Tag 027E

\(U^{3} \oplus E_{8}(-1)^{2}\)

U^3 + E8(-1)^2·II_{3,19}·L_K3·K3 lattice

Geometric objects

Invariants

Rank22
Signature \((n_+, n_-)\)(3, 19)
Determinant\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{22}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}, \dots, e_{14}\}\) \(\oplus\) \(\mathbb{Z}\{e_{15}, \dots, e_{22}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)
\(e_{1}\)0100000000000000000000
\(e_{2}\)1000000000000000000000
\(e_{3}\)0001000000000000000000
\(e_{4}\)0010000000000000000000
\(e_{5}\)0000010000000000000000
\(e_{6}\)0000100000000000000000
\(e_{7}\)000000-2010000000000000
\(e_{8}\)0000000-201000000000000
\(e_{9}\)00000010-21000000000000
\(e_{10}\)000000011-2100000000000
\(e_{11}\)0000000001-210000000000
\(e_{12}\)00000000001-21000000000
\(e_{13}\)000000000001-2100000000
\(e_{14}\)0000000000001-200000000
\(e_{15}\)00000000000000-20100000
\(e_{16}\)000000000000000-2010000
\(e_{17}\)0000000000000010-210000
\(e_{18}\)00000000000000011-21000
\(e_{19}\)000000000000000001-2100
\(e_{20}\)0000000000000000001-210
\(e_{21}\)00000000000000000001-21
\(e_{22}\)000000000000000000001-2
 
[[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((22, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{11} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 21)\, \zeta^\Sigma(s - 11)}{\zeta^\Sigma(s - 10)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 21)}{\zeta^\Sigma(s - 10)}\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)\(\geq 3\), by an embedding of \(U^{3}\) in the morphism files

Orbits of primitive vectors \(F_{L,\Gamma}(z, w) = \sum_n c_\Gamma(n)\, z^n\), with \(w^n\) for \(z^{-n}\)

\(\Gamma = \mathrm{O}(L)\)\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.
\(\Gamma = \mathrm{SO}(L)\)\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.
\(\Gamma = \widetilde{O}(L)\)\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.
\(\Gamma = \widetilde{SO}(L)\)\(1 + z^{2} + w^{2} + z^{4} + w^{4} + O(z^{5}, w^{5})\), by V. Gritsenko, K. Hulek and G. K. Sankaran, "Moduli of K3 surfaces and irreducible symplectic manifolds", arXiv:1012.4155, Lemma 7.5 (Eichler criterion): L contains U^2 and A_L = 0, so the SOtilde-orbit of a primitive vector is determined by its norm; each group contains SOtilde.

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-2, 2\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)22
\([L : R(L)]\)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\(b(r, r)\)
\(r_{1}\)\(e_{7}\)\(-2\)
\(r_{2}\)\(e_{8}\)\(-2\)
\(r_{3}\)\(e_{9}\)\(-2\)
\(r_{4}\)\(e_{10}\)\(-2\)
\(r_{5}\)\(e_{11}\)\(-2\)
\(r_{6}\)\(e_{12}\)\(-2\)
\(r_{7}\)\(e_{13}\)\(-2\)
\(r_{8}\)\(e_{14}\)\(-2\)
\(r_{9}\)\(e_{15}\)\(-2\)
\(r_{10}\)\(e_{16}\)\(-2\)
\(r_{11}\)\(e_{17}\)\(-2\)
\(r_{12}\)\(e_{18}\)\(-2\)
\(r_{13}\)\(e_{19}\)\(-2\)
\(r_{14}\)\(e_{20}\)\(-2\)
\(r_{15}\)\(e_{21}\)\(-2\)
\(r_{16}\)\(e_{22}\)\(-2\)
\(r_{17}\)\(e_{1} + e_{2}\)\(2\)
\(r_{18}\)\(e_{1} + e_{7}\)\(-2\)
\(r_{19}\)\(e_{3} + e_{4}\)\(2\)
\(r_{20}\)\(e_{3} + e_{7}\)\(-2\)
\(r_{21}\)\(e_{5} + e_{6}\)\(2\)
\(r_{22}\)\(e_{5} + e_{7}\)\(-2\)

Notes

The K3 lattice \(L_{K3}\): \(H^2(S, \mathbb{Z})\) with the intersection form, for a complex K3 surface \(S\), is the even unimodular lattice of signature \((3, 19)\), \(U^3 \oplus E_8(-1)^2\) (Gritsenko, Hulek, Sankaran, equation (3)).

Morphisms

References

  1. V. Gritsenko, K. Hulek, G. K. Sankaran, "The Kodaira dimension of the moduli of K3 surfaces", arXiv:math/0607339.