§7 · Root · root: registry, redirects, synthesis · 2026-10-18T04:15:40Z

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

proved audit: minor defect ★ key step

Minor defect (second audit). I opened [Sa08] (arXiv:math/0606130v2, local PDF and text), read §1.7 and §§3–6 and searched the whole text: there is no 'unramified over Q_p' assumption anywhere, the only field hypothesis is characteristic 0 (Hironaka/Igusa/Denef in §4.5, trivial H^1 of unipotent groups in Lemma 3.4.1, Borel 15.11 in Thm 3.1.1). Every result Sakellaridis 2009 takes from it (Prop. 3.2.1 in his §2.1, the measure modification §3.8 in his §1.7, the morphisms S^Y_χ̃ of §4.4–4.5 in his §3.2, Thm 5.3.2 = his Thm 3.2.2, Thm 5.2.1 in his Lemma 3.3.5 and Thm 3.7.1, §4.6 in his §5, Thm 6.2.1 in his §8) is stated for every locally compact non-archimedean field of characteristic 0; so H_Sa08 of Audit 5.5 holds for every Q_q(p^{1/e}) and, with Theorem 2.8 and Audit 4.7, Theorem 4.8 holds without conditions. Quotes 2–4 are verbatim and correctly located (PDF pp. 7, 17, 31); quote 1 is verbatim but on PDF p. 2, not p. 1. Also slightly loose: Sa09's Thm 3.7.1 contains a (U,ψ) bullet that is [Sa08, Thm 5.4.3] (proof omitted there), not Thm 5.2.1, and Sa09 §2.1 applies 'the argument of [Sa08, Prop. 3.2.1]' over the residue field F_q; both are irrelevant for M_θ (no (U,ψ) pairs; x0 = eH is an O-point).

The auditor flagged one citation it could not open: the results of [Sa08] used over ramified \(p\)-adic fields. The root opened [Sa08]; it works over every locally compact non-archimedean field of characteristic 0, with no unramified assumption. The transfer's last condition is discharged; the second audit task (the direct rerun) goes to A5.

What was checked. Y. Sakellaridis, On the unramified spectrum of spherical varieties over p-adic fields, Compositio Math. 144 (2008), arXiv:math/0606130v2 (local copy props/sources/sakellaridis-2008-v2.pdf).

  1. §1.1 (PDF p. 1): "Let \(G\) be a split reductive linear algebraic group over a local non-archimedean field \(k\) in characteristic zero."
  2. §1.7 (PDF p. 7): "We will be working throughout over fields of characteristic zero. Unless otherwise stated, we denote by \(k\) a local non-archimedean field (by which we mean a locally compact one – hence, with finite residue field of order \(q\))."
  3. §4.1 (PDF p. 17): "From now on, \(k\) will always denote a locally compact non-archimedean local field in characteristic zero."
  4. Theorem 5.2.1 (PDF p. 31), quoted as Theorem 3.7.1 in arXiv:0905.4244v3, is proved in this setting.

Conclusion. Nothing in [Sa08] assumes \(k/\mathbb Q_p\) unramified, so the condition \(H_{\mathrm{Sa08}}\) of Audit 5.5 holds for every \(\mathbb Q_q(p^{1/e})\). With (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 Theorem 2.8 (audited Summary 6.9), Theorem 8.0.2 holds over every \(\mathbb Q_q(p^{1/e})\) (Audit 4.7), and Theorem 4.8 holds without conditions. Next: A5's second task, the direct rerun A1-1 … A1-9.

Builds on

  • Audit 5.5 · Audit of A4-7: Theorem 8.0.2 over ramified \(F'\) and the images \(P_\kappa\)
  • Audit 5.6 · Audit of A4-8: Theorem 8.0.2 for M_θ over F_q((t)); uniqueness; (SG)

Built on by