§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:02:10Z

Audit 5.5 · Audit of A4-7: Theorem 8.0.2 over ramified \(F'\) and the images \(P_\kappa\)

proved audit: correct ★ key step

Second audit: correct. A grep of the v3 text finds exactly the three invocations of the unramified assumption (Sec. 1.1, 3.8, 5.3), all about (psi, dx); with psi of conductor o (fixed in Sec. 1.7 and 3.8) the self-dual measure gives o measure 1 for every k, and I re-derived (4.1), Tate's unramified factor (1-q^{s-1})/(1-q^{-s}) and the case-U formula (1-q^s)/(1-q^{-s-1}) delta_0(v)|w|^{-s-1}dw. The 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. (O' finite flat over Lambda, open map, Stacks 081I) and the e-independence P = R_lambda/R_0 from (7.4), (8.1) are correct, as is the remark that the paper's label sign is ambiguous but immaterial. The flagged gap H_Sa08 is genuine (Audit 4.7 step 6 wrongly calls Sec. 3.2-3.7 geometry and combinatorics). I opened Sa08 v2: its Sec. 1.7 and 4.1 assume only a locally compact non-archimedean field of characteristic 0, so H_Sa08 holds, as Audit 7.3 also found. Correct.

Against the printed text, the unramified assumption enters only through \((\psi,dx)\), which the paper itself fixes with conductor \(\mathcal O\). 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. and the \(e\)-independence of \(P_\kappa\) are correct. One citation chain is unchecked: the representation-theoretic results quoted from [Sa08] in §3.2 and §3.7.

Audit 5.5.

Outcome: holds, under one unverified citation \(H_{\mathrm{Sa08}}\). (a) The three places where the paper invokes "\(k\) unramified over \(\mathbb Q_p\)" are §1.1 p. 2, the parenthetical of §3.8 p. 23 and the end of §5.3 p. 32. Each concerns only the pair \((\psi,dx)\). With \(\psi\) of conductor \(\mathcal O\), fixed in §1.7 p. 7 and again in §3.8, and \(dx(\mathcal O)=1\), formulas (3.10), (4.1), Tate's formulas of §5.3 and Theorem 5.2.1 hold verbatim over every finite \(F'/\mathbb Q_p\). Lemma 3.9.1, Prop. 4.2.1, Thm 4.2.2, Lemma 4.2.3 and (4.5) use only \(\mu(x_0J)=1\), \(\operatorname{vol}K=\operatorname{vol}A_0=1\), \((U_0:U_1)\), [CS80, Thm 5.4] and [Ca80, Thm 3.1], all valid over every \(p\)-adic field. (b) Flat base change of Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. from \(\Lambda\) to \(\mathcal O'\) is correct. (c) \(P_\kappa:=\Phi_{F'_e}(\mathbf 1_{K'x'_\kappa})\) is independent of \(e\). (d) Correct. Gap. Step 6 calls §3.2–3.7 "geometry over \(\bar F'\) and combinatorics". In fact §3.2 and §3.7 rest on \(p\)-adic results of [Sa08]: Theorem 3.2.2, and Theorem 3.7.1 = [Sa08, Thm 5.2.1] (functional equations up to non-zero rational factors, which underlie (4.3)). Their ramification hypotheses were not checked; [Sa08] was opened neither by A4 nor by me. \(H_{\mathrm{Sa08}}\): [Sa08, Thm 5.2.1] and the construction of the morphisms \(S^Y_{\tilde\chi}\) of [Sa08, §3] hold over every finite extension of \(\mathbb Q_p\), \(p\) odd. These are qualitative statements, and §1.1 asserts the removal of the assumption is "trivial", but this is not verified here.

  1. §3.8 (pp. 22–23). The text fixes "an additive character \(\psi\) on \(k\) whose conductor is the ring of integers \(o\)" and \(o^2\) of measure \(1\). For \(\omega((x,y),(v,w))=xw-yv\), the dual lattice of \(\mathcal O^2\) is \(\mathcal O^2\) exactly when \(\psi\) has conductor \(\mathcal O\), for any ramification. So \(F\mathbf 1_{\mathcal O^2}=\mathbf 1_{\mathcal O^2}\), and the basic vector \(c_0\) is Fourier-invariant. The parenthetical "correct only under our simplifying assumption" is needed only if \(\psi\) is \(\psi_{\mathbb Q_p}\circ\operatorname{Tr}\) (conductor the inverse different). The paper does not make that choice.
  2. §5.3 (p. 32). "With respect to our fixed character \(\psi\) and the measure \(dx\) normalized as \(dx(o)=1\)": for \(\chi=|\cdot|^s\), \(\mathcal F(\chi\,d^\times x)=\frac{1-q^{-1}\chi^{-1}(\varpi)}{1-\chi(\varpi)}\chi^{-1}|x|\,d^\times x\), and the ramified case is \(q^{-m}\tau(\chi)\chi^{-1}|x|\,d^\times x\). These are Tate's local functional equations for unramified \(\psi\), valid over every non-archimedean local field. I re-derived A4-7's case-\(U\) formula \(\mathcal F(|x|^s dx\,dy)=\frac{1-q^s}{1-q^{-s-1}}\delta_0(v)|w|^{-s-1}dw\) from them.
  3. Normalizations of §3.3–§4.3. Standard eigenmeasures are fixed by \(\mu(x_0J)=1\) (Definition 3.4.1), \(\operatorname{Vol}(A_0)=1\), \(\operatorname{Vol}(K)=1\). The compatibility of eigenforms across orbits (Prop. 3.3.6) uses the \(o\)-structure of the orbits of maximal rank, i.e. Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(\mathcal O'\), which (b) supplies from \((I_\Lambda)\).
  4. (b) Base change. \(\mathcal O'\) is finite free over \(\Lambda\), so \(X_{\mathcal O'}\to X_\Lambda\) is finite flat of finite presentation, hence open. For an open map, preimages of closures are closures of preimages. Scheme-theoretic closure of the quasi-compact \(Y_{\mathbb Q_q}\hookrightarrow X_\Lambda\) commutes with flat base change (Stacks 081I). \(Y_{F'}=Y_{\mathbb Q_q}\otimes F'\), because the orbits \(Bn_wx_0\) are defined over \(\mathbb Z[1/2]\). Hence Sakellaridis's \(\mathcal O'\)-scheme \(Y\) is \(Y_\Lambda\otimes\mathcal O'\), and likewise for \(YP_\alpha\), its complement, and the fppf quotient by \(R(P_\alpha)\). Checklist item 3 is respected: the argument is about \(\mathcal O\)-schemes, not types.
  5. (c) \(e\)-independence. By (7.4), \(\Omega_{\delta(X)^{1/2}\tilde\chi}(x_{\check\lambda})/\beta=\delta_{P(X)}^{-1/2}(x_{\check\lambda})\prod_{\Theta^+}(1-\sigma q^{-r}T_{\check\theta})s_{\check\lambda}=:R_{\check\lambda}\). Since \(P_0=1\) (p. 64), \(P_{\check\lambda}=R_{\check\lambda}/R_0\), whatever the value of \(c\) in (7.3). \(R_{\check\lambda}\) is built from \(q\), \(\Theta^+=\{e_1,e_6,e_1+e_6\}\times\{(+,4)\}\), \(\check\Phi_X\) and \(\delta_{P(X)}(x_{\check\lambda})\), a power of \(q\) fixed by \(\check\lambda\). None depends on \(e\), and the label \(\check\lambda(\kappa)\) of \(K'x'_\kappa\) is the same for all \(e\).
  6. Label sign (checklist item 8). §2.1 uses \(\check\lambda\mapsto e^{\check\lambda}(\varpi)\), while Remark (2) after Theorem 4.2.2 writes \(x_{\check\lambda}=e^{-\check\lambda}(\varpi)\). The paper is thus ambiguous about the sign of labels. This is immaterial for the transfer, because \(P_\kappa\) is defined as \(\Phi_{F'_e}(\mathbf 1_{K'x'_\kappa})\) with one dictionary for all \(e\), and S4-8's use of \(P_{3\omega_1^\vee}\) is convention-robust (it allows \(\kappa\in\{(3,0),(0,3)\}\)).
  7. The gap. Theorem 3.7.1 (p. 21) is "a main result of [Sa08]" (equality up to non-zero rational functions). Theorem 4.2.2's matrices \(b_w\), defined by (4.3), exist because of it, and §5 computes their exact values. [Sa08] is a \(p\)-adic (characteristic-\(0\)) paper, so using it over \(F'_e\) is legitimate (checklist item 1). Its ramification hypotheses are the only part of the chain not checked.

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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
  • Audit 7.3 · Check: Sakellaridis's 2008 paper is stated for every local field of characteristic \(0\)

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