zeta^\Sigma(s - 4)}{\zeta^\Sigma(s - 3)}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 7)}{\zeta^\Sigma(s - 3)}\) |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(2\) |
|---|---|
| Kissing number | 16 |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 16q^{2} + 368q^{4} + 448q^{6} + 3184q^{8} + O(q^{9}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 128 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \mathrm{I}_{8,0}(2)\) |
| Rank of \(R(L)\) | 8 |
| \([L : R(L)]\) | 2 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/2\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(B_{8}\) | \(\{2, 4\}\) | 128 | \(\cong \mathrm{I}_{8,0}(2)\) |
Simple roots
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha_{1}\) | \(e_{2} + e_{3} + e_{4} + e_{5} + e_{6} + e_{7} - 2e_{8}\) | \(4\) |
| \(\alpha_{2}\) | \(e_{1} - e_{2}\) | \(4\) |
| \(\alpha_{3}\) | \(e_{2} - e_{3}\) | \(4\) |
| \(\alpha_{4}\) | \(e_{3} - e_{4}\) | \(4\) |
| \(\alpha_{5}\) | \(e_{4} - e_{5}\) | \(4\) |
| \(\alpha_{6}\) | \(e_{5} - e_{6}\) | \(4\) |
| \(\alpha_{7}\) | \(e_{6} - e_{7}\) | \(4\) |
| \(\alpha_{8}\) | \(e_{7}\) | \(2\) |
Notes
The Nikulin lattice \(N = \langle N_1, \ldots, N_8, \hat N \rangle\) with \(\hat N = \tfrac{1}{2}(N_1 + \cdots + N_8)\), \(N_i^2 = -2\) and \(N_i \cdot N_j = 0\) for \(i \neq j\) (van Geemen and Sarti, Section 1.5). The classes \(N_i\) are the exceptional curves of the resolution of the quotient of a K3 surface by a Nikulin (symplectic) involution, and \(N\) is negative definite of rank 8 with discriminant group \((\mathbb{Z}/2\mathbb{Z})^6\).
This record is \(N(-1)\), so that its form is positive definite. The basis is \(N_1, \ldots, N_7, \hat N\).
References
- B. van Geemen, A. Sarti, "Nikulin involutions on K3 surfaces", arXiv:math/0602015, Section 1.5.