§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:15:06Z

Audit 4.11 · Audit: the torus criterion and the norm-torus counterexample

proved

For tori, (SG) is equivalent to the vanishing of \(\ker(X_*(H)_\sigma\to X_*(T)_\sigma)\), while \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) always holds. For the norm torus \(\operatorname{Res}\mathbb G_m/U_1\) the kernel is \(\mathbb Z/2\); for the Galois torus \(\operatorname{Res}\mathbb G_m/\mathbb G_m\) it is \(0\). This rechecks Counterexample 1.5 and Counterexample 2.3, the simplest instance of the counterexample.

Audit 4.11.

Let \(T\) be a torus over \(\mathbb F_q\), \(\sigma\) Frobenius on \(X_*(T)\), \(H\subseteq T\) a subtorus, \(X=T/H\), \(F=\mathbb F_q(\!(t)\!)\).

  1. \(\operatorname{Tri}\{V_\lambda\ast\mathrm{IC}_0\}=\mathcal C_k\).
  2. \(K\backslash X(F)\cong X_*(X)^\sigma\), \(T_\lambda\ast\mathbf 1_{yX(\mathcal O)}=\mathbf 1_{t^\lambda yX(\mathcal O)}\) for \(\lambda\in X_*(T)^\sigma\), and (SG) \(\iff X_*(T)^\sigma\to X_*(X)^\sigma\) is onto \(\iff\ker\big(X_*(H)_\sigma\to X_*(T)_\sigma\big)=0\).
  3. \(T=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}\mathbb G_m\), \(H=U_1=T^\theta\), \(\theta(z)=\sigma(z)^{-1}\): kernel \(\mathbb Z/2\); \(\mathcal H_K\ast\varphi_0=\bigoplus_m\mathbb C\mathbf 1_{t^{2m}\mathcal O^\times}\); \((\mathrm{SG})_n\) holds iff \(n\) is even; (C), D0–D3 hold, and (R) fails only because \(T=T_0\) is not split. For \(H=\mathbb G_m\) (the Galois torus) the kernel is \(0\) and (SG) holds.

Outcome: holds for the theorem and both examples; minor defect in Counterexample 1.5 step 6, whose Weil-summand sentence needs \(\phi_1\phi_2=1\) (see Audit 4.8).

  1. Sheaves. Audit 4.2, step 7.
  2. Points. For an unramified torus \(Y\), \(Y(F)/Y(\mathcal O)\cong X_*(Y)^\sigma\): take Frobenius invariants of \(0\to L^+Y\to LY\to X_*(Y)\to0\) on \(k\)-points and use \(H^1(\mathbb F_q,L^+Y)=1\) (Lang). \(T(\mathcal O)\to X(\mathcal O)\) is onto, because \(H\)-torsors over \(\mathcal O\) are trivial (Hensel, then Lang). So the \(K\)-orbits are the cosets of \(X(\mathcal O)\), labelled by \(X_*(X)^\sigma\). As \(Kt^\lambda K=t^\lambda K\), \((T_\lambda\ast\varphi)(x)=\varphi(t^{-\lambda}x)\), and \(\mathcal H_K\ast\varphi_0=\operatorname{span}\{\mathbf 1_{t^\lambda X(\mathcal O)}:\lambda\in X_*(T)^\sigma\}\).
  3. The criterion. For the infinite cyclic group generated by \(\sigma\), \(H^0\) is the invariants and \(H^1\) the coinvariants. The long exact sequence of \(0\to X_*(H)\to X_*(T)\to X_*(X)\to0\) identifies \(\operatorname{coker}(X_*(T)^\sigma\to X_*(X)^\sigma)\) with \(\ker(X_*(H)_\sigma\to X_*(T)_\sigma)\).
  4. Norm torus. \(X_*(T)=\mathbb Z^2\), \(\sigma(a,b)=(b,a)\), \(\theta(a,b)=(-b,-a)\); \(X_*(U_1)=\mathbb Z(1,-1)\) with \(\sigma=-1\), so \(X_*(U_1)_\sigma=\mathbb Z/2\); \(X_*(T)_\sigma\cong\mathbb Z\) via \(a+b\), and \((1,-1)\mapsto0\). Concretely \(X(F)=F^\times\), \(g\cdot x=N(g)x\), \(N(\mathcal O_E^\times)=\mathcal O^\times\) (onto on residue fields, and on \(1+t\mathcal O\) by successive approximation since the trace \(\mathbb F_{q^2}\to\mathbb F_q\) is onto), \(N(t^m)=t^{2m}\), so \(T_m\ast\varphi_0=\mathbf 1_{t^{2m}\mathcal O^\times}\). Over \(F_n\): for odd \(n\) the same; for even \(n\), \(T\) splits and \(T_{(a,b)}\ast\mathbf 1_{\mathcal O_n^\times}=\mathbf 1_{t^{a+b}\mathcal O_n^\times}\). Galois torus: \(X_*(\mathbb G_m)=\mathbb Z(1,1)\) is fixed and maps to \(2\ne0\) in \(X_*(T)_\sigma\), so the kernel is \(0\), consistent with Audit 4.9, step 7.
  5. Hypotheses. \(\theta\) commutes with Frobenius; \(G^\theta=U_1\) is connected; \(S=\{(a,a)\}\) is a split maximal \(\theta\)-split torus; \(Z_{G^\theta}(S)=U_1\): (C) holds. Orbits are open and closed, the stabilizers \(L^+U_1\) are connected and every convolution map is an isomorphism onto an orbit: D0–D3 hold. (R) fails only through \(T=T_0=R\) (Audit 4.1, item 8).
  6. Nature of the failure. Here the failure is an orbit obstruction: \(X(F)\) has two \(G(F)\)-orbits (\(H^1(F,U_1)=\mathbb Z/2\)), so this example alone does not separate the leading-term mechanism from the orbit count. Hermitian \(\mathrm{SL}_2\) does (Audit 4.5, Audit 4.7).

Builds on

  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion
  • 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