Definition
In the arithmetic theory of quadratic forms, the spinor genus of a quadratic form is a subset of its genus consisting of all forms that are locally equivalent at every completion of the base field and that share the same spinor‑norm class. Two integral quadratic forms belong to the same spinor genus if they are equivalent under the action of the adelic orthogonal group modulo the kernel of the spinor norm map.
Overview
The concept of spinor genus refines the classical notion of genus, providing a finer classification of quadratic forms over number fields (most commonly over ℤ). While the genus groups together forms that are locally isometric everywhere, the spinor genus further partitions a genus according to the values of the spinor norm, an arithmetic invariant derived from the double cover Spin → O of the orthogonal group.
The spinor genus plays a central role in several results of the arithmetic of quadratic forms:
- It appears in the statement of Eichler’s theorem on the representation of integers by quadratic forms.
- Kneser’s results on the Hasse principle show that, for indefinite quadratic forms of rank ≥ 3, the genus and the spinor genus coincide, whereas for positive‑definite forms the spinor genus can be a proper subset of the genus.
- The decomposition of a genus into finitely many spinor genera is essential in the computation of representation numbers via the Siegel mass formula and in the analysis of local‑global principles.
Etymology / Origin
The term combines spinor, referring to the spin representation of the orthogonal group and the associated spinor norm, with genus, from the Latin genus meaning “kind” or “type”. The notion was introduced in the mid‑20th century by Martin Eichler and Martin Kneser in their work on quadratic forms and the arithmetic of orthogonal groups.
Characteristics
| Property | Description |
|---|---|
| Construction | For a quadratic space $ (V,q) $ over a number field $ K $, let $ O(V) $ be its orthogonal group and $ \operatorname{Spin}(V) $ its double cover. The spinor norm $ \theta : O(V) \rightarrow K^{\times} / K^{\times 2} $ extends to the adelic group $ O(V_{\mathbb{A}}) $. Two lattices $ L_1, L_2 $ lie in the same spinor genus iff there exists $ g \in O(V_{\mathbb{A}}) $ such that $ L_2 = g L_1 $ and $ \theta(g) $ belongs to the image of the global spinor norm. |
| Relation to genus | Every spinor genus is contained in a single genus; a genus splits into a finite number of spinor genera. |
| Finiteness | The number of spinor genera in a given genus is finite; it can be computed from the class group of the discriminant field and the image of the spinor norm. |
| Indefinite forms | For indefinite quadratic forms of rank ≥ 3, the genus and the spinor genus coincide (Kneser’s theorem). |
| Definite forms | For positive‑definite forms, the spinor genus can be strictly smaller than the genus; distinct spinor genera may have different representation behaviours. |
| Computational aspects | Algorithms for determining spinor genus often employ the neighbor method (Kneser) or the computation of the spinor norm via local invariants. |
Related Topics
- Quadratic form – a homogeneous polynomial of degree 2; the primary object of study.
- Genus (quadratic forms) – classification of forms that are locally isometric at all completions.
- Class (or proper equivalence class) – the finest standard equivalence relation, consisting of forms related by integral change of variables with determinant ±1.
- Spinor norm – an arithmetic invariant derived from the spin double cover of the orthogonal group.
- Orthogonal group and Spin group – algebraic groups governing the symmetries of quadratic spaces.
- Adeles and ideles – topological groups used in the adelic formulation of spinor genus.
- Siegel mass formula – expresses the weighted count of classes in a genus, often refined by spinor genera.
- Eichler’s theorem – links representation of numbers by quadratic forms to spinor genera.
- Kneser’s neighbor method – a technique for traversing classes within a genus and determining spinor genera.