Tag 02A9

\(\langle -1 \rangle^{3} \oplus \langle 2 \rangle\)

<-1>^3 + <2>·invariant lattice of the symplectic K3 group G_56 of Hashimoto

Invariants

Rank	4
Signature \((n_+, n_-)\)	(1, 3)
Determinant	\(-2\)

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

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)
\(e_{1}\)	-1	0	0	0
\(e_{2}\)	0	-1	0	0
\(e_{3}\)	0	0	-1	0
\(e_{4}\)	0	0	0	2

[[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 2]]

Integral lattice

Parity	odd
Discriminant group	\(\mathbb{Z}/2\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\)	\(\chi_{-8} = \left(\frac{-8}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, L^\Sigma(s - 2, \chi_{-8})}{L^\Sigma(s - 1, \chi_{-8})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{L^\Sigma(s - 1, \chi_{-8})}\)

Indefinite lattice

Isotropic	yes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	yes
\(L = \mathbb{Z}\Phi_S(L)\) for	\(S = \{-1, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)	\(L\)
Rank of \(R(L)\)	4
\([L : R(L)]\)	1
\(R(L)\) is primitive in \(L\)	yes
Proof	Each \(r_i\) 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 \(L\).

Root	Coordinates	\(b(r, r)\)
\(r_{1}\)	\(e_{1}\)	\(-1\)
\(r_{2}\)	\(e_{2}\)	\(-1\)
\(r_{3}\)	\(e_{3}\)	\(-1\)
\(r_{4}\)	\(e_{4}\)	\(2\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 56 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the groups \(\mathfrak{G}_{56} \cong \Gamma_{25} a_1\), \(\mathfrak{G}_{41} \cong \Gamma_7 a_1\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(-4)\); the corpus records \(M\) and not its twist.

Morphisms

References

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