Tag 02AL

\(\begin{pmatrix} 4 & 2 & 2 \\ 2 & 4 & 1 \\ 2 & 1 & 4 \end{pmatrix}\)

(4 2 2; 2 4 1; 2 1 4)·invariant lattice of the symplectic K3 group G_78 of Hashimoto

Invariants

Rank3
Signature \((n_+, n_-)\)(3, 0)
Determinant\(36\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}, e_{3}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)
\(e_{1}\)422
\(e_{2}\)241
\(e_{3}\)214
 
[[4, 2, 2], [2, 4, 1], [2, 1, 4]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/3 \oplus \mathbb{Z}/12\)
Unimodularno
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 2)\, L^\Sigma(s - 1, \chi_{-36n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number10

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 10q^{4} + 4q^{6} + 8q^{10} + 12q^{12} + O(q^{13}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots10
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \mathrm{I}_{2,0}(6) \oplus \langle 4 \rangle\)
Rank of \(R(L)\)3
\([L : R(L)]\)2
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong \mathbb{Z}/2\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(B_{2}\)\(\{6, 12\}\)8\(\cong \mathrm{I}_{2,0}(6)\)
\(\Phi_{2}\)\(A_{1}\)\(\{4\}\)2\(\cong \langle 4 \rangle\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{1} - 2e_{2}\)\(12\)
\(\alpha^{(1)}_{2}\)\(e_{2} - e_{3}\)\(6\)
\(\alpha^{(2)}_{1}\)\(e_{1}\)\(4\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 78 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{78} \cong \mathfrak{A}_{4,4}\), \(\mathfrak{G}_{52} \cong 2^2 (C_2 \times C_6)\), \(\mathfrak{G}_{53} \cong 2^2 Q_{12}\), \(\mathfrak{G}_{68} \cong 2^3 D_{12}\), \(\mathfrak{G}_{72} \cong \mathfrak{A}_4^2\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(2)\); the corpus records \(M\) and not its twist.

  • 02AT \(\Lambda_{\mathfrak{G}_{78}}(-1)\) A lattice \(M\) with \(M(-1) \cong L(2)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(2) = \Lambda^G\) for the group \(\mathfrak{G}_{78}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(2)\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{78}\) of Table 10.2 of Hashimoto.

Morphisms

References

  1. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.