Tag 0128

\(U \oplus E_{8}(-1)\)

U + E8(-1)·II_{1,9}·(r, a, delta) = (10, 0, 0)·Enriques lattice·E10

Invariants

Rank	10
Signature \((n_+, n_-)\)	(1, 9)
Determinant	\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{10}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, \dots, e_{10}\}\), an orthogonal sum.

	\(e_{1}\)	\(e_{2}\)	\(e_{3}\)	\(e_{4}\)	\(e_{5}\)	\(e_{6}\)	\(e_{7}\)	\(e_{8}\)	\(e_{9}\)	\(e_{10}\)
\(e_{1}\)	0	1	0	0	0	0	0	0	0	0
\(e_{2}\)	1	0	0	0	0	0	0	0	0	0
\(e_{3}\)	0	0	-2	0	1	0	0	0	0	0
\(e_{4}\)	0	0	0	-2	0	1	0	0	0	0
\(e_{5}\)	0	0	1	0	-2	1	0	0	0	0
\(e_{6}\)	0	0	0	1	1	-2	1	0	0	0
\(e_{7}\)	0	0	0	0	0	1	-2	1	0	0
\(e_{8}\)	0	0	0	0	0	0	1	-2	1	0
\(e_{9}\)	0	0	0	0	0	0	0	1	-2	1
\(e_{10}\)	0	0	0	0	0	0	0	0	1	-2

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

Integral lattice

Parity	even
Discriminant group	\(0\)
Unimodular	yes
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Nikulin invariants \((r, a, \delta)\)	\((10, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{5} \det L\)	trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 9)\, \zeta^\Sigma(s - 5)}{\zeta^\Sigma(s - 4)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 9)}{\zeta^\Sigma(s - 4)}\)
Genus symbol	\(\mathrm{II}_{1,9}\)
Class number of the genus	1
Hyperbolic index \(\max\{n : L \cong U^n \oplus L'\}\)	\(\geq 1\), by an embedding of \(U^{1}\) 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 \(R(L)\)	10
\([L : R(L)]\)	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	\(b(r, r)\)
\(r_{1}\)	\(e_{3}\)	\(-2\)
\(r_{2}\)	\(e_{4}\)	\(-2\)
\(r_{3}\)	\(e_{5}\)	\(-2\)
\(r_{4}\)	\(e_{6}\)	\(-2\)
\(r_{5}\)	\(e_{7}\)	\(-2\)
\(r_{6}\)	\(e_{8}\)	\(-2\)
\(r_{7}\)	\(e_{9}\)	\(-2\)
\(r_{8}\)	\(e_{10}\)	\(-2\)
\(r_{9}\)	\(e_{1} + e_{2}\)	\(2\)
\(r_{10}\)	\(e_{1} + e_{3}\)	\(-2\)

Notes

This is an even hyperbolic unimodular lattice, so \(A_L = 0\) and its invariants are \((r, a, \delta) = (10, 0, 0)\).

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.

The Enriques lattice \(E_{10} = U \oplus E_8(-1)\): \(\mathrm{Num}(S)\) for an Enriques surface \(S\) (Dolgachev, Section 1).

Morphisms

References

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
I. V. Dolgachev, "A brief introduction to Enriques surfaces", arXiv:1412.7744, Section 1.