Tag 0013
\(U(2)\)
U(2)·(r, a, delta) = (2, 2, 0)
Invariants
| Rank | 2 |
|---|---|
| Signature \((n_+, n_-)\) | (1, 1) |
| Determinant | \(-4\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}\).
| \(e_{1}\) | \(e_{2}\) | |
|---|---|---|
| \(e_{1}\) | 0 | 2 |
| \(e_{2}\) | 2 | 0 |
[[0, 2], [2, 0]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{2}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 4 |
| Nikulin invariants \((r, a, \delta)\) | \((2, 2, 0)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{1} \det L\) | trivial |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 1)\, \zeta^\Sigma(s - 1)}{\zeta^\Sigma(s)}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 1)}{\zeta^\Sigma(s)}\) |
| Genus symbol | \(\mathrm{II}_{1,1}\ \bigl(2\colon 2^{2}\bigr)\) |
| Class number of the genus | 1 |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong\) 0001 \(\langle 4 \rangle\) \(\oplus\) 0001 \(\langle -4 \rangle\) |
| Rank of \(R(L)\) | 2 |
| \([L : R(L)]\) | 2 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/2\) |
| Proof | Each \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root. The notes prove that each root of \(L\) is in \(\mathbb{Z}\{r_i\}\). |
Roots that generate \(\mathbb{Z}\Phi(L)\)
| Root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{1} + e_{2}\) | \(4\) |
| \(r_{2}\) | \(e_{1} - e_{2}\) | \(-4\) |
Embedding \(M_1 \oplus M_2 \oplus \dots \to L\) with image \(\mathbb{Z}\Phi(L)\)
Image of each basis vector \(e_j\) of each summand \(M_i\).
Notes
This is the even hyperbolic 2-elementary lattice with invariants \((r, a, \delta) = (2, 2, 0)\).
Since \(a = r\), \(L^* = \tfrac{1}{2} L\), and \(L\) is the twist \(M(2)\) of the unimodular lattice \(M = U\), the record 0016. The corpus records a twist \(M(n)\), \(|n| \geq 2\), only for the 13 rows \(a = r\) of Nikulin’s table, because the classification names the twist and not \(M\).
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.
\(U\) is the lattice with basis \(e, f\) and \(b(e, e) = b(f, f) = 0\), \(b(e, f) = 1\). \(L(k)\) is the module \(L\) with the form \(k b\). Roots: \(L(k)\) has the roots of \(L\) for \(k \neq 0\), because the reflection \(s_r\) of \(L(k)\) is the reflection \(s_r\) of \(L\). The roots of \(U\) are \(\pm(e + f)\) and \(\pm(e - f)\); the record of \(U\) has the proof. In \(U(2)\), \(b(e + f, e + f) = 4\) and \(b(e - f, e - f) = -4\). \(\mathbb{Z}\Phi(L)\) is \(\{a e + c f : a \equiv c \bmod 2\}\), which has index 2 in \(L\) and the orthogonal basis \(e + f\), \(e - f\).
Related lattices
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