Tag 029G

\(\mathrm{Kum}(-1)\)

Kum(-1)·Kummer lattice with the sign reversed

Invariants

Rank16
Signature \((n_+, n_-)\)(16, 0)
Determinant\(64\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{16}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)
\(e_{1}\)2000000100010111
\(e_{2}\)0200000100010100
\(e_{3}\)0020000100010010
\(e_{4}\)0002000100010001
\(e_{5}\)0000200100000110
\(e_{6}\)0000020100000101
\(e_{7}\)0000002100000011
\(e_{8}\)1111111400020222
\(e_{9}\)0000000020010110
\(e_{10}\)0000000002010101
\(e_{11}\)0000000000210011
\(e_{12}\)1111000211140222
\(e_{13}\)0000000000002111
\(e_{14}\)1100110211021422
\(e_{15}\)1010101210121242
\(e_{16}\)1001011201121224
 
[[2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1], [0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0], [0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1], [0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 1], [1, 1, 1, 1, 1, 1, 1, 4, 0, 0, 0, 2, 0, 2, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1], [1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 4, 0, 2, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1], [1, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 2, 1, 4, 2, 2], [1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 4, 2], [1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{6}\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)514
Nikulin invariants \((r, a, \delta)\)\((16, 6, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{8} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 15)\, \zeta^\Sigma(s - 8)}{\zeta^\Sigma(s - 7)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 15)}{\zeta^\Sigma(s - 7)}\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number32

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 32q^{2} + 8160q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots32
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 2 \rangle^{16}\)
Rank of \(R(L)\)16
\([L : R(L)]\)32
\(R(L)\) is primitive in \(L\)no: \(L / R(L) \cong (\mathbb{Z}/2)^{5}\)

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{2}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{3}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{4}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{5}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{6}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{7}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{8}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{9}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{10}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{11}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{12}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{13}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{14}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{15}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)
\(\Phi_{16}\)\(A_{1}\)\(\{2\}\)2\(\cong \langle 2 \rangle\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{13}\)\(2\)
\(\alpha^{(2)}_{1}\)\(e_{11}\)\(2\)
\(\alpha^{(3)}_{1}\)\(e_{10}\)\(2\)
\(\alpha^{(4)}_{1}\)\(e_{9}\)\(2\)
\(\alpha^{(5)}_{1}\)\(e_{7}\)\(2\)
\(\alpha^{(6)}_{1}\)\(e_{6}\)\(2\)
\(\alpha^{(7)}_{1}\)\(e_{5}\)\(2\)
\(\alpha^{(8)}_{1}\)\(e_{4}\)\(2\)
\(\alpha^{(9)}_{1}\)\(e_{3}\)\(2\)
\(\alpha^{(10)}_{1}\)\(e_{2}\)\(2\)
\(\alpha^{(11)}_{1}\)\(e_{1}\)\(2\)
\(\alpha^{(12)}_{1}\)\(e_{1} + e_{4} + e_{6} + e_{7} + e_{10} + e_{11} + e_{13} - 2e_{16}\)\(2\)
\(\alpha^{(13)}_{1}\)\(e_{1} + e_{3} + e_{5} + e_{7} + e_{9} + e_{11} + e_{13} - 2e_{15}\)\(2\)
\(\alpha^{(14)}_{1}\)\(e_{1} + e_{2} + e_{5} + e_{6} + e_{9} + e_{10} + e_{13} - 2e_{14}\)\(2\)
\(\alpha^{(15)}_{1}\)\(e_{1} + e_{2} + e_{3} + e_{4} + e_{9} + e_{10} + e_{11} - 2e_{12}\)\(2\)
\(\alpha^{(16)}_{1}\)\(e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6} + e_{7} - 2e_{8}\)\(2\)

Notes

The Kummer lattice \(K\) is the minimal primitive sublattice of \(H^2(\mathrm{Km}(A), \mathbb{Z})\) that contains the classes of the 16 rational curves \(K_p\), \(p \in A[2] \cong (\mathbb{Z}/2\mathbb{Z})^4\), of a Kummer surface (Garbagnati and Sarti, Definition 2.1). It is negative definite, even, of rank 16 and discriminant \(2^6\) (Proposition 2.2). A K3 surface is a Kummer surface if and only if \(K\) embeds primitively in its Néron–Severi lattice (Nikulin).

This record is \(K(-1)\), so that its form is positive definite. The basis is a Hermite normal form basis of the span, in \(\tfrac{1}{2}\bigoplus_p \mathbb{Z} K_p\), of the classes \(K_p\), the class \(\tfrac{1}{2}\sum_p K_p\) and the 30 classes \(\tfrac{1}{2}\sum_{p \in W} K_p\) for the affine hyperplanes \(W\) of \((\mathbb{Z}/2\mathbb{Z})^4\) (Remark 2.3 (1)–(2)). The determinant of this span is \(2^6\), the discriminant of \(K\), so the span is \(K\).

References

  1. A. Garbagnati, A. Sarti, "Kummer surfaces and K3 surfaces with (Z/2Z)^4 symplectic action", arXiv:1305.3514, Definition 2.1, Proposition 2.2 and Remark 2.3.