Completing Weyl's reduction theory for arithmetical equivalence: optimal constants and the structure of the reduction-cell pattern

In 1940 Hermann Weyl developed a unified reduction theory for arithmetical equivalence of lattices: a reduction domain $Z$ in the cone of positive (gauge) forms whose images under $GL_n(\mathbb{Z})$ (his 'modular group') tile the cone, with two finiteness theorems, and an extension to complex and quaternion coefficient rings governed by what he calls Hypothesis P (every one-sided ideal of the coefficient ring is principal). The paper left four unfinished items: (i) the constant $\lambda_n$ in the discriminant inequality $\lambda_n\, g_{11} \cdots g_{nn} \le D$ for reduced forms, bounding the product of diagonal coefficients by the determinant $D$, is, in Weyl's own words, 'certainly not optimal'; (ii) the rings satisfying Hypothesis P were not classified; (iii) the stated ultimate goal — that the cell pattern form a combinatorial complex that is the skeleton of a manifold — was not achieved; (iv) the parameter sets and generating substitutions behind the finiteness theorems were not made explicit or minimal. Subsequent literature establishes: item (ii) is essentially resolved — for imaginary quadratic rings it is the class-number-one problem (Stark 1967: exactly nine fields), and Brzezinski (1995) proved there are exactly 25 definite quaternion orders over $\mathbb{Z}$ of class number one; item (iii) was answered negatively in its literal form — Tammela (1988) proved the partition into Minkowski reduction domains is not normal for $n \ge 7$ — while Voronoi's secondary-cone decomposition and the well-rounded retract realize the topological goal in modified form; item (iv) is subsumed by modern quantitative reduction theory. The surviving open core is item (i) in general form: the sharp constants (equivalently, sharp bounds on the orthogonality defects of reduced bases), known exactly only for $n \le 5$ in the real Minkowski-reduced setting, and settled for the separate Hermite–Korkin–Zolotarev reduction through $n = 4$ together with the asymptotic law $D_n = n^{(2+o(1))n}$ (Li 2026), and open in general dimension and for the Hermitian and Hamiltonian analogs, together with the residual part of item (iii).

Background

Reduction theory for arithmetical equivalence. A lattice $L \subset \mathbb{R}^n$ is a discrete subgroup of rank $n$; choosing a basis $a_1, \dots, a_n$ presents its vectors as $\sum x_i a_i$ with integer $x_i$, and the associated positive quadratic form $f(x) = g[x] = \sum_{i,k} g_{ik} x_i x_k$ (Weyl's gauge function, $G = (g_{ik})$ positive definite) records squared lengths of lattice vectors. Two forms are arithmetically equivalent when they differ by a unimodular integer change of basis, i.e. by the action of $GL_n(\mathbb{Z})$ (integer matrices of determinant $\pm 1$), which Weyl calls the modular group. A reduction theory chooses one representative per equivalence class: Weyl constructs a cell $Z$ (a reduction domain) inside the open cone of positive forms such that the modular-group translates of $Z$ cover the cone with disjoint interiors. His two finiteness theorems state that suitable sets $G(\rho,\sigma)$ of forms, with values pinched between determinant-normalized bounds, meet only finitely many cells (Theorem 13), and that the substitutions carrying $Z$ onto adjacent cells (cells sharing a wall, i.e. $N-1$ linearly independent points) form, up to the group fixing $Z$, a finite generating set of the whole modular group (Theorems 14–15).

The constants. For reduced forms Weyl proves the discriminant inequality $\lambda_n\, g_{11} \cdots g_{nn} \le D$ with $D = \det G$, first for real integral forms (Theorem 11) and then for reduced Hermitian and Hamiltonian forms with the same shape of constant, $\lambda_n = (\omega_n/\rho_n)^2$, built from volume constants of ellipsoidal gauge bodies. He states that these values are 'certainly not optimal, but fairly good', and the first open item is the optimal value in each dimension. In modern language the reciprocal of the square root of the optimal $\lambda_n$ is the maximal orthogonality defect of a reduced basis: for a lattice basis $b_1, \dots, b_n$ the orthogonality defect is $\prod_i \|b_i\| / \det L$, the factor by which the basis falls short of being orthogonal. This is a relative of the classical Hermite constant problem: the Hermite constant $\gamma_n$ is the supremum of $\lambda_1(L)^2 / \det L$ over all rank-$n$ lattices, where $\lambda_1$ is the length of the shortest nonzero vector, and the successive minima $\lambda_1 \le \dots \le \lambda_n$ are the smallest radii at which the lattice spans $i$ independent vectors; sharp inequalities between a determinant and a product of basis-vector lengths (equivalently, between $D$ and the $g_{kk}$, which are squared basis-vector lengths) are of the same Hermite-constant type, whose exact values are unknown in general dimension. The notion parallel to Weyl's is Minkowski reduction: a basis is Minkowski-reduced when, for each $k$, the vector $b_k$ is no longer than every lattice vector $b$ for which $b_1, \dots, b_{k-1}, b$ are linearly independent. Replacing it by Hermite–Korkin–Zolotarev (HKZ) reduction — every $b_k$ is a shortest vector of the projection of the lattice onto the orthogonal complement of the span of $b_1, \dots, b_{k-1}$ — gives the actively studied HKZ version of the defect.

Complex and quaternion coefficients; Hypothesis P. Weyl extends the theory to complex vector spaces, i.e. Hermitian forms over an imaginary quadratic ring, and to quaternion spaces, i.e. Hamiltonian forms (Hermitian forms over Hamilton's real quaternions; arithmetically, over an order — a lattice subring — of a definite quaternion algebra). Extending his lemmas to these coefficient rings requires a Bézout-type ideal theory, which he isolates as Hypothesis P: every left or right ideal in the coefficient ring $J$ is a principal ideal. When a division-with-remainder parameter is below 1 the reduction becomes algorithmic; his examples include the Gaussian integers and the Hurwitz quaternion order. For an imaginary quadratic maximal order (the largest order of the field, consisting of all algebraic integers in it), Hypothesis P is the principal-ideal-domain, i.e. class number one, condition; the class-number-one problem for imaginary quadratic fields was settled in the 1960s (Stark's determination lists exactly nine fields). For definite quaternion orders over $\mathbb{Z}$ the analogous classification is due to Brzezinski: exactly 25 isomorphism classes of class number one, listed explicitly. (An order is called Gorenstein when its dual with respect to the reduced trace, $\{x : \mathrm{tr}(x\,\mathcal{O}) \subseteq \mathbb{Z}\}$, is an invertible — equivalently locally principal — ideal of $\mathcal{O}$; the two non-Gorenstein orders on Brzezinski's list carry one-sided ideals that are not invertible, which is where the standard class-number-one notion, counting only invertible ideals, can in principle diverge from Weyl's literal one-sided-ideal reading.)

The topological program and its later fate. The paper's stated ultimate goal is to show that the pattern of mutually bordering cells 'is a complex in the combinatorial topological sense, of such particular structure as to form the skeleton of a manifold' — a combinatorial cell complex modeling the topology of the space of lattices up to basis change. This goal has since been answered in both directions. Tammela proved that the partition of the positivity cone into translates of the Minkowski reduction domain is not normal for $n \ge 7$ — a partition being normal when each face of a domain borders exactly one other domain — so the literal expectation fails for the classical reduction domains in high dimension. Positively, two modern constructions realize the goal in modified form: Voronoi's second reduction, which decomposes the cone of positive forms into finitely many polyhedral cones indexed by Delaunay polytope types (a Delaunay polytope of a lattice being the convex hull of the lattice points nearest to a given point of space, with no lattice points inside their common circumsphere; this is the secondary-cone or L-type decomposition) and yields a normal, face-to-face cell complex, computationally developed by Schürmann and Vallentin; and the well-rounded retract, a canonical deformation retract of the space of lattices onto a finite simplicial complex built from well-rounded lattices (lattices whose shortest nonzero vectors span the ambient space), introduced by Soulé and developed by Ash and coauthors, whose minimality was proved by Pettet and Souto. The quaternionic thread has been revived by Parkkonen and Paulin, who follow Weyl's terminology for Hamiltonian forms and build their reduction theory with hyperbolic geometry, and by Chenevier and Paulin, who determined the analog of the Hermite constant for binary positive definite Hamiltonian forms. A parallel quantitative line (Li–Margulis; Daw–Orr) develops effective versions of reduction and generation, the modern descendants of the finiteness theorems.

Problem Statement

The problem is to complete the unfinished parts of Weyl's reduction theory for arithmetical equivalence, in the coefficient settings he considered (real integral forms, complex / imaginary-quadratic Hermitian forms, and quaternionic Hamiltonian forms):

1. Optimal constants. Determine, for each dimension $n$ and each coefficient setting, the optimal value of the constant in the discriminant inequality $\lambda_n\, g_{11} \cdots g_{nn} \le D$ for the reduced forms of Weyl's reduction domain $Z$ (in the real setting, the Minkowski-reduced forms), where $D$ is the determinant of the form and $g_{11}, \dots, g_{nn}$ are the diagonal coefficients (squared lengths of the reduced basis vectors) — equivalently, the supremum of the orthogonality defect $\prod_i \|b_i\| / \det L$ over bases reduced with respect to $Z$ (in the real setting, Minkowski-reduced bases) of $n$-dimensional lattices over that coefficient ring (the square of that supremum being the reciprocal of the optimal $\lambda_n$). Settled cases for this constant: dimensions $n \le 5$ in the real Minkowski-reduced setting (van der Waerden 1969 for $n = 5$). (Chenevier–Paulin 2020 determine the binary quaternionic Hermite-type constant — the shortest-vector minimum, a different invariant — so the binary quaternionic orthogonality-defect constant is not settled.) For the different notion of Hermite–Korkin–Zolotarev (HKZ) reduction, whose optimal orthogonality defect is a separate quantity, the corresponding constants are settled through $n = 4$ together with the asymptotic law $D_n = n^{(2+o(1))n}$ (Zhixing Li 2026). Open: all larger dimensions in the real setting; the binary quaternionic orthogonality-defect constant; the Hermitian and Hamiltonian analogs in rank at least 3.

2. Structure of the cell pattern. Determine the combinatorial-topological structure of the pattern of Weyl's reduction cells, including whether the partition of the positivity cone into translates of Weyl's reduction domain $Z$ is normal — a partition being normal when each face of a domain borders exactly one other domain — in the dimensions and coefficient settings where it is not settled: in the real setting the dimensions $n \le 6$ (Tammela's theorem rules out the Minkowski partition for $n \ge 7$), and the complex and quaternion coefficient settings in every rank $n \ge 2$ (no non-normality theorem is known in those settings). Weyl's stated goal that the pattern of mutually bordering cells be a complex in the combinatorial topological sense forming the skeleton of a manifold is the objective of this item. (Voronoi's secondary-cone decomposition and the well-rounded retract realize this goal in modified form for the classical reduction domains.)

References

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