Tag 02AC
\(\langle 1 \rangle \oplus A_{2}(3)\)
<1> + A2(3)·invariant lattice of the symplectic K3 group G_63 of Hashimoto
Invariants
| Rank | 3 |
|---|---|
| Signature \((n_+, n_-)\) | (3, 0) |
| Determinant | \(27\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}, e_{3}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}, e_{3}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | |
|---|---|---|---|
| \(e_{1}\) | 1 | 0 | 0 |
| \(e_{2}\) | 0 | 6 | 3 |
| \(e_{3}\) | 0 | 3 | 6 |
[[1, 0, 0], [0, 6, 3], [0, 3, 6]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \(\mathbb{Z}/3 \oplus \mathbb{Z}/9\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 2 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2, 3\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 2)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 2)\, L^\Sigma(s - 1, \chi_{-27n})\) |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(1\) |
|---|---|
| Kissing number | 2 |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 2q + 2q^{4} + 6q^{6} + 12q^{7} + 2q^{9} + 12q^{10} + O(q^{13}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{1, 6\}\) |
| Number of roots | 14 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong A_{2}(3) \oplus \langle 1 \rangle\) |
| Rank of \(R(L)\) | 3 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(G_{2}\) | \(\{6, 18\}\) | 12 | \(\cong A_{2}(3)\) |
| \(\Phi_{2}\) | \(A_{1}\) | \(\{1\}\) | 2 | \(\cong \langle 1 \rangle\) |
Simple roots
\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha^{(1)}_{1}\) | \(e_{3}\) | \(6\) |
| \(\alpha^{(1)}_{2}\) | \(e_{2} - 2e_{3}\) | \(18\) |
| \(\alpha^{(2)}_{1}\) | \(e_{1}\) | \(1\) |
Notes
Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 63 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{63} \cong M_9\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(2)\); the corpus records \(M\) and not its twist.
Related lattices
Morphisms
References
- K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.