Definition. Full adelic groups and the real factor [fcap-004H]

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Let \(Q\) be a nondegenerate quadratic form on a finite-dimensional rational vector space \(V\), and choose a compatible compact-open reference tuple. Extend scalars to \(V_\infty =\mathbb R\otimes _{\mathbb Q}V\), with quadratic form \(Q_\infty \). Write \(O_\infty \), \(SO_\infty \) and \(\operatorname {Spin}_\infty \) for its real groups with the canonical topologies of Definition [fcap-0044]. In particular \(\operatorname {Spin}_\infty \) uses the actual vector-generated, even, reverse-norm-one carrier of Lemma [fcap-0045].

The full adelic orthogonal, special orthogonal and Spin groups relative to the chosen references are \[\begin {aligned} O_{\mathbb A}(U)&=O_\infty \times O_f(U),\\ SO_{\mathbb A}(U)&=SO_\infty \times SO_f(U),\\ \operatorname {Spin}_{\mathbb A}(U)&=\operatorname {Spin}_\infty \times \operatorname {Spin}_f(U). \end {aligned}\] The finite factors are those of Definition [fcap-004F]. In each case take the ordinary product topology and coordinatewise group operations. There is no reference-subgroup condition at the real place: its factor is unrestricted. More generally write \(G_{\mathbb A}(U)=G_\infty \times G_f(U)\) for any of these three constructions. They are Hausdorff locally compact topological groups.

Indeed, Lemma [fcap-0045] gives those properties for the real factor, and Definition [fcap-004F] gives them for the finite factor. Multiplication and inversion are continuous in each component, hence in the product topology. Products of separating neighborhoods give the Hausdorff property. At each point choose compact neighborhoods in the two factors; their product is a compact neighborhood, proving local compactness. The finite-coordinate projection is the continuous group homomorphism \[\begin {aligned} \pi _f:G_{\mathbb A}(U)&\longrightarrow G_f(U),\\ (g_\infty ,g_f)&\longmapsto g_f. \end {aligned}\] Continuity is the defining property of the product topology, and the homomorphism identity follows from coordinatewise multiplication.

For example, take the hyperbolic plane \(Q(x,y)=xy\). For \(t>0\), the real matrix \(h_t=\operatorname {diag}(t,t^{-1})\) preserves \(Q\) and has determinant one. Thus \((h_t,1)\) is a point of \(SO_{\mathbb A}(U)\) for every \(t>0\), with identity at every finite prime. The nonidentity matrices \(h_t\) approach the identity as \(t\to 1\), exhibiting a nondiscrete real factor.

Let \(K(U)=\prod _pU_p\) be the compact open reference subgroup of the finite factor. The subgroup \(\{1\}\times K(U)\) of the full group is compact, but is not open when \(G_\infty \) is nondiscrete. Compactness follows from compactness of both factors. If the subgroup were open, its inverse image under the continuous map \(g\mapsto (g,1)\) would make \(\{1\}\) open in \(G_\infty \). Translations would then make every singleton open, contradicting nondiscreteness. Instead, for each open identity neighborhood \(W_\infty \subseteq G_\infty \), the set \(W_\infty \times K(U)\) is an open identity neighborhood in the full group. A compact real identity neighborhood \(C_\infty \) gives the compact identity neighborhood \(C_\infty \times K(U)\), without asserting that it is open. In dimension zero all three real and finite groups are singletons, so the nondiscreteness qualification is essential.

The unrestricted archimedean factors and the finite-place conditions occur together in the classical affine adelic stage formula of [conrad2011adelic, Theorem 3.6, equation (3.6.1), pp. 6-8]. Here the product construction is explicit for the chosen group references; it makes no additional identification with integral points of an affine scheme. TauCeti's TauCeti.RestrictedProductGroupWithFactor is the general carrier with an extra unrestricted factor. The three specializations are TauCeti.QuadraticMap.OrthogonalCompactOpens.fullAdelicOrthogonal, TauCeti.QuadraticMap.OrthogonalCompactOpens.fullAdelicSpecialOrthogonal and TauCeti.QuadraticMap.OrthogonalCompactOpens.fullAdelicSpin. The corresponding continuous finite projections are TauCeti.QuadraticMap.OrthogonalCompactOpens.continuous_fullAdelicOrthogonalToFinite, TauCeti.QuadraticMap.OrthogonalCompactOpens.continuous_fullAdelicSpecialOrthogonalToFinite and TauCeti.QuadraticMap.OrthogonalCompactOpens.continuous_fullAdelicSpinToFinite.