§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:04:28Z

Audit 5.8 · Audit of A3-1, A3-2, A3-3: integral model, \(H^1\) and (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). over \(\mathbb F_q((t))\)

proved audit: correct

Second audit: correct. Re-derived Proposition 3.2 and Lemma 3.3: Z/2 = mu_2 over Z[1/2], and Conrad Lemma 2.2.4 and Thm 2.3.1 (checked in the Conrad text) give smoothness of H and quasi-affineness of G/H. My scripts confirm that the theta-fixed roots are Phi_L, that no root has theta b = -b, that no root is orthogonal to the black coroots (so Z_G(T_8) = T), elementary divisors (1,1), S[2] inside T_8 (e1 = a2v+a5v, e6 = a2v+a3v mod 2) and <rho_L, alpha_j^v> = (-3,1,1,1,1,-3). BT III Thm 4.7, Hilbert 90 for split tori and split unipotent groups, and Richardson's characteristic-free affineness criterion are correctly used. I also read S4-3...S4-13 in S4.json/S4b.json: Audit 3.1's table lists every characteristic-0 input. Correct.

The integral model (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7). of \(\mathcal M_\theta\) over \(\mathbb F_q[\![t]\!]\) and the Galois cohomology behind (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). are proved using only that \(2\) is invertible. The cited Conrad results are as stated in the local PDF. The audit table Audit 3.1 is accurate once the two repairs of Audit 5.10 and Objection 5.11 are made.

Audit 5.8.

Outcome: Proposition 3.2 and Lemma 3.3 hold. Audit 3.1 holds as an audit table, except that its rows "S4-5(3), S4-10: only types G, U" and "S4-3 step 6: colours" need the repairs of Audit 5.10 and Objection 5.11. Characteristic \(p\) enters only as follows: \(H\), \(B_8\), \(L^\theta\) are smooth because the fixed points are taken under the étale \(\mu_2\cong\mathbb Z/2\) (\(2\in\mathcal O^\times\)); \(H^1\) is taken only for smooth connected groups (\(\mathrm{Spin}_8\), \(F_4\), split tori, split unipotent groups, \(B_8\)); \(p=3\) causes nothing, since \(Z(G)^\theta=\mu_3\cap\mu_2=1\) and \(X^*(T^\theta)\) is torsion-free.

  1. Conrad, checked in the local PDF. Lemma 2.2.4 (pp. 64–65): for \(Y\) of multiplicative type in a smooth \(S\)-affine group \(G\), \(Z_G(Y)\) is smooth with \(\mathrm{Lie}=\mathrm{Lie}(G)^Y\). Theorem 2.3.1 (p. 67): for \(G\) smooth \(S\)-affine and \(H\) of multiplicative type, \(G/Z_G(H)\) is a smooth quasi-affine \(S\)-scheme. Both hold over any base. Applied to \(G'=G\rtimes\mathbb Z/2\) and \(\mu_2=\langle\varepsilon\rangle\): \(Z_{G'}(\mu_2)=H\times\mu_2\), and \(G'/(H\times\mu_2)=G/H\), since \(G\to G'/(H\times\mu_2)\) is onto with stabilizer \(G\cap(H\times\mu_2)=H\).
  2. Lie algebra and roots (recomputed, Audit 5.2). The \(\theta\)-fixed roots are exactly \(\Phi_L\) (24 roots), no root has \(\theta\beta=-\beta\), and \(\theta(\Phi^+\smallsetminus\Phi_L)\subseteq\Phi^-\). So \(\dim\mathfrak g^\theta=4+24+24=52\).
  3. Connected \(F_4\) fibres. \(X^*(T^\theta)=X^*(T)/(1-\theta^*)X^*(T)\) is free of rank \(4\) (elementary divisors \((1,1)\), recomputed), so \(T^\theta=T_8\) is a torus. \(Z_G(T_8)=T\) because no root is orthogonal to the black coroots. Hence \(T_8\) is maximal in \(H_k^\circ\). \(H^\circ_k\) is reductive (Prasad–Yu, order \(2\) prime to \(p\)) with \(48\) one-dimensional roots in rank \(4\), hence of type \(F_4\). Every automorphism of \(F_4\) is inner (Conrad Thm 7.1.9). So \(g\in H(k)\) factors as \(hz\) with \(z\in Z_G(T_8)\cap H=T_8\), and \(H_k\) is connected. Valid for every \(p\ne2\), including \(p=3\).
  4. Open orbit. \(B\cap H=B_L\cap L^\theta\) by the big-cell argument, functorial in \(\mathcal O\)-algebras. \(L^\theta=\mathrm{Spin}_8\): \(\theta(sm)=s^{-1}m\) forces \(s\in S[2]\), and \(S[2]\subseteq T_8\) since \(e_1\equiv\alpha_2^\vee+\alpha_5^\vee\) and \(e_6\equiv\alpha_2^\vee+\alpha_3^\vee\) modulo \(2X_*(T)\). Equality as schemes holds since both are smooth with the same fibres. \(B/B_8\to X\) is a fibrewise open immersion, hence an open immersion (EGA IV 17.9.1). Purity gives \(\mathring X=U_P\times A_X\). The special fibre is one \(B\)-orbit with smooth stabilizer, so the orbit map is separable (checklist item 5).
  5. Measures and \(X(\mathcal O)\). \(X^*(H)=0\), so the relative canonical bundle is \(G\)-trivial. On \(\mathring X\), \((2\rho-2\rho_L)|_{T_8}=0\), using \(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\), re-derived by summing the \(\alpha_3\)-coefficients of the \(D_4\) positive roots. \(H^1(\mathcal O,H)=H^1(\mathbb F_q,H)=1\) by Hensel and Lang.
  6. Lemma 3.3. \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\): Bruhat–Tits III, Thm 4.7 (cited, not opened; the characteristic-free replacement of Kneser for semisimple simply connected groups over henselian fields with perfect residue field of dimension \(\le1\)). \(H^1(F,U_8)=1\) uses the splitting of \(U_8\) (additive Hilbert 90 in characteristic \(p\)). A wound unipotent group would not do, but none occurs. \(X(F)=G(F)x_0\), \(\mathring X(F)=B(F)x_0\), and \(T(\mathcal O)\to A_X(\mathcal O)\) is onto. \(X_F\) is affine by Richardson–Haboush (cited), or directly from Audit 4.3 item 1 (\(\tau\) a closed immersion).
  7. Checklist item 6. The only finite group schemes involved are \(S[2]\) (étale, \(p\ne2\)) and the constant \(N_H(S)/Z_H(S)\). The descent of Proposition 3.4 lands in \(\mathrm{Spin}_8\), so \(H^1(F,S_3)\), infinite for \(p=3\), never enters.

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  • Summary 5.13 · Verdict of audit A5: transfer route A4-2…A4-9 and char-p hypotheses A3-1…A3-10

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