Tag 0152
\(U \oplus A_{1}(-1)^{10}\)
U + A1(-1)^10·(r, a, delta) = (12, 10, 1)
Invariants
| Rank | 12 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 11) |
| Determinant | \(-1024\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\) \(\oplus\) \(\mathbb{Z}\{e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}\}\) \(\oplus\) \(\mathbb{Z}\{e_{12}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | \(e_{6}\) | \(e_{7}\) | \(e_{8}\) | \(e_{9}\) | \(e_{10}\) | \(e_{11}\) | \(e_{12}\) | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(e_{1}\) | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{2}\) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{6}\) | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 |
| \(e_{7}\) | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 |
| \(e_{8}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 |
| \(e_{9}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 |
| \(e_{10}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 |
| \(e_{11}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 |
| \(e_{12}\) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 |
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{10}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | not decided |
| Nikulin invariants \((r, a, \delta)\) | \((12, 10, 1)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{6} \det L\) | \(\chi_{-4} = \left(\frac{-4}{\cdot}\right)\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 11)\, L^\Sigma(s - 6, \chi_{-4})}{L^\Sigma(s - 5, \chi_{-4})}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 11)}{L^\Sigma(s - 5, \chi_{-4})}\) |
| Genus symbol | \(\mathrm{II}_{1,11}\ \bigl(2\colon 1^{2}\,[2^{10}]_{6}\bigr)\) |
| Class number of the genus | 1 |
| 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 |
|---|
Hyperbolic lattice
| Reflective | yes declared |
|---|
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 \(R(L)\) | 12 |
| \([L : R(L)]\) | 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 | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{3}\) | \(-2\) |
| \(r_{2}\) | \(e_{4}\) | \(-2\) |
| \(r_{3}\) | \(e_{5}\) | \(-2\) |
| \(r_{4}\) | \(e_{6}\) | \(-2\) |
| \(r_{5}\) | \(e_{7}\) | \(-2\) |
| \(r_{6}\) | \(e_{8}\) | \(-2\) |
| \(r_{7}\) | \(e_{9}\) | \(-2\) |
| \(r_{8}\) | \(e_{10}\) | \(-2\) |
| \(r_{9}\) | \(e_{11}\) | \(-2\) |
| \(r_{10}\) | \(e_{12}\) | \(-2\) |
| \(r_{11}\) | \(e_{1} + e_{2}\) | \(2\) |
| \(r_{12}\) | \(e_{1} + e_{3}\) | \(-2\) |
Notes
This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (12, 10, 1)\).
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.
Its invariants satisfy \(r + a = 22\), and it is reflective (Alexeev, Theorem 1.1).
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. Alexeev, "Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler manifolds", arXiv:2209.09110v4, Theorem 1.1. https://arxiv.org/abs/2209.09110v4