Tag 0036
\(U \oplus U(2)\)
U + U(2)
Invariants
| Rank | 4 |
|---|---|
| Signature \((n_+, n_-)\) | (2, 2) |
| Determinant | \(4\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{4}\).
In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}, e_{4}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | |
|---|---|---|---|---|
| \(e_{1}\) | 0 | 1 | 0 | 0 |
| \(e_{2}\) | 1 | 0 | 0 | 0 |
| \(e_{3}\) | 0 | 0 | 0 | 2 |
| \(e_{4}\) | 0 | 0 | 2 | 0 |
[[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 2], [0, 0, 2, 0]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \((\mathbb{Z}/2)^{2}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 4 |
| Nikulin invariants \((r, a, \delta)\) | \((4, 2, 0)\) |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2\}\) |
| Character of the discriminant \(\chi_D\), \(D = (-1)^{2} \det L\) | trivial |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 3)\, \zeta^\Sigma(s - 2)}{\zeta^\Sigma(s - 1)}\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 3)}{\zeta^\Sigma(s - 1)}\) |
| Genus symbol | \(\mathrm{II}_{2,2}\ \bigl(2\colon 1^{2}:2^{2}\bigr)\) |
| 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: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong\) 0001 \(\langle 2 \rangle\) \(\oplus\) 0001 \(\langle -2 \rangle\) \(\oplus\) 0016 \(U(2)\) |
| Rank of \(R(L)\) | 4 |
| \([L : R(L)]\) | 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 | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{1} + e_{2}\) | \(2\) |
| \(r_{2}\) | \(e_{1} - e_{2}\) | \(-2\) |
| \(r_{3}\) | \(e_{1} + e_{2} + e_{3}\) | \(2\) |
| \(r_{4}\) | \(e_{1} + e_{2} + e_{4}\) | \(2\) |
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
Roots: let \(e, f\) be the basis of \(U\) and \(g, h\) the basis of \(U(2)\), and let \(r = a e + c f + p g + q h\) be a root. \(b(r, r) = 2ac + 4pq\), and \(b(r, L) = \delta\mathbb{Z}\) with \(\delta = \gcd(a, c, 2p, 2q)\). \(r\) is primitive, so \(\gcd(a, c, p, q) = 1\), and \(\delta \in \{1, 2\}\). \(b(r, r)\) divides \(2 b(r, x)\) for each \(x\) in \(L\), so it divides \(2\delta\). When \(\delta = 2\), \(a\) and \(c\) are even. When \(\delta = 1\), \(b(r, r)\) divides 2 and is even, so \(ac + 2pq = \pm 1\), and \(a\) and \(c\) are odd. Each root is therefore in \(M = \{a \equiv c \bmod 2\}\), which has index 2 in \(L\) and the basis \(e + f\), \(e - f\), \(g\), \(h\). \(e + f\), \(e - f\), \(e + f + g\) and \(e + f + h\) are roots, and they generate \(M\): \(\mathbb{Z}\Phi(L) = M = \mathbb{Z}(e + f) \oplus \mathbb{Z}(e - f) \oplus U(2)\), an orthogonal sum.