Tag 026Y

\(U(2) \oplus D_{6}(-1) \oplus D_{4}(-1)\)

U(2) + D6(-1) + D4(-1)·(r, a, delta) = (12, 6, 1)

Invariants

Rank12
Signature \((n_+, n_-)\)(1, 11)
Determinant\(-64\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{12}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, \dots, e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}, \dots, e_{12}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)
\(e_{1}\)020000000000
\(e_{2}\)200000000000
\(e_{3}\)00-2100000000
\(e_{4}\)001-210000000
\(e_{5}\)0001-21000000
\(e_{6}\)00001-2110000
\(e_{7}\)000001-200000
\(e_{8}\)0000010-20000
\(e_{9}\)00000000-2100
\(e_{10}\)000000001-211
\(e_{11}\)0000000001-20
\(e_{12}\)00000000010-2
 
[[0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -2, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{6}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)122
Nikulin invariants \((r, a, \delta)\)\((12, 6, 1)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{6} \det L\)\(\chi_{-4} = \left(\frac{-4}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 11)\, L^\Sigma(s - 6, \chi_{-4})}{L^\Sigma(s - 5, \chi_{-4})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 11)}{L^\Sigma(s - 5, \chi_{-4})}\)

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-2, 4\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)12
\([L : R(L)]\)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\(b(r, r)\)
\(r_{1}\)\(e_{3}\)\(-2\)
\(r_{2}\)\(e_{4}\)\(-2\)
\(r_{3}\)\(e_{5}\)\(-2\)
\(r_{4}\)\(e_{6}\)\(-2\)
\(r_{5}\)\(e_{7}\)\(-2\)
\(r_{6}\)\(e_{8}\)\(-2\)
\(r_{7}\)\(e_{9}\)\(-2\)
\(r_{8}\)\(e_{10}\)\(-2\)
\(r_{9}\)\(e_{11}\)\(-2\)
\(r_{10}\)\(e_{12}\)\(-2\)
\(r_{11}\)\(e_{1} + e_{2}\)\(4\)
\(r_{12}\)\(e_{1} + e_{3}\)\(-2\)

Notes

This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (12, 6, 1)\).

Nikulin lists it in Table 1 of Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections (1983), the table of the even hyperbolic lattices \(L\) with \(2 A_L = 0\) that have a primitive embedding in the K3 lattice. These are the invariant lattices of the nonsymplectic involutions of K3 surfaces (Remark 4.5.3), and Theorem 4.3.2 states that \((r, a, \delta)\) determines each of them.

Morphisms

References

  1. V. V. Nikulin, "Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications", J. Soviet Math. 22 (1983), 1401-1475, §4, Table 1 and Theorem 4.3.2. https://doi.org/10.1007/BF01094757
  2. V. V. Nikulin, "Integral symmetric bilinear forms and some of their applications", Math. USSR-Izv. 14 (1980), 103-167, DOI 10.1070/IM1980v014n01ABEH001060.