Tag 026L
\(U \oplus A_{1}(-1)^{7}\)
U + A1(-1)^7·(r, a, delta) = (9, 7, 1)
Invariants
| Rank | 9 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 8) |
| Determinant | \(128\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{9}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | |
|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 |
[[0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{7}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 514 |
| Nikulin invariants \((r, a, \delta)\) | \((9, 7, 1)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 8)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 8)\, L^\Sigma(s - 4, \chi_{128n})\) |
| Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\) | \(\geq 1\), by an embedding of \(U^{1}\) in the morphism files |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-2, 2\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 9 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
| Proof | Each \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\). |
Roots that generate \(\mathbb{Z}\Phi(L)\)
The \(r_i\) are a basis of \(L\).
| Root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{3}\) | \(-2\) |
| \(r_{2}\) | \(e_{4}\) | \(-2\) |
| \(r_{3}\) | \(e_{5}\) | \(-2\) |
| \(r_{4}\) | \(e_{6}\) | \(-2\) |
| \(r_{5}\) | \(e_{7}\) | \(-2\) |
| \(r_{6}\) | \(e_{8}\) | \(-2\) |
| \(r_{7}\) | \(e_{9}\) | \(-2\) |
| \(r_{8}\) | \(e_{1} + e_{2}\) | \(2\) |
| \(r_{9}\) | \(e_{1} + e_{3}\) | \(-2\) |
Notes
This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (9, 7, 1)\).
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.
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
- V. V. Nikulin, "Integral symmetric bilinear forms and some of their applications", Math. USSR-Izv. 14 (1980), 103-167, DOI 10.1070/IM1980v014n01ABEH001060.