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

1. (a). Read in chen-yi-v4.pdf. §2.1: "Let \(G\) be a connected reductive group over \(k=\overline{\mathbb F}_p\) of characteristic \(p\ne2\), and \(\theta\) an involution. We fix a θ-stable Borel pair \((B_0,T_0)\)"; the θ-split pair \((B,T)\) with \(S\subset T\subset B\subset P\) is introduced separately. §4: "We will assume \(K\) is connected from Section 4–Section 10." §8.2: "We assume \(G,B_0,T_0,\theta\) are defined over a finite field \(\mathbb F_q\) and \(T_0\) is split over \(\mathbb F_q\)." A split maximal torus inside an \(\mathbb F_q\)-Borel makes \(G\) split. The Iwahori \(I_0\) is the preimage of \(B_0\), and the Weil categories of §8.2.1–8.2.2 are built on \(I_0\backslash LX\).

2. (b). \(\operatorname{Ad}\operatorname{diag}(1,-1)\begin{pmatrix}a&b\\0&d\end{pmatrix}=\begin{pmatrix}a&-b\\0&d\end{pmatrix}\): the upper triangular Borel and the diagonal torus are θ-stable and split over every \(\mathbb F_q\); \(G^\theta=T\) is connected.

3. (c). \(JgJ^{-1}=(g^{\mathsf T})^{-1}\) for \(g\in\mathrm{SL}_2\). A Borel \(B=\operatorname{Stab}(\ell)\) is θ-stable iff \(J\in N(B)=B\), iff \(J\ell=\ell\); the eigenlines of \(J\) are \((1,\pm i)\), eigenvalues \(\pm i\). If \(T'\subseteq B_\pm\) is θ-stable, then \(J\) normalizes \(T'\) and \(J\in B_\pm\), so \(J\in N_{B_\pm}(T')=T'\). As \(J\) is regular semisimple (\(p\) odd), \(T'=Z_G(J)=\{aI+bJ:a^2+b^2=1\}=\mathrm{SO}_2\). Frobenius sends \((1,i)\) to \((1,i^q)\), so \(B_\pm\) are \(\mathbb F_q\)-rational iff \(i\in\mathbb F_q\); when \(q\equiv3\bmod4\) they are exchanged and no θ-stable Borel subgroup is rational. \(\mathrm{SO}_2\) is the norm-one torus of \(\mathbb F_q(i)\), split iff \(i\in\mathbb F_q\). For \(D\): \(\theta(d)=d^{-1}\), and θ maps the upper to the lower triangular Borel, with intersection \(D=Z_G(D)\).

4. (d). An \(\mathbb F_q\)-isomorphism of pairs is \(\operatorname{Ad}(g)\) with \(g\in\mathrm{PGL}_2(\mathbb F_q)=\mathrm{GL}_2(\mathbb F_q)/\mathbb F_q^\times\) (Hilbert 90) and \(g\operatorname{diag}(1,-1)g^{-1}=cJ\), \(c\in\mathbb F_q^\times\). Comparing eigenvalues, \(\{1,-1\}=\{ci,-ci\}\), which forces \(i\in\mathbb F_q\). As \(G\)-varieties: every rational point of \(X_{-1}\) has split stabilizer (binary forms over \(\mathbb F_q\) of discriminant \(-1\) are hyperbolic), while \(\operatorname{Stab}(I)=\mathrm{SO}_2\) in \(X_1\) is anisotropic when \(q\equiv3\bmod4\). Conversely, put \(P=\begin{pmatrix}1&1\\ i&-i\end{pmatrix}\), with \(\det P=-2i\) a unit as \(p\) is odd. Then \(P^{-1}JP=\operatorname{diag}(i,-i)\) and \(\operatorname{Ad}\operatorname{diag}(i,-i)=\operatorname{Ad}\operatorname{diag}(1,-1)\), so \(\operatorname{Ad}(P^{-1})\) carries \((G,\operatorname{Ad}J,\mathrm{SO}_2)\) to \((G,\operatorname{Ad}\operatorname{diag}(1,-1),T)\). If \(i\in\mathcal O\), then \(P\in\mathrm{GL}_2(\mathcal O)\), and \(\operatorname{Ad}(P^{-1})\) preserves \(K\) and each \(Kt^\lambda K\) (elementary divisors are \(\mathrm{GL}_2(\mathcal O)\)-invariant); it sends \(x_0\mapsto x_0\), hence \(X(\mathcal O)=Kx_0\) to \(Kx_0\).

5. (e). In the \(\mathrm{SO}_2\)-form \(x_m=\operatorname{diag}(t^m,t^{-m})\) is \(\mathbb F_q\)-rational, and \(\operatorname{Stab}_{L^+G}(x_m)=\{g_{a,b}\}\) (Computation 2.1; the computation uses only \(2\in\mathcal O^\times\)). This is \(\{\pm I\}\times(\)connected pro-unipotent\()\) for \(m\ge1\) and \(L^+\mathrm{SO}_2\) (connected) for \(m=0\), all over \(\mathbb F_q\). Lang's theorem for connected pro-algebraic groups (D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q.) gives \(K\backslash\mathcal O_m(\mathbb F_q)=H^1(\mathbb F_q,\{\pm I\})=\mathbb Z/2\) for \(m\ge1\), and a point for \(m=0\). The split form is Computation 2.8(a). Only the rationality of \(S\) is used: \(S=D\) is split in the \(\mathrm{SO}_2\)-form. This is the hypothesis "\(T\supseteq S\) split" of Setup 4.1, which is weaker than §8.2.

6. (f), purity transfer. If \(x\) is \(\mathbb F_{q^n}\)-rational, the eigenvalues of \(\Phi_n\) on a stalk at \(x\) have absolute value \((q^n)^{(w+m)/2}\) iff those of \(\Phi_{2n}=\Phi_n^2\) have absolute value \((q^{2n})^{(w+m)/2}\). Over \(\mathbb F_{q^2}\) the \(\mathrm{SO}_2\)-form is the split form, by (d). Its canonical Weil structures agree with the transported ones up to the signs \(\chi(c)=\pm1\), which arise when the normalising point moves to the other \(K\)-orbit, and signs do not change absolute values. So Thm. 81 for the split form over \(\mathbb F_{q^2}\) gives it for the \(\mathrm{SO}_2\)-form.

7. Correction 10.5 checked. All four parts are re-derived above and correct. One addition: "the function statements were proved for the split model" is accurate, but the run left (SG) for the \(\mathrm{SO}_2\)-form with \(q\equiv3\bmod4\) open (Summary 2.9, gap; Audit 7.18, item 6). Proposition X2.6 settles it. \(\square\)

Effect. The false sentence of Audit 7.16 is the claim that §8.2 needs only \(T_0\) split and that the \(\mathrm{SO}_2\)-form satisfies this with \(S=T\) diagonal. It confuses the θ-split torus with the θ-stable \(T_0\). The stated conclusion of that sentence is true by (e).