§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:15:58Z

Lemma 5.2 · Spheres: the \(\Spin_{2n}(\cO)\)-orbits on \(Q(F)\) are the pole-order shells, for every local field

proved audit: minor defect

Minor defect (second audit). All claims are true and every step re-derived (the pair \(s_0,t_0\) exists for every \(n\ge2\): an isotropic vector of \(\bar y^\perp\) exists by Chevalley–Warning and is not in the radical; torsor lift; Dickson correction; the product of the two reflections is \(\operatorname{diag}(u,u^{-1})\) with spinor norm \(u\)). Two slips that change no claim: (i) after Hensel, \(B(y,z')=1+2a\varpi^{2m}\), not \(1\), so the stated map \(y\mapsto y_m\), \(z'\mapsto e_1\) is an isometry only after dividing \(z'\) by this unit; (ii) for \(m=0\) the original asks for \(N(z)\equiv\bar a^2+\bar a\), whereas a root of \(f\) needs \(N(z)\equiv-(\bar a^2+\bar a)\) with \(2\bar a+1\ne0\); since \(N(z+w)\) takes every residue, the right choice (e.g. residue \(0\), root \(0\)) is available.

For the hyperbolic quadric \(Q=\{N=1\}\), a model of \(\Spin_{2n}/\Spin_{2n-1}\) that is smooth over \(\Z\), the \(K\)-orbits on \(Q(F)\) are exactly the shells of fixed pole order; each is compact open, and \(Q(\cO)\) is the shell of order \(0\). This holds in every residue characteristic, \(2\) included, and supplies the orbit classification needed for \((\mathrm{SG})\).

Lemma 5.2.

Let \(n\ge2\), let \(N(x)=\sum_{i\le n}x_ix_{2n+1-i}\) on \(\mathbb A^{2n}\) (a hyperbolic form, smooth over \(\Z\)), \(B(x,y)=N(x+y)-N(x)-N(y)\) its polar form, \(Q=\{N=1\}\) and \(X(F):=Q(F)\); let \(K=\Spin_{2n}(\cO)\) act through \(\SO_{2n}(\cO)\). For \(m\ge0\) put

$$O_m:=\{x\in Q(F):\ \min_i\val(x_i)=-m\},\qquad x_m:=\varpi^me_1+\varpi^{-m}e_{2n}\in O_m .$$

Then \(K\backslash Q(F)=\{O_m\}_{m\ge0}\), every \(O_m\) is compact open, and \(Q(\cO)=O_0\). There is no restriction on \(q\) or on \(\operatorname{char}F\).

Write \(L=\cO^{2n}\). The polar form \(B\) is perfect on \(L\) in every characteristic.

  1. The pole order is an invariant and is nonnegative. If every \(x_i\) lies in \(\varpi\cO\) then \(N(x)\in\varpi^2\cO\), so \(N(x)\ne1\); hence \(m:=-\min_i\val(x_i)\ge0\) on \(Q(F)\). The group \(\SO_{2n}(\cO)\) preserves every lattice \(\varpi^kL\), hence the pole order. The shells \(O_m\) are open (the pole order is locally constant) and closed and bounded, hence compact, and \(Q(\cO)=O_0\).
  1. A hyperbolic plane through \(y\). Let \(x\in O_m\) and \(y:=\varpi^mx\); it is primitive in \(L\) with \(N(y)=\varpi^{2m}\). Since \(B\) is perfect and \(y\) is primitive there is \(z\in L\) with \(B(y,z)=1\), and
$$N(z+ay)=N(z)+a+\varpi^{2m}a^2 .$$

For \(m\ge1\) the polynomial \(f(a)=\varpi^{2m}a^2+a+N(z)\) has \(f'\equiv1\bmod\varpi\), so Hensel's lemma gives a root and \(z':=z+ay\) is isotropic. For \(m=0\) first replace \(z\) by \(z+w\) with \(w\in y^\perp\): the reduction \(\bar y^\perp\), of dimension \(2n-1\ge3\), contains a pair \(s_0,t_0\) with \(B(s_0,t_0)=1\), and \(N(z+ss_0+tt_0)\) is a quadratic polynomial in \((s,t)\) with nondegenerate quadratic part, which takes every value of \(\F_q\) (also for \(q\) even). So \(N(z+w)\) can be given a residue for which \(f\) has a simple root modulo \(\varpi\), and Hensel's lemma applies again. The plane \(P=\cO y\oplus\cO z'\) is then a hyperbolic plane, with an isometry onto \(\langle e_1,e_{2n}\rangle\) sending \(y\mapsto y_m:=\varpi^{2m}e_1+e_{2n}\) and \(z'\mapsto e_1\).

  1. The complement and the lift. \(P\) is unimodular, so \(L=P\oplus P^\perp\) with \(P^\perp\) regular of rank \(2n-2\), and \(\overline{P^\perp}\cong H^{n-1}\) over \(\F_q\) because the discriminant (resp. the Arf invariant) is additive. The scheme of isometries \(P^\perp\to H^{n-1}\) is a torsor under the smooth group \(\mathrm O(H^{n-1})\) (cited: Conrad, Reductive group schemes, Appendix C), so the isometry over \(\F_q\) lifts over the complete ring \(\cO\). This gives \(g\in\mathrm O(L)\) with \(gy=y_m\). Composing with the reflection in a vector of unit norm in \(P^\perp\), which fixes \(y\), corrects the determinant (resp. the Dickson invariant), so that \(g\in\SO(L)\) and \(gx=x_m\). Hence \(\SO_{2n}(\cO)\) is transitive on \(O_m\).
  1. \(K\)-orbits equal \(\SO_{2n}(\cO)\)-orbits. From the fppf sequence \(1\to\mu_2\to\Spin\to\SO\to1\) over \(\cO\), the image of \(\Spin_{2n}(\cO)\) in \(\SO_{2n}(\cO)\) is the kernel of the spinor norm \(\SO_{2n}(\cO)\to H^1_{\mathrm{fppf}}(\cO,\mu_2)=\cO^\times/\cO^{\times2}\) (cited: Kummer theory, the connecting map being the spinor norm). The stabilizer of \(y_m\) contains \(\SO(W)(\cO)\), \(W=\langle e_2,\dots,e_{2n-1}\rangle\), and in it the product of reflections \(r_{e_2+e_{2n-1}}r_{e_2+ue_{2n-1}}\) acts as \(\operatorname{diag}(u,u^{-1})\) on \(\langle e_2,e_{2n-1}\rangle\), with spinor norm \(u\). So \(\SO_{2n}(\cO)=\operatorname{im}(K)\cdot\Stab(y_m)\), and the \(K\)-orbit of \(x_m\) is the whole of \(O_m\). \(\square\)

Builds on

  • Strategy 5.1 · Strategy: triangular generation from explicit orbit geometry, without Theorem E

Built on by

  • Theorem 5.3 · \((\mathrm{SG})\) for \(\Spin_{2n}/\Spin_{2n-1}\), every \(n\ge2\), every non-archimedean local field
  • Audit 7.20 · Audit of the sphere results: the residue-characteristic-2 case is genuinely covered
Original record, as the agent wrote it at 03:15 UTC

Sphere: Spin(O)-orbits on Q(F) are the pole-order shells, for every local field F

Model (families/spin.html §1): \(Q=\{x\in\mathbb A^{2n}:q(x)=1\}\), \(q(x)=\sum_{i\le n}x_ix_{2n+1-i}\) (hyperbolic, smooth over \(\mathbb Z\)), \(X(F):=Q(F)\), \(K=\mathrm{Spin}_{2n}(\mathcal O)\) acting through \(\mathrm{SO}_{2n}(\mathcal O)\), \(n\ge2\).

Claim. \(K\backslash Q(F)=\{O_m\}_{m\ge0}\), \(O_m=\{x\in Q(F):\min_i\operatorname{val}x_i=-m\}\ni x_m:=\varpi^me_1+\varpi^{-m}e_{2n}\); each \(O_m\) is compact open; \(Q(\mathcal O)=O_0\).

Proof. 1. \(m<0\) impossible (\(q(x)\in\varpi^2\mathcal O\)); \(\min\operatorname{val}\) is \(\mathrm{SO}_{2n}(\mathcal O)\)-invariant; shells are open, \(Kx\) compact.

  1. \(\mathrm{SO}_{2n}(\mathcal O)\) is transitive on \(O_m\). \(y=\varpi^mx\) is primitive in \(L=\mathcal O^{2n}\), \(q(y)=\varpi^{2m}\); \(L\) unimodular gives \(z\in L\), \(B(y,z)=1\). \(q(z+ay)=q(z)+a+\varpi^{2m}a^2\): for \(m\ge1\) Hensel (\(f'\equiv1\)) gives \(q(z')=0\); for \(m=0\) first move \(z\) by \(w\in y^\perp\) (\(\bar y^\perp\) has a hyperbolic pair, \(2n-1\ge3\), and \(s^2\alpha+st+t^2\beta+\gamma s+\delta t\) is onto \(\mathbb F_q\), also for \(q\) even) to make \(q(z)\equiv\bar a^2+\bar a\), then Hensel. So \(P=\mathcal Oy\oplus\mathcal Oz'\cong H\) via \(y\mapsto y_m:=\varpi^{2m}e_1+e_{2n}\), \(z'\mapsto e_1\). \(P^\perp\) is unimodular of rank \(2n-2\) with \(\overline{P^\perp}\cong H^{n-1}\) (Arf/discriminant additive), and the isometry lifts to \(\mathcal O\) since \(\mathrm{Isom}(P^\perp,H^{n-1})\) is a torsor under the smooth \(\mathrm O(H^{n-1})\) (cited: Conrad, Reductive group schemes, App. C). Get \(g\in\mathrm O(L)\), \(gy=y_m\); correct det/Dickson by the reflection in a \(q\)-unit vector of \(P^\perp\).
  2. \(K\)-orbits \(=\mathrm{SO}_{2n}(\mathcal O)\)-orbits: \(\mathrm{im}(\mathrm{Spin}(\mathcal O))=\ker(\mathrm{sn}:\mathrm{SO}(\mathcal O)\to H^1_{\mathrm{fppf}}(\mathcal O,\mu_2)=\mathcal O^\times/\mathcal O^{\times2})\) (cited: Kummer, connecting map = spinor norm). \(\mathrm{Stab}(y_m)\supseteq\mathrm{SO}(W)(\mathcal O)\), \(W=\langle e_2,\dots,e_{2n-1}\rangle\), and \(r_{e_2+e_{2n-1}}r_{e_2+ue_{2n-1}}=\mathrm{diag}(u,u^{-1})\) on \(\langle e_2,e_{2n-1}\rangle\) has \(\mathrm{sn}=u\); so \(\mathrm{SO}(\mathcal O)=\mathrm{im}(K)\cdot\mathrm{Stab}(y_m)\). \(\square\)

No restriction on \(q\) or \(\operatorname{char}F\).

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