Tag 0289

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

U^2 + E8(-1)^2 + <-2>·L_2d for d = 1

Invariants

Rank	21
Signature \((n_+, n_-)\)	(2, 19)
Determinant	\(-2\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{21}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}, \dots, e_{12}\}\) \(\oplus\) \(\mathbb{Z}\{e_{13}, \dots, e_{20}\}\) \(\oplus\) \(\mathbb{Z}\{e_{21}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)	\(e_{11}\)	\(e_{12}\)	\(e_{13}\)	\(e_{14}\)	\(e_{15}\)	\(e_{16}\)	\(e_{17}\)	\(e_{18}\)	\(e_{19}\)	\(e_{20}\)	\(e_{21}\)
\(e_{1}\)	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{2}\)	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{3}\)	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{4}\)	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{5}\)	0	0	0	0	-2	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{6}\)	0	0	0	0	0	-2	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{7}\)	0	0	0	0	1	0	-2	1	0	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{8}\)	0	0	0	0	0	1	1	-2	1	0	0	0	0	0	0	0	0	0	0	0	0
\(e_{9}\)	0	0	0	0	0	0	0	1	-2	1	0	0	0	0	0	0	0	0	0	0	0
\(e_{10}\)	0	0	0	0	0	0	0	0	1	-2	1	0	0	0	0	0	0	0	0	0	0
\(e_{11}\)	0	0	0	0	0	0	0	0	0	1	-2	1	0	0	0	0	0	0	0	0	0
\(e_{12}\)	0	0	0	0	0	0	0	0	0	0	1	-2	0	0	0	0	0	0	0	0	0
\(e_{13}\)	0	0	0	0	0	0	0	0	0	0	0	0	-2	0	1	0	0	0	0	0	0
\(e_{14}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	-2	0	1	0	0	0	0	0
\(e_{15}\)	0	0	0	0	0	0	0	0	0	0	0	0	1	0	-2	1	0	0	0	0	0
\(e_{16}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	-2	1	0	0	0	0
\(e_{17}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	-2	1	0	0	0
\(e_{18}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	-2	1	0	0
\(e_{19}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	-2	1	0
\(e_{20}\)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	-2	0
\(e_{21}\)	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], [1, 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, 1, 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, -2, 0, 1, 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, 1, 1, -2, 1, 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, 1, -2, 1, 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, 1, -2, 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, -2, 0, 1, 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, 1, 1, -2, 1, 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, 1, -2, 1, 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, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2]]

Integral lattice

Parity	even
Discriminant group	\(\mathbb{Z}/2\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Nikulin invariants \((r, a, \delta)\)	\((21, 1, 1)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 20)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 20)\, L^\Sigma(s - 10, \chi_{-2n})\)
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 2\), by an embedding of \(U^{2}\) in the morphism files

Indefinite lattice

Isotropic	yes

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 \(R(L)\)	21
\([L : R(L)]\)	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	\(b(r, r)\)
\(r_{1}\)	\(e_{5}\)	\(-2\)
\(r_{2}\)	\(e_{6}\)	\(-2\)
\(r_{3}\)	\(e_{7}\)	\(-2\)
\(r_{4}\)	\(e_{8}\)	\(-2\)
\(r_{5}\)	\(e_{9}\)	\(-2\)
\(r_{6}\)	\(e_{10}\)	\(-2\)
\(r_{7}\)	\(e_{11}\)	\(-2\)
\(r_{8}\)	\(e_{12}\)	\(-2\)
\(r_{9}\)	\(e_{13}\)	\(-2\)
\(r_{10}\)	\(e_{14}\)	\(-2\)
\(r_{11}\)	\(e_{15}\)	\(-2\)
\(r_{12}\)	\(e_{16}\)	\(-2\)
\(r_{13}\)	\(e_{17}\)	\(-2\)
\(r_{14}\)	\(e_{18}\)	\(-2\)
\(r_{15}\)	\(e_{19}\)	\(-2\)
\(r_{16}\)	\(e_{20}\)	\(-2\)
\(r_{17}\)	\(e_{21}\)	\(-2\)
\(r_{18}\)	\(e_{1} + e_{2}\)	\(2\)
\(r_{19}\)	\(e_{1} + e_{5}\)	\(-2\)
\(r_{20}\)	\(e_{3} + e_{4}\)	\(2\)
\(r_{21}\)	\(e_{3} + e_{5}\)	\(-2\)

Notes

\(L_{2d}\) for \(d = 1\): the orthogonal complement \(h^\perp \cong U^2 \oplus E_8(-1)^2 \oplus \langle -2 \rangle\) of a primitive vector \(h\) with \(b(h, h) = 2\) in the K3 lattice \(\mathrm{II}_{3,19} \cong U^3 \oplus E_8(-1)^2\), by equation (4) of Gritsenko, Hulek and Sankaran. The period domain of K3 surfaces with a polarization of degree \(2d\) is defined from this lattice.

Morphisms

References

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