Definition. The finite adelic spinor norm and its reference images [fcap-004N]

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Let \(Q\) be nondegenerate on a finite-dimensional rational space \(V\), and choose a compatible compact-open reference tuple \(U\). To collect the local spinor obstructions of one restricted SO tuple, the target must retain the images of its chosen reference subgroups. For each prime \(p\), put \[C_p=\mathbb Q_p^{\times }/(\mathbb Q_p^{\times })^2,\qquad \theta _p=\nu _{Q_p}:SO_p\longrightarrow C_p,\] using the reverse-norm spinor norm of Theorem [fcap-0023], with multiplicative notation. Give \(C_p\) its quotient topology. Define the spinor-norm reference image and the finite adelic square-class target by \[I_p=\theta _p(U_p^{SO}),\qquad C_f(U)=\prod _p'(C_p,I_p).\] The restricted-product notation and topology are those of Definition [fcap-004F]; explicitly \((c_p)_p\in C_f(U)\) means \(c_p\in I_p\) for all but finitely many \(p\).

Each \(C_p\) is discrete, including at \(p=2\). Indeed, Proposition [fcap-0046] proves that the square subgroup is open, using [lam2005quadratic, VI.2, Theorem 2.19 and Corollary 2.20, p. 161]. Every coset is then open, so every singleton in the quotient is open. The same local result proves that \(\theta _p\) is continuous. Since \(U_p^{SO}\) is compact, its image \(I_p\) is compact. A compact discrete set is finite: its open singleton cover has a finite subcover. Thus \(I_p\) is a finite open subgroup of \(C_p\), and the restricted product has the compact-open references required by Definition [fcap-004F].

The finite adelic spinor norm is the continuous homomorphism \[\begin {aligned} \Theta :SO_f(U)&\longrightarrow C_f(U),\\ (g_p)_p&\longmapsto (\theta _p(g_p))_p. \end {aligned}\] For if \(g_p\in U_p^{SO}\) outside a finite set \(S\), then \(\theta _p(g_p)\in I_p\) outside that same set. This proves restricted membership, and the local homomorphism laws give the global homomorphism law. Continuity is the component-map argument of Proposition [fcap-004J]: on each principal stage the map is the product of the continuous maps \(SO_p\to C_p\) for \(p\in S\) and their restrictions \(U_p^{SO}\to I_p\) elsewhere, followed by the continuous target-stage inclusion. The open stages cover the domain.

For example, a tuple equal to \(g\in U_\ell ^{SO}\) at one prime \(\ell \) and to the identity elsewhere maps to the tuple with \(\theta _\ell (g)\in I_\ell \) at \(\ell \) and identity elsewhere. It therefore lies in the target's product reference \(\prod _p I_p\). This describes the actual reference condition without replacing \(I_p\) by the square classes of local units or asserting that it is all of \(C_p\).

If \(\dim V=0\) or \(1\), every local \(SO_p\) is trivial. In dimension one a linear automorphism is scalar multiplication, and its determinant-one condition forces that scalar to be one; dimension zero has only the identity. Therefore every \(I_p=\{1\}\) and \(\Theta \) is the trivial homomorphism. The ambient square-class groups \(C_p\) need not be trivial. No surjectivity of \(\Theta \) is asserted in any dimension.

This construction combines the local square theorem with the finite-stage functoriality described in [conrad2011adelic, Theorem 3.6 and the following paragraph, pp. 6-8]. The actual-image choice is TauCeti.QuadraticMap.OrthogonalCompactOpens.localSpinorNormImage, and its openness is TauCeti.QuadraticMap.OrthogonalCompactOpens.isOpen_localSpinorNormImage. The target and continuous map are TauCeti.QuadraticMap.OrthogonalCompactOpens.finiteAdelicSquareClasses and TauCeti.QuadraticMap.OrthogonalCompactOpens.continuous_adelicSpinorNorm. These use the same chosen references and nondegeneracy assumption. The low-rank image statement is TauCeti.QuadraticMap.OrthogonalCompactOpens.localSpinorNormImage_eq_bot_of_finrank_le_one. The compactness and finiteness of the images are the continuous-image argument given above, not a separate square-class count.