Tag 02A6

\(U \oplus \langle -6 \rangle^{2}\)

U + <-6>^2·invariant lattice of the symplectic K3 group G_51 of Hashimoto

Invariants

Rank4
Signature \((n_+, n_-)\)(1, 3)
Determinant\(-36\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)
\(e_{1}\)0-100
\(e_{2}\)-1000
\(e_{3}\)00-60
\(e_{4}\)000-6
 
[[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, -6, 0], [0, 0, 0, -6]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/6)^{2}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\)\(\chi_{-4} = \left(\frac{-4}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-4})}{L^\Sigma(s - 1, \chi_{-4})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-4})}\)

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-6, -2, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)4
\([L : R(L)]\)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\(b(r, r)\)
\(r_{1}\)\(e_{3}\)\(-6\)
\(r_{2}\)\(e_{4}\)\(-6\)
\(r_{3}\)\(e_{1} + e_{2}\)\(-2\)
\(r_{4}\)\(e_{1} - e_{2}\)\(2\)
\(r_{5}\)\(3e_{1} + e_{3}\)\(-6\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 51 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{51} \cong C_2 \times \mathfrak{S}_4\), \(\mathfrak{G}_{35} \cong C_2 \times \mathfrak{A}_4\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(-2)\); the corpus records \(M\) and not its twist.

Roots: \(e_{3}\) and \(e_{4}\) have \(b(r, r) = -6\) and \(b(r, L) = 6\mathbb{Z}\), and \(-6\) divides \(12\). \(e_{1} + e_{2}\) has \(b(r, r) = -2\) and \(b(r, L) = \mathbb{Z}\), and \(-2\) divides \(2\). \(e_{1} - e_{2}\) has \(b(r, r) = 2\) and \(b(r, L) = \mathbb{Z}\), and \(2\) divides \(2\). \(3 e_{1} + e_{3}\) has \(b(r, r) = -6\) and \(b(r, L) = 3\mathbb{Z}\), and \(-6\) divides \(6\). These roots generate \(L\), so \(L = \mathbb{Z}\Phi(L)\).

  • 02B6 \(\Lambda_{\mathfrak{G}_{51}}(-1)\) A lattice \(M\) with \(M(-1) \cong L(-2)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-2) = \Lambda^G\) for the group \(\mathfrak{G}_{51}\) 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 \(\mathfrak{G}_{51}\) 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.