§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:11:23Z

Audit 4.3 · Audit: \(K\)-orbits on \(X(F)\) for the twists, one per Frobenius-stable orbit label

proved

Rechecks Setup 2.4(3),(4): every \(L^+G\)-orbit \(\mathcal O_c\) has connected stabilizer \(J_c\); in the unitary twist each \(\mathcal O_c\) contains the rational point \(t^c\), in the Galois twist only \(c=-w_0c\) occur; Lang gives one \(K\)-orbit per such label and \(X(\mathcal O)=Kx_0\). A truncated finite check for \(\mathrm{GL}_2\) and \(\mathrm{SL}_2\), \(q=3\), agrees.

Audit 4.3.

  1. Unitary twist: \(K\backslash X_u(F)\cong\Lambda_H^+\), \(c\mapsto O_c=K\cdot t^c\), and the \(K\)-orbit of \(x\in X_u(F)\subseteq H(E)\) is its Cartan type in \(H(\mathcal O_E)\backslash H(E)/H(\mathcal O_E)\).
  2. Galois twist: \(K\backslash X_g(F)\cong\{c\in\Lambda_H^+:c=-w_0c\}\); the orbits \(\mathcal O_c\) with \(c\ne-w_0c\) have no rational point.
  3. In both, \(X(\mathcal O)=K\cdot x_0\).

Outcome: holds.

  1. Orbits over \(k\). \(\vartheta(L^+H)=L^+H\), so the \(L^+G\)-orbit of \(x\in LH\) is \(L^+H\,x\,L^+H\); by the Cartan decomposition of \(H(k(\!(t)\!))\) these are the \(\mathcal O_c\), \(c\in\Lambda_H^+\).
  2. Stabilizers. In the unitary twist \(g_1t^c\vartheta(g_2)^{-1}=t^c\iff\vartheta(g_2)=t^{-c}g_1t^c\), which forces \(g_1\in J_c=L^+H\cap t^cL^+Ht^{-c}\) and determines \(g_2\). So \(\operatorname{Stab}(t^c)\cong J_c\), which is \(Z_H(c)\ltimes\)(pro-unipotent): its reduction modulo \(t\) is a parabolic subgroup of \(H\) and the kernel is pro-unipotent; in particular it is connected. The Galois twist is the case \(\vartheta=1\).
  3. Rational points. Unitary: \(\mathrm{Fr}_X(t^c)=\mathrm{Fr}(\vartheta(t^c))^{-1}=\mathrm{Fr}(t^{-c})^{-1}=t^c\), since \(T_H\) is split. Galois: \(\mathrm{Fr}_X(x)=\mathrm{Fr}(x)^{-1}\) maps \(\mathcal O_c\) onto \(\mathcal O_{-w_0c}\), so a rational point forces \(c=-w_0c\); conversely a Frobenius-stable \(\mathcal O_c\) is a homogeneous space of the connected group \(L^+G\), hence has a rational point by Lang's theorem.
  4. One \(K\)-orbit. For rational \(x\in\mathcal O_c\) and \(y=gx\) rational, \(g\in L^+G(k)\), the class of \(g^{-1}\mathrm{Fr}(g)\) in \(H^1(\mathbb F_q,\operatorname{Stab}(x))\) depends only on \(y\) and classifies its \(K\)-orbit, since \(H^1(\mathbb F_q,L^+G)=1\). \(\operatorname{Stab}(x)\) is a Frobenius-stable form of \(J_c\), connected, so this \(H^1\) vanishes (Lang for connected pro-algebraic groups, D3). As \(X(F)=LX(\mathbb F_q)=\bigsqcup_c\mathcal O_c(\mathbb F_q)\), items 1–2 follow; \(c=0\) gives item 3. Since \(\vartheta\sigma(K)=K\), the orbit \(K\cdot x\) lies in \(KxK\), which identifies the label with the Cartan type.
  5. Finite truncated check (not used in the proofs). For \(q=3\), all \(3^{12}\) hermitian \(2\times2\) matrices over \(\mathcal O_E/t^3\) were sorted by elementary-divisor type \((c_1,c_2)\), \(c_1\le2\) (the types determined modulo \(t^3\)). For each of the six types the set is a single orbit of \(\mathrm{GL}_2(\mathcal O_E/t^3)\) under \(x\mapsto gxg^*\); the sizes are \(393660,87480,4860,29160,1080,60\) for \((0,0),(1,0),(1,1),(2,0),(2,1),(2,2)\). The \(14580\) matrices of type \((2,0)\) with \(\det\equiv t^2\bmod t^3\), i.e. \(t\) times hermitian matrices of determinant \(1\) and type \(\alpha^\vee\), form a single \(\mathrm{SL}_2(\mathcal O_E/t^3)\)-orbit. This agrees with step 4 and with the classical Jordan-splitting classification of hermitian lattices over an unramified quadratic extension.
  6. \(H=\mathbb G_m\) (Counterexample 2.3, step 5): \(K\) acts on \(F^\times\) through \(N(\mathcal O_E^\times)=\mathcal O^\times\), so the \(K\)-orbits are the \(t^c\mathcal O^\times\), \(c\in\mathbb Z=\Lambda_{\mathbb G_m}^+\).

Builds on

  • Setup 2.4 · Setup: the unitary and Galois \(\mathbb F_q\)-forms of the group case
  • Counterexample 2.3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm

Built on by

  • Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor

Sources