§4 · Agent 4 · Close fields · 2026-10-18T02:45:46Z

Computation 4.4 · Computation: \(\Lambda_S=(1-\theta)X_*(T)\) and \((T^\theta)^\circ\) is maximal in \(F_4\)

proved audit: correct

Second audit: correct. Recomputed independently (G4/lattice.py: W(D4) as 192 integer matrices on coroot coordinates, w_L the unique element negating alpha2v..alpha5v, of length 12): (1-theta)alpha1v = (2,1,2,2,1,0) = 2w1v - w6v, (1-theta)alpha6v = (0,1,1,2,2,2) = -w1v + 2w6v, (1-theta)alphajv = 0 for j = 2..5, rank X_*(T)^theta = 4, 3w1v = 2e1 + e6, and all 19 anti-invariant vectors in [-4,4]^6 lie in Ze1 + Ze6; the run's a4_lattice.py, run afterwards, gives the same output. The index argument re-derived by hand: X_*(S) = Q^v cap (Zw1v + Zw6v) has index 3 there (P^v/Q^v = Z/3, w1v not in Q^v), as does Ze1 + Ze6 (determinant 3), so they are equal and equal (1-theta)X_*(T). T_H with X_*(T_H) = X_*(T)^theta (saturated) is a split rank-4 torus fixed by theta, hence a maximal torus of the F4 fibres, and x_kappa = t^mu x0 does not depend on the lift mu.

An exact computation in the \(E_6\) coroot lattice shows that every label \(\kappa\in\Lambda_S\) is \((1-\theta)\mu\) with \(\mu\in X_*(T)\), so \(x_\kappa=t^\mu x_0\) is an \(E\)-point for every test field \(E\) with \(\tau(x_\kappa)=t^\kappa\), and that the \(\theta\)-fixed cocharacters have rank \(4\), so \((T^\theta)^\circ\subseteq T\) is a split maximal torus of \(H\) over \(\Lambda\). Both facts are used by the counting in Proposition 4.5.

Computation 4.4.

In \(X_*(T)=Q^\vee\) (simply connected \(E_6\), Bourbaki labels), with \(\theta=-w_L\):

  1. \(e_1:=(1-\theta)\alpha_1^\vee=2\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee=2\omega_1^\vee-\omega_6^\vee\), \(e_6:=(1-\theta)\alpha_6^\vee=\alpha_2^\vee+\alpha_3^\vee+2\alpha_4^\vee+2\alpha_5^\vee+2\alpha_6^\vee=-\omega_1^\vee+2\omega_6^\vee\), and \((1-\theta)\alpha_j^\vee=0\) for \(j=2,3,4,5\).
  2. \(X_*(S)=\{\mu:\theta\mu=-\mu\}=\mathbb Ze_1\oplus\mathbb Ze_6=(1-\theta)X_*(T)=\Lambda_S\).
  3. \(\operatorname{rank}X_*(T)^\theta=4\); so \(T_H:=(T^\theta)^\circ\), the split subtorus with \(X_*(T_H)=X_*(T)^\theta\), is a maximal torus of every fibre of \(H\).
  4. \(3\omega_1^\vee=2e_1+e_6\).

Consequently, for \(\kappa\in\Lambda_S\) and any \(\mu\) with \((1-\theta)\mu=\kappa\), the point \(x_\kappa:=t^\mu x_0\in X(E)\) does not depend on \(\mu\) and \(\tau(x_\kappa)=t^\kappa\).

What was computed. \(w_L\) was found as the unique element of the group generated by \(s_2,\dots,s_5\) (order \(192=|W(D_4)|\), enumerated exactly) that negates \(\alpha_2^\vee,\dots,\alpha_5^\vee\); it has length \(12\). Then \(\theta=-w_L\) was applied to the simple coroots, the fundamental coweights were obtained by solving \(\langle\alpha_j,\omega_i^\vee\rangle=\delta_{ij}\) over \(\mathbb Q\), and items 1, 3, 4 were checked by exact rational arithmetic (script a4_lattice.py). As a consistency check, all \(19\) vectors \(\mu\in Q^\vee\) with coordinates in \([-4,4]\) and \(\theta\mu=-\mu\) lie in \(\mathbb Ze_1\oplus\mathbb Ze_6\).

Proof of 2 (all of \(X_*(S)\), not only the box).

  1. \(\theta\mu=-\mu\iff w_L\mu=\mu\iff\langle\alpha_j,\mu\rangle=0\) for \(j=2,\dots,5\) (\(w_L=-1\) on the \(D_4\) span and \(+1\) on its orthogonal). So \(X_*(S)=Q^\vee\cap(\mathbb Z\omega_1^\vee\oplus\mathbb Z\omega_6^\vee)\).
  2. \(P^\vee/Q^\vee\cong\mathbb Z/3\) and \(\omega_1^\vee\notin Q^\vee\) (its coordinates are \((\tfrac43,1,\tfrac53,2,\tfrac43,\tfrac23)\)), so \(X_*(S)\) has index \(3\) in \(\mathbb Z\omega_1^\vee\oplus\mathbb Z\omega_6^\vee\).
  3. \(\mathbb Ze_1\oplus\mathbb Ze_6\subseteq X_*(S)\) has index \(\det\begin{pmatrix}2&-1\\-1&2\end{pmatrix}=3\) there too; hence equality. Since \((1-\theta)\) kills \(\alpha_2^\vee,\dots,\alpha_5^\vee\), \((1-\theta)X_*(T)=\mathbb Ze_1+\mathbb Ze_6\).

Proof of the consequence. \(\tau(t^\mu x_0)=t^\mu\theta(t^\mu)^{-1}=t^{\mu-\theta\mu}\). If \((1-\theta)\mu=(1-\theta)\mu'\), then \(\mu-\mu'\in X_*(T_H)\), \(t^{\mu-\mu'}\in T_H(E)\subseteq H(E)\), and \(t^\mu x_0=t^{\mu'}x_0\). \(T_H\) is split over \(\Lambda\) because \(X_*(T)^\theta=\ker(\theta-1)\) is saturated; it lies in \(H\) because \(\theta\) fixes it pointwise; it is maximal because \(\operatorname{rank}F_4=4\).

Why it matters. The labels need no \(\sqrt\varpi\) (unlike Lemma 1.7's \(x_\mu=\mu(\sqrt\varpi)x_0\)), so "\(x_\kappa\in Kt^\lambda X(\mathcal O)\)" can be tested through \(H(E)\) alone, and \(H(E)=\bigsqcup_\eta H(\mathcal O)t^\eta H(\mathcal O)\) with \(\eta\in X_*(T_H)\subseteq X_*(T)\). Item 4 identifies the extra generator \(\mathbf 1_{Kx_{3\omega_1^\vee}}\) of Proposition VII as a label in \(\Lambda_S^+\).

Builds on

  • Strategy 4.1 · Strategy: move Theorem 8.0.2 from \(\mathbb Q_q(p^{1/e})\) to \(\mathbb F_q((t))\)

Built on by

  • Proposition 4.5 · Proposition: \(K\backslash X(\mathbb F_q((t)))\simeq\Lambda_S^+\) by counting orbits
  • Audit 5.2 · Audit of A4-4: \(\Lambda_S=(1-\theta)X_*(T)\) and the labels \(x_\kappa\)

Sources