Tag 029M

\(U \oplus U(3)^{2} \oplus A_{2}(-1)^{2}\)

U + U(3)^2 + A2(-1)^2·invariant lattice of the symplectic K3 group G_2 of Hashimoto

Invariants

Rank10
Signature \((n_+, n_-)\)(3, 7)
Determinant\(-729\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{10}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}, e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}, e_{10}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)
\(e_{1}\)0100000000
\(e_{2}\)1000000000
\(e_{3}\)0003000000
\(e_{4}\)0030000000
\(e_{5}\)0000030000
\(e_{6}\)0000300000
\(e_{7}\)000000-2100
\(e_{8}\)0000001-200
\(e_{9}\)00000000-21
\(e_{10}\)000000001-2
 
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, -2]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/3)^{6}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)393
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{5} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 9)\, \zeta^\Sigma(s - 5)}{\zeta^\Sigma(s - 4)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 9)}{\zeta^\Sigma(s - 4)}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)\(\geq 1\), by an embedding of \(U^{1}\) in the morphism files

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-2, 2, 6\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)10
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes
ProofEach \(r_i\) 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 \(L\).

RootCoordinates\(b(r, r)\)
\(r_{1}\)\(e_{7}\)\(-2\)
\(r_{2}\)\(e_{8}\)\(-2\)
\(r_{3}\)\(e_{9}\)\(-2\)
\(r_{4}\)\(e_{10}\)\(-2\)
\(r_{5}\)\(e_{1} + e_{2}\)\(2\)
\(r_{6}\)\(e_{1} + e_{7}\)\(-2\)
\(r_{7}\)\(e_{3} + e_{4}\)\(6\)
\(r_{8}\)\(e_{3} + e_{7}\)\(-2\)
\(r_{9}\)\(e_{5} + e_{6}\)\(6\)
\(r_{10}\)\(e_{5} + e_{7}\)\(-2\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 2 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{2} \cong C_3\) of Table 10.2, and this record is \(\Lambda^G\).

  • 0150 \(K_{12}\) A lattice \(M\) with \(M(-1) \cong L^\perp\) in \(\mathrm{II}_{3,19}\), where \(L = \Lambda^G\) for the group \(\mathfrak{G}_{2}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{2}\) 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.