§4 · Agent 4 · Close fields · 2026-10-18T02:48:03Z

Audit 4.7 · Check: Theorem 8.0.2 over ramified \(F'\) needs only Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(W(\mathbb F_q)\)

proved audit: minor defect ★ key step

Minor defect (second audit). Checked against the printed text: the unramified assumption is invoked only in §1.1, the §3.8 parenthetical (in the same sentence that fixes psi of conductor o, as §1.7 does) and §5.3; with psi of conductor O and dx(O) = 1 the measure is self-dual, and (3.10), (4.1) (F 1_{p x O} = q^-1 1_{O x p^-1}), Tate's unramified and ramified formulas, (5.2), (5.3), Lemma 3.9.1, Lemma 4.2.3 and (4.5) hold verbatim over every finite F'/Q_p; P_kappa = c^-1 times the right side of (7.4) (Def. 7.2.3: L_X^{1/2} = c beta) depends only on q; S4-3/4/5/10/12/13 are ramification-free; base change of (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. from Lambda to the finite free O'_e is right (flat closures, open map). Defect: step 6 calls §3.2-3.7 'geometry over the algebraic closure and combinatorics', but §3.2 (Thm 3.2.2), §3.3 (Lemma 3.3.5) and §3.7 (Thm 3.7.1 = [Sa08, Thm 5.2.1]) are p-adic harmonic analysis from [Sa08], whose hypotheses the step never checks. I opened [Sa08] (math/0606130v2): §1.7 and §4.1 fix k a locally compact non-archimedean field of characteristic 0, with no unramified, conductor or residue-characteristic assumption anywhere, so every claim (a)-(d) holds; this is Audit 7.3's check, which I confirmed. Fine point: 'quotients by R(P_alpha) commute with flat base change' needs the quotient map to be universal (e.g. flat), as it is for the models of Theorem 2.8; Sakellaridis only says 'geometric quotient'.

Every place where the paper uses '\(k\) unramified over \(\mathbb Q_p\)' (§1.1, §3.8, §5.3) and every harmonic-analysis formula entering types \(G\) and \(U\) is checked: with \(\psi'\) of conductor \(\mathcal O'\), as §1.7 already prescribes, each holds verbatim for ramified \(F'\). The other hypotheses for \(\mathcal M_\theta\) over \(\mathcal O'\) are verified in S4/S4b by ramification-free arguments, and Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(\mathcal O'\) follows from Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(W(\mathbb F_q)\) by flat base change. The image \(P_\kappa\) of each orbit indicator is an explicit polynomial depending only on \(q\).

Audit 4.7.

Let \(F'\) be a finite extension of \(\mathbb Q_q\) with residue field \(\mathbb F_q\) (any ramification), \(p\) odd. (a) Fix \(\psi'\) of conductor \(\mathcal O'\) (exists: \(\psi'(x)=\psi_{\mathbb Q_p}(\operatorname{Tr}(cx))\) with \(c\mathcal O'=\mathfrak d_{F'/\mathbb Q_p}^{-1}\)) and \(dx(\mathcal O')=1\). Then (3.10), (4.1), Lemma 3.9.1, Lemma 4.2.3, (4.5), Tate's formulas of §5.3, (5.2) (case \(U\)) and (5.3) (case \((U,\psi)\)) hold over \(F'\) exactly as printed. (b) (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)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). hold for \(\mathcal M_\theta\) over \(\mathcal O'\). If Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. holds for \(\mathcal M_\theta\) over \(\Lambda=W(\mathbb F_q)\) as \(\Lambda\)-schemes — call this \((I_\Lambda)\) — it holds over \(\mathcal O'\). (c) Under \((I_\Lambda)\), Theorem 8.0.2 holds for \(\mathcal M_\theta\) over \(F'\), and the isomorphism \(\Phi_{F'}\) sends \(\mathbf 1_{K'x'_\kappa}\) to $$P_\kappa=c^{-1}\,\delta_{P(X)}^{-1/2}(x_\kappa)\prod_{\check\theta\in\Theta^+}\big(1-q^{-4}T_{\check\theta}\big)s_{\check\lambda(\kappa)},\qquad c=\beta(\delta_{P(X)}^{1/2})^{-1},\quad\beta=\prod_{\check\gamma\in\check\Phi_X^+}\frac{1-e^{\check\gamma}}{1-q^{-4}e^{\check\gamma}},$$ \(\Theta^+=\check\Phi_X^+\), \(s\) the Schur polynomial of lowest weight \(\check\lambda(\kappa)\) (Sakellaridis's label of \(Kx_\kappa\)). \(P_\kappa\) depends only on \(q\), not on \(F'\). (d) The statement of Theorem 8.0.2 involves no choice of \(\psi\) or of measures beyond \(\operatorname{vol}K=1\).

(a) Normalizations.

  1. Where the paper invokes the assumption. §1.1 (p. 2): "we will assume that \(k\) is unramified over \(\mathbb Q_p\); the modifications needed to remove this assumption are trivial". §3.8 (p. 23): "\(o^2\) has measure \(1\) (this is the correct choice only under our simplifying assumption...)". §5.3 (p. 32): "here we are using our simplifying assumption...; these equations would have to be modified otherwise". In each place the computation uses only (i) \(\mathcal O\) is the largest fractional ideal on which \(\psi\) is trivial (conductor \(\mathcal O\)), (ii) \(dx(\mathcal O)=1\); by (i) the dual lattice of \(\mathcal O\) is \(\mathcal O\), so (ii) is the self-dual measure. §1.7 (p. 7), and §3.8 in the same sentence as the parenthetical, fix \(\psi\) "whose conductor is equal to the ring of integers"; with that \(\psi\) the parenthetical is not needed. The assumption would matter only for the standard character \(\psi_{\mathbb Q_p}\circ\operatorname{Tr}\), whose conductor is the inverse different, equal to \(\mathcal O\) exactly when \(k/\mathbb Q_p\) is unramified (Serre, Local Fields, III §3; not opened).
  2. (3.10). For \(\psi'\) of conductor \(\mathcal O'\) and \(\omega=xw-yv\) on \(\mathcal O'^2\), the dual lattice of \(\mathcal O'^2\) is \(\mathcal O'^2\); so \(dx\,dy(\mathcal O'^2)=1\) is the self-dual measure, \(F\mathbf 1_{\mathcal O'^2}=\mathbf 1_{\mathcal O'^2}\), and the basic vector \(c_0\) is Fourier-invariant, as §3.8 requires.
  3. (4.1). \(F\mathbf 1_{\mathfrak p\times\mathcal O}(v,w)=\int_{\mathfrak p}\psi(-xw)dx\int_{\mathcal O}\psi(yv)dy=q^{-1}\mathbf 1_{\mathfrak p^{-1}}(w)\mathbf 1_{\mathcal O}(v)\): uses (i), (ii).
  4. (5.2), case \(U\). \(F(|x|^{s}dx\,dy)=\frac{1-q^{s}}{1-q^{-s-1}}\,\delta_0(v)|w|^{-s-1}dw\) is Tate's unramified formula of §5.3 with \(\chi=|\cdot|^{s+1}\); it uses (i), (ii). (5.3), case \((U,\psi)\): the \(y\)-integral gives \(\delta(v-1/x)\) for the self-dual \(dy\), then \(x=1/v\). Tate's ramified formula (Gauss sums, §5.3) uses (i), (ii) as well, and is needed only for ramified \(\tilde\chi\) in cases \(T\), \(N\), which do not occur for \(\mathcal M_\theta\) (S4-3, S4-5(3), S4-10).
  5. Lemma 3.9.1, Lemma 4.2.3, (4.5). These use \(\mu(x_0J)=1\), \(\operatorname{vol}K=1\), \(\operatorname{vol}A_0=1\), \(Q=(U_0:U_1)\operatorname{Vol}(J)\) and Casselman–Shalika [CS80, Thm 5.4] (any \(p\)-adic field, \(\psi\) of conductor \(\mathcal O\)); all are expressed through \(q\).
  6. The rest of the proof (§3.2–3.7, §6, §7, §8) is geometry over \(\bar F'\) (characteristic \(0\)) and combinatorics. So the paper's remark on p. 2 holds in the precise sense: no modification is needed once \(\psi'\) has conductor \(\mathcal O'\).

(b) Hypotheses over \(\mathcal O'\).

  1. S4-4 (\(\mathcal O'\)-model, open orbit, \(X(\mathcal O')=K'x_0\), §8 conditions), S4-5 (Axiom 2.4.2 via Iwahori factorization, open cells and a square root of \(\zeta(\varpi')\); distinct labels; types \(G\), \(U\) only), S4-3 and S4-10 (A4, statements over \(\bar F'\)) and S4-12, S4-13 (Axiom 2.4.1 via Delorme–Sécherre for any non-archimedean locally compact field of odd residue characteristic, \(|J|=1\) by Kneser for any finite extension of \(\mathbb Q_p\)) use \(p\ne2\), Hensel and Lang, never \(e(F')=1\).
  2. Base change of (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary.. \(\mathcal O'\) is finite free over \(\Lambda\), with \(\mathcal O'\otimes_\Lambda\mathbb Q_q=F'\). The maximal-rank \(B\)-orbits are defined over the base field (Sa06, Prop. 3.6.1, quoted on p. 9), so \(Y_{F'}=Y_{\mathbb Q_q}\otimes F'\). Scheme-theoretic closures commute with flat base change (Stacks, Tag 081I), and for the flat, open, generizing morphism \(X_{\mathcal O'}\to X_\Lambda\) closures of subsets pull back to closures. Hence Sakellaridis's \(\mathcal O'\)-scheme \(Y\) ("complement of the closure of the complement of \(Y_{F'}\) in its closure") is \(Y_\Lambda\otimes\mathcal O'\), likewise \(YP_\alpha\) and its complement; and quotients by \(R(P_\alpha)\) commute with flat base change. An isomorphism \(YP_\alpha/R(P_\alpha)\cong\) (one of the four models) over \(\Lambda\) therefore base-changes to \(\mathcal O'\), and so does the complement clause.

(c) The images \(P_\kappa\).

  1. The proof of Theorem 8.0.2 (p. 64) defines the isomorphism by \(\mathbf 1_{x_{\check\lambda}K}\mapsto P_{\check\lambda}=\Omega_{\chi\delta(X)^{1/2}}(x_{\check\lambda})/L_X^{1/2}(\chi)\) (8.1). By (7.4) (p. 60) and Definition 7.2.3 (p. 62), \(P_{\check\lambda}=c^{-1}\delta_{P(X)}^{-1/2}(x_{\check\lambda})\prod_{\Theta^+}(1-\sigma_{\check\theta}q^{-r_{\check\theta}}T_{\check\theta})s_{\check\lambda}\), \(c\) as in (7.3). For \(\mathcal M_\theta\), \(\Theta^+=\check\Phi_X^+\times\{(+,4)\}\) (S4-3). \(\delta_{P(X)}(x_{\check\lambda})\) is a power of \(q\) fixed by \(\check\lambda\); the torus \(A_X^*\), the point \(\delta(X)^{1/2}\), \(W_X\) and the dictionary \(\check\lambda\leftrightarrow\kappa\) (S4-5: Sakellaridis's \(x_0a\), \(a=t^{-\mu}\), is our \(t^\mu x_0\)) are the same for every \(F'\) with residue field \(\mathbb F_q\). So \(P_\kappa\) depends only on \(q\).
  2. (d). \(\mathcal H_K\) (with \(\operatorname{vol}K=1\)), \(\mathcal S\), \(r\) and \(\mathbf 1_{X(\mathcal O)}\) are canonical; \(\psi\), Fourier transforms and eigenmeasures occur only inside the proof. For \(\mathcal M_\theta\) the isomorphism with \(\varphi_0\mapsto1\) is moreover unique (Theorem 4.8, step 4). \(\square\)

Builds on

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

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