Example. Compact and split binary Spin groups over the 3-adic field [fcap-0048]
Example. Compact and split binary Spin groups over the 3-adic field [fcap-0048]
Over \(\mathbb Q_3\), with \(|3|_3=1/3\), consider \[\begin {aligned} Q_+(x,y)&=x^2+y^2,\\ Q_-(x,y)&=x^2-y^2. \end {aligned}\] Both forms are nondegenerate. In the canonical topologies, \(\operatorname {Spin}(Q_+)\) is compact and \(\operatorname {Spin}(Q_-)\) is not. This is a concrete application of Theorem [fcap-0047]. The local-field convention is Definition [fcap-0044]; the \(p\)-adic fields and their residue fields are included in [lam2005quadratic, VI.2, Definition 2.1 and discussion, pp. 151--152]. The calculations below establish the contrast directly.
If \(x^2+y^2=0\) and \(y=0\), then \(x=0\). If \(y\ne 0\), set
\(z=x/y\); then \(z^2=-1\). Its 3-adic valuation is zero, so \(z\) is a
unit in \(\mathbb Z_3\). Reduction modulo \(3\) would give a square
root of \(-1=2\) in \(\mathbb F_3\). But the two nonzero residues both
square to \(1\). Hence no such \(z\) exists, and \(Q_+\) is
anisotropic. Its Spin group is compact by
Theorem [fcap-0047]. The nonsquare input is also
Padic.not_isSquare_neg_one_of_mod_four_eq_three
at \(p=3\); TauCeti.anisotropic_binary_one_one_iff
identifies anisotropy of \(\langle 1,1\rangle \) with this nonsquare
condition over any field. These support the calculation, rather than
asserting a formalization of every displayed lift below.
For \(Q_-\), take \[\begin {gathered} u=(1,1),\\ v=(1/4,-1/4). \end {gathered}\] They satisfy \(Q_-(u)=Q_-(v)=0\) and \(B_{Q_-}(u,v)=1\), in the unhalved convention of Convention [fcap-0027]. They form a basis, and \(Q_-(au+bv)=ab\). Thus \(Q_-\) is a split hyperbolic plane, agreeing with [lam2005quadratic, I.3, Theorem 3.2, p. 9].
There is an explicit unbounded family in the Spin image. In the ordered basis \((u,v)\), let \(h_t=\operatorname {diag}(t,t^{-1})\), where \(t\in \mathbb Q_3^\times \). Write \(\widehat w=\iota _{Q_-}(w)\). Both \(tu+t^{-1}v\) and \(u+v\) have quadratic value one, so \[s_t=\widehat {(tu+t^{-1}v)}\,\widehat {(u+v)}\] lies in the actual vector-generated Spin group by Theorem [fcap-000C] and the reverse-norm convention in Remark [fcap-0028]. The reflection formula gives the following matrices in \((u,v)\): \[\begin {gathered} r_{tu+t^{-1}v}=\begin {pmatrix}0&-t^2\\-t^{-2}&0\end {pmatrix},\\ r_{u+v}=\begin {pmatrix}0&-1\\-1&0\end {pmatrix},\\ \rho (s_t)=r_{tu+t^{-1}v}r_{u+v},\\ \rho (s_t)=\begin {pmatrix}t^2&0\\0&t^{-2}\end {pmatrix}=h_t^2. \end {gathered}\] The product order is that of the two Clifford factors. Thus these are actual normalized lifts, not only abstract square-class witnesses.
Set \(t=3^{-r}\) for \(r\geq 1\). The first matrix entry has \(|t^2|_3=3^{2r}\), which is unbounded. The continuous image of a compact Spin group would have bounded matrix entries, proving that \(\operatorname {Spin}(Q_-)\) is not compact. In particular, binary noncompactness already follows from this torus family, without an orthogonal complement in which to construct transvections.