An orthogonal perspective on Gauss composition

John Voight; Haochen Wu

Home/Archive/arXiv

arXiv · 2511.03987

An orthogonal perspective on Gauss composition

John Voight·Haochen Wu

View original source ↗Read source document ↗Export metadata ↓

Abstract

We revisit Gauss composition over a general base scheme, with a focus on orthogonal groups. We show that the Clifford and norm functors provide an equivalence of stacks between binary quadratic modules and pseudoregular modules over quadratic algebras. As a consequence, we exhibit a composition law for coprimitive forms over a general base, including a universal version of Dirichlet composition. This perspective synthesizes the constructions of Kneser and Wood, reconciles algebraic and geometric approaches, and clarifies the role of orientations and the natural emergence of narrow class groups.

Explore related subjects

math.RA math.AG math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

John Voight, Haochen Wu. 2026-06-22. An orthogonal perspective on Gauss composition. https://arxiv.org/abs/2511.03987

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On solutions of singular Sylvester equations in quaternions

The quaternionic equations ax-xb=0 and ax-xb=c are investigated, which are called homogeneous and inhomogeneous Sylvester equations, respectively. Conditions for the existence of solutions are provided. In addition, the general and nonzero solutions to these equations are derived applying quaternion square roots.

math.RA↗

arXiv2026

Positivity preservers over finite fields II

We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\equiv1\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.

math.RA↗

arXiv2026

Graphs of Moore-Penrose inverse of matrices possessing the treeangle property

It is known that the inverse of an invertible real square matrix satisfying the treeangle property, is a treediagonal matrix. A converse statement also holds. We show that the verbatim analogues are not true for the Moore-Penrose inverse, and obtain the precise structure of graphs corresponding to the Moore-Penrose inverse of matrices possessing the treeangle property.

math.RA↗