\(
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ter— title: Orbits of isotropic vectors and the Tits building of a cusp configuration unit: method status: reusable tags: - isotropic-orbits - cusps - baily-borel - tits-buildings - method notes: |-
Orbits of isotropic vectors and the Tits building of a cusp configuration
The enumeration of \(\Orth(T)\)- and \(\Gamma\)-orbits of primitive isotropic vectors in \(T_\Co\) is left open in [[open-problems|open problems]], where it blocks the uniqueness of the Baily–Borel \(0\)-cusp. Two published algorithm families answer that question for indefinite lattices; this page states what each computes and under which hypotheses.
Why the orbit count is the cusp count
Deciding orbit equivalence of two vectors
Proposition 73.4 (Conditions necessary for equivalence) Let \(L\) be a nondegenerate lattice, \(\Gamma\leq\Orth(L)\), and \(v_1, v_2\in L\) primitive with \(\phi(v_1) = v_2\) for some \(\phi\in\Gamma\). Then
\(v_1^2 = v_2^2\);
\(\div_L(v_1) = \div_L(v_2)\), and the isometry induced by \(\phi\) on \(A_L\) carries \([v_1^*]\) to \([v_2^*]\);
\(\phi\) restricts to an isometry \(v_1^{\perp L}\to v_2^{\perp L}\), so the two complements are isometric and in particular have equal discriminants.
Each holds a fortiori for any subgroup of \(\Orth(L)\), so a failure of any one of them refutes equivalence under every group considered here.
Example 73.7 (Two specimens in $U\oplus A_3$) Both specimens are stated in \(L = U\oplus A_3\) with the Gram matrices \[
G_U = \begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad
G_{A_3} = \begin{pmatrix}-2&-1&0\\-1&-2&-1\\0&-1&-2\end{pmatrix},
\] the coordinates below being taken in that basis.
For Example 2.2 of (Dawes 2022), \(v_1 = (4,4,1,2,-1)\) and \(v_2 = (36,144,5,-30,83)\), and both have \(v^2 = 20\): the \(U\)-part contributes \(32\) and \(10368\), the \(A_3\)-part \(-12\) and \(-10348\). The paper’s conclusion is \(v_1\sim v_2\) under the group it denotes \(\wh{\Orth}^+(L)\).
For Example 2.6, \(v_1 = (1,-1,0,0,0)\) and \(v_2 = (1,0,1,0,0)\), and both have \(v^2 = -2\). The paper’s conclusion is \(v_1\sim v_2\) under the group it denotes \(\wh{S\Orth}^+(L)\), and the in-tree computation reproduces it.
Question 73.8 (An unreconciled outcome on the first specimen) Run on Example 2.2 of Example 73.7, the in-tree computation returns \(v_1\not\sim v_2\), on the ground that \(v_1^{\perp L}\) and \(v_2^{\perp L}\) have different discriminants. The record attributes the disagreement to a simplified isometry test, but a discriminant is an isometry invariant, so by Proposition 73.4(3) unequal discriminants refute equivalence under \(\Orth(L)\) and under every subgroup of it, whatever isometry test is used downstream. The disagreement therefore lies in one of three places: the conclusion of (Dawes 2022), the transcription of the vectors or of the Gram matrix, or the computed complements. The squares recorded in Example 73.7 are consistent with the stated Gram matrices, which settles the transcription of the vectors; recomputing the two complements and their discriminants settles the rest.
The polarized Enriques computation and its lattice
Observation 73.11 (The lattice of that computation is $\tdp$) The specification records the target lattice by the shorthand \(U\oplus 2U\oplus 2E_8(-1)\), in which the prefix \(2\) has two different meanings, and by two unambiguous invariants: rank \(20\) and signature \((2,18)\). Those invariants force the reading \[
M \;=\; U\oplus U(2)\oplus E_8^{\oplus 2},
\] with \(E_8\) negative definite, since no other assignment of the shorthand to scalings and multiplicities of \(U\) and \(E_8\) has rank \(20\). This is the lattice \(L_{20,2,0} = \tdp\) of two-elementary type \((20,2,0)\) in which the root configuration \(\Phi_{(18,2,0)}\) and the folded Sterk diagrams are written ([[root-vectors-and-folded-sterk-diagrams|root vectors and folded Sterk diagrams]]), and in which the embedding chain \(T_\Co\injects \ten\injects \tdp\) terminates ([[coble-lattice-isotropic-candidates|Coble lattice isotropic candidates]]). The isotropic-orbit algorithms are therefore already stated for the ambient lattice of this project, and the residue is their transport along that chain.
What remains for the Coble lattices
Related: [[scattone-cusp-method|Scattone cusp method]], [[coble-lattice-isotropic-candidates|Coble lattice isotropic candidates]], [[computational-toolchain-and-recipe|computational toolchain and recipe]], [[cusp-correspondence-morphism-chain|cusp-correspondence morphism chain]], [[open-problems|open problems]].
Dawes, Matthew. 2022.
Orbits in Lattices.
https://doi.org/10.48550/arXiv.2205.10601.
Nikulin, V. V. 1980.
“Integral Symmetric Bilinear Forms and Some of Their Applications.” Mathematics of the USSR-Izvestiya 14 (1): 103.
https://doi.org/10.1070/IM1980v014n01ABEH001060.
Sikirić, Mathieu Dutour, and Klaus Hulek. 2023.
Moduli of Polarized Enriques Surfaces – Computational Aspects.
https://doi.org/10.48550/arXiv.2302.01679.