§10 · Audit · Second audit, then re-checks · 2026-10-17T07:04:06Z

Correction 10.5 · The \(\SO_2\)-model of \(\SL_2/\SO_2\) over \(\F_q\) and Chen–Yi's splitness hypothesis

proved correction ★ key step

Correction. Corrects Audit 7.16 · Audit: the comparability objection is correct — three fields, two bridges, and what each bridge needs.

For \(\theta(g)=(g^{\mathsf T})^{-1}\) the \(\theta\)-stable Borel pairs are \((B_\pm,\SO_2)\), defined and split over \(\F_q\) only when \(q\equiv1\bmod4\); the diagonal torus is the \(\theta\)-split torus, not the torus of a \(\theta\)-stable pair. So the split model is genuinely needed for Chen–Yi's Weil-structure setting, while Lang's count of \(K\)-orbits holds in both models.

Correction 10.5.

Let \(G=\SL_2\) over \(\F_q\), \(q\) odd, with \(\theta(g)=(g^{\mathsf T})^{-1}\), \(X=G/G^\theta=G/\SO_2\), and \(J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\).

  1. \(\theta=\operatorname{Ad}(J)\). The \(\theta\)-stable Borel subgroups of \(G_{\overline{\F}_q}\) are the stabilizers \(B_\pm\) of the lines spanned by \((1,\pm i)\), and the only \(\theta\)-stable maximal torus contained in \(B_\pm\) is \(\SO_2=Z_G(J)\).
  2. A \(\theta\)-stable Borel pair defined over \(\F_q\) with split torus exists if and only if \(q\equiv1\bmod4\). For \(q\equiv3\bmod4\) the standing hypothesis of Chen–Yi, §8.2 (\(G,B_0,T_0,\theta\) defined over \(\F_q\), \(T_0\) split, \((B_0,T_0)\) a \(\theta\)-stable Borel pair) fails for this model. The diagonal torus is the maximal \(\theta\)-split torus \(S\); it belongs to Chen–Yi's \(\theta\)-split Borel pair \((B,T)\), not to \((B_0,T_0)\).
  3. Independently of 2, Lang's theorem counts the \(K\)-orbits in both models: in the \(\SO_2\)-model \(x_m=\operatorname{diag}(t^m,t^{-m})\) is \(\F_q\)-rational and \(\pi_0\Stab_{L^+G}(x_m)=\{\pm I\}\) with trivial Frobenius action, so \(\mathcal O_m(\F_q)\) is two \(K\)-orbits for \(m\ge1\) and one for \(m=0\).
  4. Hence the recommendation of Objection 1.8 to use the split model \(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\) — for which (upper triangular, diagonal) is a \(\theta\)-stable Borel pair over \(\F_q\) with split torus — stands. Sheaf-level statements over \(\overline{\F}_q\) do not depend on the model, and over \(\F_{q^2}\) the two models become isomorphic.
  1. \(\theta\) is inner. For \(g=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\SL_2\), \(JgJ^{-1}=\begin{pmatrix}d&-c\\-b&a\end{pmatrix}=(g^{\mathsf T})^{-1}\). A Borel subgroup \(B=\Stab(\ell)\), \(\ell\) a line, is \(\theta\)-stable iff \(J\ell=\ell\), i.e. iff \(\ell\) is an eigenline of \(J\); since \(J(1,\pm i)^{\mathsf T}=\pm i\,(1,\pm i)^{\mathsf T}\), these are \(B_\pm\). If \(T'\subseteq B_\pm\) is a \(\theta\)-stable maximal torus, then \(J\) normalizes \(T'\); as \(J\in B_\pm\) and \(N_{B_\pm}(T')=T'\), we get \(J\in T'\). Since \(J\) is regular semisimple (eigenvalues \(\pm i\)), \(T'=Z_G(J)=\{a+bJ:a^2+b^2=1\}=\SO_2\).
  2. Rationality. Frobenius maps the line spanned by \((1,i)\) to the line spanned by \((1,i^q)\). So \(B_\pm\) are defined over \(\F_q\) iff \(i\in\F_q\), i.e. \(q\equiv1\bmod4\); otherwise Frobenius exchanges them, and no \(\theta\)-stable Borel subgroup is defined over \(\F_q\). Likewise \(\SO_2\) (the norm-one torus of \(\F_q(i)\)) is split iff \(-1\) is a square in \(\F_q\). The diagonal torus \(D\) satisfies \(\theta(d)=d^{-1}\), so it is a maximal \(\theta\)-split torus; it lies in the upper triangular Borel \(B\), and \(\theta(B)\) is the lower triangular Borel, so \(B\cap\theta(B)=D\): \((B,D)\) is a \(\theta\)-split Borel pair in the sense of Chen–Yi, §2.1, not a \(\theta\)-stable one. In the split model \(\theta=\operatorname{Ad}\operatorname{diag}(1,-1)\), on the other hand, the upper triangular Borel and the diagonal torus are \(\theta\)-stable and split over \(\F_q\).
  3. Lang's count. In the \(\SO_2\)-model, as in Computation 2.1 (the computation only uses that \(2\) is invertible), \(\Stab_{L^+G}(x_m)=\{g_{a,b}:a^2+b^2=1,\ a\in\cO,\ b\in t^m\cO\}\), which for \(m\ge1\) is \(\{\pm I\}\) times a connected pro-unipotent group, and for \(m=0\) is \(L^+\SO_2\), connected. These groups and \(x_m\) are defined over \(\F_q\). By Lang's theorem for the connected pro-algebraic groups \(L^+G\) and \(\Stab(x_m)^\circ\), the \(K\)-orbits in \(\mathcal O_m(\F_q)\) correspond to \(H^1(\F_q,\pi_0\Stab(x_m))=\operatorname{Hom}(\widehat\Z,\Z/2)=\Z/2\) for \(m\ge1\), and to a single class for \(m=0\). The same count holds in the split model (Computation 2.8, part (a)).
  4. Comparison over \(\F_{q^2}\). Over \(\F_{q^2}\) the element \(J\) is diagonalised by an element of \(\SL_2(\F_{q^2})\) (its eigenvectors \((1,\pm i)\), with one column rescaled to make the determinant \(1\)), so \(\operatorname{Ad}(J)\) becomes \(\operatorname{Ad}\operatorname{diag}(i,-i)=\operatorname{Ad}\operatorname{diag}(1,-1)\); the two models are isomorphic over \(\F_{q^2}\), and over \(\overline{\F}_q\) the categories \(\mathcal C_\ell\) coincide. So the false remark of Audit 7.16 affects no conclusion of the run: the function statements were proved for the split model, and the sheaf statements are geometric.

Builds on

  • Audit 7.16 · Audit: the comparability objection is correct — three fields, two bridges, and what each bridge needs
  • Objection 1.8 · Comparability: the same pair \((G,X)\) in every row, but three different fields
  • Computation 2.1 · Stabilizers for \(\SL_2/\SO_2\): \(\mu_2\) times a pro-unipotent group

Built on by

  • Correction (repair of A7-16) X2.5 · Chen–Yi §8.2 holds for the \(\mathrm{SO}_2\)-form iff \(q\equiv1\bmod4\); Lang’s count holds in both
  • Summary 9.5 · Verdict after the second audit: \(E_6/F_4\) is still the one divergence