Tag 02AJ

\(\langle 1 \rangle \oplus \langle 2 \rangle \oplus \langle 3 \rangle\)

<1> + <2> + <3>·invariant lattice of the symplectic K3 group G_76 of Hashimoto

Invariants

Rank3
Signature \((n_+, n_-)\)(3, 0)
Determinant\(6\)

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, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)
\(e_{1}\)100
\(e_{2}\)020
\(e_{3}\)003
 
[[1, 0, 0], [0, 2, 0], [0, 0, 3]]

Integral lattice

Parityodd
Discriminant group\(\mathbb{Z}/6\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)1
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 2)\, L^\Sigma(s - 1, \chi_{-6n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(1\)
Kissing number2

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 2q + 2q^{2} + 6q^{3} + 6q^{4} + 4q^{5} + 12q^{6} + 4q^{7} + 2q^{8} + 14q^{9} + 8q^{11} + 18q^{12} + O(q^{13}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{1, 2, 3\}\)
Number of roots6
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 3 \rangle \oplus \langle 2 \rangle \oplus \langle 1 \rangle\)
Rank of \(R(L)\)3
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{1}\)\(\{3\}\)2\(\cong \langle 3 \rangle\)
\(\Phi_{2}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{3}\)\(A_{1}\)\(\{1\}\)2\(\cong \langle 1 \rangle\)

Simple roots

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

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

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 76 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{76} \cong H_{192}\), \(\mathfrak{G}_{66} \cong 2^4 C_6\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(4)\); the corpus records \(M\) and not its twist.

  • 02AU \(\Lambda_{\mathfrak{G}_{76}}(-1)\) A lattice \(M\) with \(M(-1) \cong L(4)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(4) = \Lambda^G\) for the group \(\mathfrak{G}_{76}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(4)\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{76}\) 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.