Tag 02AY

\(\Lambda_{\mathfrak{G}_{63}}(-1)\)

Lambda_G(-1), G = G_63 of Hashimoto·coinvariant lattice of the symplectic K3 group G_63 of Hashimoto, twisted by -1

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(216\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{19}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)42-22-21-2-211-2-21-12122-2
\(e_{2}\)24020-1-1-1-1-1-1-1-1-212220
\(e_{3}\)-204-22-1020-10001-1-10-22
\(e_{4}\)22-24-1-1-1-1-1-1-1-1-1-201220
\(e_{5}\)-202-14-2020-2000-10-1-102
\(e_{6}\)1-1-1-1-24-1-201112101-10-2
\(e_{7}\)-2-10-10-140-1022-21-1-1-1-11
\(e_{8}\)-2-12-12-204000001-1-2-1-11
\(e_{9}\)1-10-100-1042-2-2211-110-1
\(e_{10}\)1-1-1-1-2100240-1120000-2
\(e_{11}\)-2-10-10120-2042-11-20-200
\(e_{12}\)-2-10-10120-2-124-11-10-2-10
\(e_{13}\)1-10-102-2021-1-1410000-1
\(e_{14}\)-1-21-2-1111121114-1-1-1-2-1
\(e_{15}\)21-1000-1-110-2-10-14001-1
\(e_{16}\)12-11-11-1-2-10000-10411-1
\(e_{17}\)2202-1-1-1-110-2-20-101410
\(e_{18}\)22-2200-1-1000-10-21114-1
\(e_{19}\)-20202-211-1-200-1-1-1-10-14
 
[[4, 2, -2, 2, -2, 1, -2, -2, 1, 1, -2, -2, 1, -1, 2, 1, 2, 2, -2], [2, 4, 0, 2, 0, -1, -1, -1, -1, -1, -1, -1, -1, -2, 1, 2, 2, 2, 0], [-2, 0, 4, -2, 2, -1, 0, 2, 0, -1, 0, 0, 0, 1, -1, -1, 0, -2, 2], [2, 2, -2, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -2, 0, 1, 2, 2, 0], [-2, 0, 2, -1, 4, -2, 0, 2, 0, -2, 0, 0, 0, -1, 0, -1, -1, 0, 2], [1, -1, -1, -1, -2, 4, -1, -2, 0, 1, 1, 1, 2, 1, 0, 1, -1, 0, -2], [-2, -1, 0, -1, 0, -1, 4, 0, -1, 0, 2, 2, -2, 1, -1, -1, -1, -1, 1], [-2, -1, 2, -1, 2, -2, 0, 4, 0, 0, 0, 0, 0, 1, -1, -2, -1, -1, 1], [1, -1, 0, -1, 0, 0, -1, 0, 4, 2, -2, -2, 2, 1, 1, -1, 1, 0, -1], [1, -1, -1, -1, -2, 1, 0, 0, 2, 4, 0, -1, 1, 2, 0, 0, 0, 0, -2], [-2, -1, 0, -1, 0, 1, 2, 0, -2, 0, 4, 2, -1, 1, -2, 0, -2, 0, 0], [-2, -1, 0, -1, 0, 1, 2, 0, -2, -1, 2, 4, -1, 1, -1, 0, -2, -1, 0], [1, -1, 0, -1, 0, 2, -2, 0, 2, 1, -1, -1, 4, 1, 0, 0, 0, 0, -1], [-1, -2, 1, -2, -1, 1, 1, 1, 1, 2, 1, 1, 1, 4, -1, -1, -1, -2, -1], [2, 1, -1, 0, 0, 0, -1, -1, 1, 0, -2, -1, 0, -1, 4, 0, 0, 1, -1], [1, 2, -1, 1, -1, 1, -1, -2, -1, 0, 0, 0, 0, -1, 0, 4, 1, 1, -1], [2, 2, 0, 2, -1, -1, -1, -1, 1, 0, -2, -2, 0, -1, 0, 1, 4, 1, 0], [2, 2, -2, 2, 0, 0, -1, -1, 0, 0, 0, -1, 0, -2, 1, 1, 1, 4, -1], [-2, 0, 2, 0, 2, -2, 1, 1, -1, -2, 0, 0, -1, -1, -1, -1, 0, -1, 4]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/2 \oplus \mathbb{Z}/6 \oplus \mathbb{Z}/18\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)8
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2, 3\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 18)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-216n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number8100

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 8100q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots0
\(R(L) = \mathbb{Z}\Phi(L)\)\(0\)
Rank of \(R(L)\)0
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. The coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) is negative definite of rank 19 and has no vector of norm \(-2\); this record is \(\Lambda_G(-1)\) for the group \(\mathfrak{G}_{63} \cong M_9\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 63 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[5,20] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 5 that is number 20 of rank 5 in their Table 1. It satisfies their K3 criterion \(\operatorname{rk} B \geq 5\), \(\operatorname{rk} B - \operatorname{rk} A_B \geq 2\), and among the coinvariant lattices that satisfy it, it is the only one whose rank and discriminant form are those of \(\Lambda_G(-1)\): its discriminant form is the form \(q\) of row 63 of Table 10.3. The sum of its twist by \(-1\) and \(\Lambda^G\) therefore glues to an even unimodular lattice of signature \((3, 19)\), on which the stabilizer of \(B\) in \(\mathrm{Co}_0\) acts with this coinvariant lattice.

  • 02AC \(\langle 1 \rangle \oplus A_{2}(3)\) A lattice \(M\) with \(M(2) \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{63}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(-1)\) is the coinvariant lattice \(\Lambda_G\) of the group \(\mathfrak{G}_{63}\) of Table 10.2 of Hashimoto.
  • 028S \(\Lambda_{24}\) The Leech lattice \(\Lambda\), in which \(L\) is the orthogonal complement of the sublattice fixed by a subgroup of \(\mathrm{Co}_0\).

Morphisms

References

  1. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.
  2. G. Höhn and G. Mason, "The 290 fixed-point sublattices of the Leech lattice", arXiv:1505.06420, Table 1 and its ancillary Magma file lattices.txt, entry lattices[5,20].