Pith. sign in

REVIEW 4 major objections 3 minor 44 references

A local converse theorem for quasi-split even special orthogonal groups

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Local twisted gamma factors up to GL_l determine supercuspidal representations of quasi-split nonsplit SO_{2l} up to the outer automorphism.

desk verdict A genuinely new direct Bessel-function proof strategy for quasi-split non-split SO_{2l}, but load-bearing results are cited to an unpublished manuscript; referee if the missing pieces can be supplied. read the letter →

arxiv 2501.16339 v1 pith:PRU3YMYG submitted 2025-01-13 math.RT math.NT

Alexander Hazeltine This is my paper · ORCID
classification math.RTmath.NT MSC 11F7022E5011F85
keywords localconversetheoremquasi-splitevenspecialorthogonalgroupsgammafactorsHowevectorspartialBesselfunctionssupercuspidalrepresentationsgenericouterautomorphism
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a local converse theorem for the quasi-split non-split even special orthogonal group SO_{2l} over a non-Archimedean local field of characteristic p ≠ 2. It shows that two irreducible generic supercuspidal representations with the same central character are isomorphic, or become isomorphic after applying the outer automorphism, whenever their twisted gamma factors agree for all twists by generic representations of GL_n with n ≤ l. The proof is direct, using Howe vectors and partial Bessel functions to extract information cell by cell from the Bruhat decomposition, rather than importing the result through Langlands functoriality. The paper also states a generic-case version over characteristic-zero fields and an automorphic weak-rigidity consequence, but those proofs are omitted.

What carries the argument

The engine of the proof is the partial Bessel function B_m(g; f), defined by averaging a Whittaker function W^f over the Howe-vector subgroup U_m; it transforms under the generic character on upper-triangular unipotents and under an auxiliary character ψ_m on a compact subgroup H_m. The Bessel support of these functions is the set of Weyl elements supporting nonzero values, and it is partitioned into Bruhat cells B_n(SO_{2l}) for n = 1, ..., l−1. Equality of gamma factors for GL_k twists is shown to erase all cells with n ≤ k; the GL_{l−1} twist erases the surviving cell on c-fixed elements, and the GL_l twist shows that on the remaining elements B_m + B_m^c = 0. Because B_m^c is the partial Bessel function of π^c, uniqueness of Whittaker models forces π ≅ π′ or π ≅ π′^c.

What would settle it

Take l = 2 (the quasi-split non-split group SO_4) and compute B_m(t w̃_1, f_{w̃_1}) and B_m^c(t w̃_1, f_{w̃_1}) for a non-c-fixed torus element t; if the equality of GL_2 gamma factors does not force the sum to vanish, Theorem 6.11 is false. Equivalently, one could search for two supercuspidal ψ-generic representations of quasi-split non-split SO_4(F) with equal gamma factors against all GL_1 and GL_2 twists that are neither isomorphic nor outer-conjugate.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: over a non-Archimedean local field of characteristic p ≠ 2, if π and π′ are irreducible ψ-generic supercuspidal representations of quasi-split non-split SO_{2l}(F) with the same central character, and if γ(s, π × τ, ψ) = γ(s, π′ × τ, ψ) for every irreducible generic representation τ of GL_n(F) with n ≤ l, then π is isomorphic to π′ or to π′^c, where c is the outer automorphism. The proof compares partial Bessel functions attached to matrix coefficients of π and π′, shows that equality of gamma factors erases the Bruhat cells below rank l, and uses the GL_l twist to force the surviving cell terms to cancel after adding the outer-conjugate term. Uniqueness of Whittaker models then yields the dichotomy. The generic case and the automorphic weak-rigidity theorem are stated as consequences, with proofs referred to a future writeup.

Load-bearing premise

The proof leans on three structural claims about which Weyl cells can support the special functions and how torus elements land in them, all taken from the author's own manuscript in preparation; if any of those claims is wrong, the reduction to the final Bruhat cell and the closing GL_l computation collapse.

Editorial extensions

If this is right

  • For quasi-split non-split SO_{2l}, equality of gamma factors against GL_n twists for n ≤ l identifies a generic supercuspidal representation up to the two-to-one ambiguity π ↔ π^c.
  • The gamma factors of π and π^c coincide for all GL_n twists with n ≤ l, so this family of invariants cannot separate an outer-conjugate pair; any finer uniqueness statement needs an additional invariant.
  • Over characteristic-zero fields, the statement extends from supercuspidals to all irreducible generic representations, giving a local converse theorem at the full generic level.
  • The generic local theorem yields a weak rigidity statement for cuspidal automorphic representations: agreement of local components at almost all places forces agreement or outer-conjugacy at every place.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • [Editorial inference] The three unpublished Bessel-support statements are the natural target for independent verification; a self-contained proof of Proposition 4.5, Lemma 6.3(1), and Proposition 6.9 would remove the main external dependency of the argument.
  • [Editorial inference] The same partial-Bessel-function framework may extend to the positive-characteristic generic case, since Theorem 1.2 already works for p ≠ 2 and the missing step is the analogue of the supercuspidal-to-generic reduction.
  • [Editorial inference] Because the twist family leaves exactly the two-to-one ambiguity, adding a single invariant that changes sign under the outer automorphism on every pair π ≠ π^c would complete the classification; the paper notes that twisted exterior-square gamma factors are insufficient for SO_6, so the needed invariant must be something else.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper proves a local converse theorem for irreducible generic supercuspidal representations of quasi-split non-split even special orthogonal groups over non-Archimedean local fields: if two such representations have the same central character and the same family of local gamma factors twisted by all irreducible generic representations of GL_n for n ≤ l, then they are isomorphic up to the outer automorphism. The proof uses Howe vectors and partial Bessel functions, following the strategy of Zhang, Jo, and Hazeltine–Liu for other classical groups, with a new treatment of the nonsplit torus and the outer automorphism. The paper also states, without proof, an analogous generic-case theorem and a weak rigidity theorem for automorphic representations.

Significance. If the central argument is correct, the paper gives a direct, Arthur-independent proof of the supercuspidal local converse theorem for quasi-split non-split SO_{2l}, a case where the previously available proofs used theta correspondence or Langlands functoriality. The use of Howe vectors and partial Bessel functions is conceptually valuable and yields, as a byproduct, intrinsic proofs of the equality of gamma factors for π and π^c. The paper is honest about its dependencies, but several of those dependencies are unpublished statements from the author's own manuscript [15], and one corollary central to the induction is stated without proof. The generic-case theorem is independently known from Haan–Kim–Kwon and from Arthur's work, so the main novelty rests on Theorem 1.2, whose proof is not self-contained at load-bearing points.

major comments (4)
  1. [§4.4, Corollary 4.11] The proof of Corollary 4.11 is omitted ('The proof is an adaptation of [44, Corollary 4.7] which we omit'). This corollary supplies the initial decomposition Equation (4.2), from which every later vanishing statement in Theorem 6.1 and Theorems 6.7 and 6.11 proceeds. Since the Bruhat-cell structure and Bessel support in the quasi-split non-split case are exactly what the paper develops, a proof or a complete reference must be provided; an appeal to an unpublished manuscript is not sufficient for this load-bearing step.
  2. [§6.1, Lemma 6.3(1)] Lemma 6.3(1) is used in Propositions 6.5 and 6.7 and again in Theorem 6.11 to identify the partial Bessel functions of π and π′ on the Levi cell t_{l−1}(a). The text says only 'This follows from the proof of [15, Proposition 4.8]', where [15] is listed as 'In prepartion'. If this vanishing/equality statement fails, the comparison of the non-intertwined zeta integrals in those propositions collapses. The proof needs to be included in this paper or the statement must be verifiable in a publicly accessible reference.
  3. [§6.3, Proposition 6.9] Proposition 6.9 is quoted from [15, Proposition 7.2] and is the only mechanism in Theorem 6.11 that restricts the torus integration to T_l = {t : a ≠ 1}, produces the matrix A used to realize the zeta integral over GL_l, and sets up the two-to-one map tw ↔ ctcw. If the characterization is wrong for some w or some t with a ≠ 1, the identity B_m(g,f_{\tilde w_{l−1}}) + B^c_m(g,f_{\tilde w_{l−1}}) = 0 is unsupported. This is a load-bearing external dependency and should be proved in full in this paper or replaced by a checkable argument.
  4. [Theorems 1.3 and 1.5] Theorem 1.3 (generic case) and Theorem 1.5 (weak rigidity) are stated without proof. The paper notes that Theorem 1.3 is independently due to Haan–Kim–Kwon and also follows from Arthur, but the manuscript presents the theorem as one of its main results and gives only 'similar arguments as in [24, §3.2]'. Since the generic case is not the paper's novel contribution, the authors should either include the proof or clearly state that these are recorded for completeness and that the paper's direct proof covers the supercuspidal case only.
minor comments (3)
  1. [Introduction, §2] The abstract and Theorem 1.2 state characteristic p ≠ 2, while Section 2 begins 'Let n,l ∈ N and F be a non-Archimedean local field of characteristic 0.' Please clarify the precise characteristic assumptions consistently across the paper, especially for the supercuspidal theorem versus the generic theorem.
  2. [References] Reference [15] is listed as 'In prepartion'; this should be 'In preparation'.
  3. [Theorem 6.11 proof] The proof of Theorem 6.11 refers to 'Equation (5.4)' in two places when evaluating the image of sections under the intertwining operator; the intended displayed equation appears to be (5.2). Please correct the cross-references.

Circularity Check

0 steps flagged · score 2.0 of 10

No definitional or fitted-input circularity: the theorem is not assumed and no fitted parameter is renamed as a prediction. The proof does rely on the author's own unpublished [15] for several technical lemmas, which is a verifiability gap rather than a circular reduction.

full rationale

The central derivation is not circular. The input is the assumed equality of twisted gamma factors; the proof computes zeta integrals via Howe vectors and partial Bessel functions, deriving Bessel-function identities (Theorems 6.1, 6.7, 6.11), and finally invokes uniqueness of Whittaker models. No fitted parameter is renamed as a prediction, no object is defined in terms of the theorem's conclusion, and Theorem 1.2 is not used as a hypothesis. The skeptical concerns are about reliance on the author's unpublished manuscript [15] for Proposition 4.5, Lemma 6.3(1), and Proposition 6.9, together with omitted proofs (Corollary 4.11, stated as an adaptation of [44, Corollary 4.7]; Theorems 1.3 and 1.5, stated without proof). These are load-bearing dependencies and a genuine completeness risk: if a cited result in [15] were wrong, the Bruhat-cell reduction in Section 6.1 or the GL_l computation in Section 6.3 could fail. However, those cited statements are structural facts about Weyl groups, Bruhat cells, and embeddings, or finite-field analogues; they do not assert the local converse theorem and are not equivalent to the gamma-factor equality input. The generic case also has independent support from Haan-Kim-Kwon [14] and Arthur's classification. Thus the derivation chain does not reduce to its own inputs; the self-citation burden is real but not definitional, so score 2 is appropriate.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof is a standard local-converse proof in the Langlands program; it introduces no free parameters or invented entities. It depends on several external theorems and, critically, on the author's own unpublished results.

assumptions (5)
  • standard math Uniqueness of Whittaker models for generic representations of SO_{2l}(F) and GL_n(F).
    Invoked in Section 2 and at the end of Section 6.3 to conclude pi is isomorphic to pi' or pi'^c from equality of partial Bessel functions.
  • standard math The local gamma factors gamma(s,pi x tau,psi) exist as proportionality factors satisfying the local functional equation.
    Used throughout Sections 3 and 6; based on multiplicity-one theorems [1] and [34] as stated in Section 3.
  • standard math Cogdell-Shahidi-Tsai theorem: partial Bessel functions can be replaced by functions supported on lower Bruhat cells when they vanish on tori (Theorem 4.10).
    This is the main reduction tool in Section 4.4 and Corollary 4.11; the paper says the proof generalizes from [44].
  • ad hoc to paper The Bessel support partition and preimage criterion of Propositions 4.5 and 6.9, and Lemma 6.3(1), from the author's unpublished manuscript [15].
    These are load-bearing black boxes cited as [15], which is listed as 'In prepartion' with no public version; the proof of the present paper relies on them.
  • domain assumption The generic case (Theorem 1.3) follows from the supercuspidal case by the same argument as in [24, Section 3.2] using multiplicativity of normalized twisted gamma factors [29].
    The paper explicitly says 'We omit the proof' for Theorem 1.3; no derivation is provided in the text, so this is an unproved asserted step for the generic-case claim.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A local converse theorem for quasi-split even special orthogonal groups." pith.science (2026). https://pith.science/paper/PRU3YMYG

@misc{pith2026250116339,
  author       = {Pith},
  title        = {Pith review of: A local converse theorem for quasi-split even special orthogonal groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PRU3YMYG}},
  note         = {Machine review of arXiv:2501.16339}
}
abstract

We give a direct proof of the local converse theorem for quasi-split non-split $\mathrm{SO}_{2l}$ over a local non-Archimedean field of characteristic $p\neq 2$, applying the theory of Howe vectors and partial Bessel functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 43 canonical work pages

  1. [14]

    Jaeho Haan, Yeansu Kim, and Sanghoon Kwon, A local converse theorem for quasi-split O2n and SO 2n : the generic case , 2023, arXiv 2301.12693. 1, 4

    work page arXiv 2023

  2. [15]

    5, 12, 13, 18, 26, 32 LOCAL CONVERSE THEOREM FOR QUASI-SPLIT SO 2l 33

    Alexander Hazeltine, A converse theorem for quasi-split even special orthogonal groups over finite fields, 2021, In prepartion. 5, 12, 13, 18, 26, 32 LOCAL CONVERSE THEOREM FOR QUASI-SPLIT SO 2l 33

    work page 2021

  3. [1]

    Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis, an d G´ erard Schiffmann,Multiplicity one theorems, Ann. of Math. (2) 172 (2010), no. 2, 1407–1434. 8

    work page 2010

  4. [2]

    61, American Mathematical S ociety, Providence, RI, 2013, Orthog- onal and symplectic groups

    James Arthur, The endoscopic classification of representations , American Mathematical Society Colloquium Publications, vol. 61, American Mathematical S ociety, Providence, RI, 2013, Orthog- onal and symplectic groups. 1, 2, 4, 22, 32

    work page 2013

  5. [3]

    4, 9, 10, 15

    Ehud Moshe Baruch, Local factors attached to representations of p-adic groups and strong multi- plicity one , ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.)–Yale Uni versity. 4, 9, 10, 15

    work page 1995

  6. [4]

    Kyle Petersen, William Slofstra, and Bridget Tenner, Parabolic double cosets in Coxeter groups , Electron

    Sara Billey, Matjaˇ z Konvalinka, T. Kyle Petersen, William Slofstra, and Bridget Tenner, Parabolic double cosets in Coxeter groups , Electron. J. Combin. 25 (2018), no. 1, Paper No. 1.23, 66. 14

    work page 2018

  7. [5]

    Hautes ´Etudes Sci

    Armand Borel and Jacques Tits, Groupes r´ eductifs, Inst. Hautes ´Etudes Sci. Publ. Math. (1965), no. 27, 55–150. 14

    work page 1965

  8. [6]

    Jingsong Chai, Bessel functions and local converse conjecture of Jacquet , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 6, 1703–1728. 1, 19

    work page 2019

Show all 44 references
  1. [7]

    Number Theory 120 (2006), no

    Jiang-Ping Jeff Chen, The n × (n − 2) local converse theorem for GL(n) over a p-adic field , J. Number Theory 120 (2006), no. 2, 193–205. 19

    2006

  2. [8]

    James Cogdell, Ilya Piatetski-Shapiro, and Freydoon Sha hidi, Partial Bessel functions for quasi- split groups , Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 9...

    2005

  3. [9]

    James Cogdell, Freydoon Shahidi, and Tung-Lin Tsai, Local Langlands correspondence for GLn and the exterior and symmetric square ε-factors, Duke Math. J. 166 (2017), no. 11, 2053–2132. 3, 14

    2017

  4. [10]

    Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, cen- tral critical L values, and restriction problems in the representation the ory of classical groups , Ast´ erisque (2012), no. 346, 1–109, Sur les conjectures de G ross et Prasad. I. 8

    2012

  5. [11]

    1254, Springer-Verlag , Berlin, 1987

    Stephen Gelbart, Ilya Piatetski-Shapiro, and Stephen R allis, Explicit constructions of automorphic L-functions, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag , Berlin, 1987. 8

    1987

  6. [12]

    David Ginzburg, Stephen Rallis, and David Soudry, L-functions for symplectic groups , Bull. Soc. Math. France 126 (1998), no. 2, 181–244. 1

    1998

  7. [13]

    Jaeho Haan, A local converse theorem for M p2n : the generic case , 2022, arXiv 2212.05234. 1

    2022

  8. [16]

    Alexander Hazeltine and Baiying Liu, A converse theorem for split SO2l over finite fields , Acta Math. Sin. (Engl. Ser.) 40 (2024), no. 3, 731–771. 5, 12, 32

    2024

  9. [17]

    Theory, to appear

    , On the local converse theorem for split SO2l, 2025, Represent. Theory, to appear. 1, 2, 3, 4, 5, 9, 12, 13, 14, 15, 16, 28, 31

    2025

  10. [18]

    Math., vol

    Herv´ e Jacquet,Germs for Kloosterman integrals, a review , Advances in the theory of automorphic forms and their L-functions, Contemp. Math., vol. 664, Amer. Math. Soc., Pro vidence, RI, 2016, pp. 173–185. 13

    2016

  11. [19]

    5, 1399 – 1422

    Herv´ e Jacquet and Baiying Liu, On the local converse theorem for p-adic GLn , American Journal of Mathematics 140 (2018), no. 5, 1399 – 1422. 1

    2018

  12. [20]

    2, 367–464

    Herv´ e Jacquet, Ilya Piatetski-Shapiro, and Joseph Sha lika, Rankin-Selberg convolutions, American Journal of Mathematics 105 (1983), no. 2, 367–464. 1

    1983

  13. [21]

    Herv´ e Jacquet and Joseph Shalika, A lemma on highly ramified ǫ-factors, Math. Ann. 271 (1985), no. 3, 319–332. 19

    1985

  14. [22]

    Chris Jantzen and Baiying Liu, The generic dual of p-adic split SO2n and local Langlands param- eters, Israel J. Math. 204 (2014), no. 1, 199–260. 22, 32

    2014

  15. [23]

    1, World Scientific, 2006, pp

    Dihua Jiang, On local γ-factors, Arithmetic Geometry and Number Theory, Series on Number Theory and Applications, vol. 1, World Scientific, 2006, pp. 1–28. 1

    2006

  16. [24]

    Dihua Jiang and David Soudry, The local converse theorem for SO2m+1 and applications , Ann. Math. 157 (2003), 743–806. 1, 2, 4

    2003

  17. [25]

    Dihua Jiang and David Soudry, Appendix: On the local descent from GL(n) to classical groups , Amer. J. Math. 134 (2012), no. 3, 767–772. 2

    2012

  18. [26]

    1, 2, 3, 9, 12

    Yeongseong Jo, The local converse theorem for odd special orthogonal group s and symplectic groups in positive characteristic , 2022, arXiv 2205.09004. 1, 2, 3, 9, 12

    2022 arXiv

  19. [27]

    Number Theory 130 (2010), no

    Eyal Kaplan, An invariant theory approach for the unramified computation of Rankin-Selberg in- tegrals for quasi-split SO2n × GLn, J. Number Theory 130 (2010), no. 8, 1801–1817. 1

    2010

  20. [28]

    , The unramified computation of Rankin-Selberg integrals for SO2l × GLn, Israel J. Math. 191 (2012), no. 1, 137–184. 1

    2012

  21. [29]

    1, 2, 6, 7, 8, 9

    Eyal Kaplan, Multiplicativity of the gamma factors of Rankin-Selberg in tegrals for SO2l × GLn, Manuscripta Math 142 (2013), 307–346. 1, 2, 6, 7, 8, 9

    2013

  22. [30]

    Number Theory 146 (2015), 390–447

    Eyal Kaplan, Complementary results on the Rankin-Selberg gamma factors of classical groups , J. Number Theory 146 (2015), 390–447. 1, 6

    2015

  23. [31]

    Number Theory 234 (2022), 285–332

    Baiying Liu and Qing Zhang, Gamma factors and converse theorems for classical groups ov er finite fields, J. Number Theory 234 (2022), 285–332. 4, 12, 13

    2022

  24. [32]

    , On a converse theorem for G2 over finite fields , Math. Ann. 383 (2022), no. 3-4, 1217–

    2022

  25. [33]

    Nadir Matringe, Local converse theorems and Langlands parameters , 2024, arXiv:2409.20240. 4

    2024

  26. [34]

    347, 2012, Sur les conjectures de Gross et Prasad

    Colette Mœglin and Jean-Loup Waldspurger, La conjecture locale de Gross-Prasad pour les groupes sp´ eciaux orthogonaux: le cas g´ en´ eral, no. 347, 2012, Sur les conjectures de Gross et Prasad. II, pp. 167–216. 8

    2012

  27. [35]

    Chung Pang Mok, Endoscopic classification of representations of quasi-spl it unitary groups , Mem. Amer. Math. Soc. 235 (2015), no. 1108, vi+248. 1

    2015

  28. [36]

    Kazuki Morimoto, On the irreducibility of global descents for even unitary gr oups and its applica- tions, Trans. Amer. Math. Soc. 370 (2018), no. 9, 6245–6295. 1

    2018

  29. [37]

    3, 653–674

    Chufeng Nien, A proof of the finite field analogue of Jacquet’s conjecture , American Journal of Mathematics 136 (2014), no. 3, 653–674. 4

    2014

  30. [38]

    Edva-Aida Roditty, On gamma factors and Bessel functions for representations o f general linear groups over finite field , Master’s thesis, Tel-Aviv University, 2010. 4

    2010

  31. [39]

    Freydoon Shahidi, Fourier transforms of intertwining operators and Plancher el measures for GL(n), Amer. J. Math. 106 (1984), no. 1, 67–111. 1

    1984

  32. [40]

    , A proof of Langlands’ conjecture on Plancherel measures; co mplementary series for p-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. 1

    1990

  33. [41]

    David Soudry, Rankin-Selberg convolutions for SO2l+1 × GLn: local theory , Mem. Amer. Math. Soc. 105 (1993), no. 500, vi+100. 1

    1993

  34. [42]

    Number Theory 146 (2015), 557–626

    David Soudry and Yaacov Tanay, On local descent for unitary groups , J. Number Theory 146 (2015), 557–626. 1

    2015

  35. [43]

    Pan Yan and Qing Zhang, On a refined local converse theorem for SO(4), Proc. Amer. Math. Soc. 152 (2024), no. 11, 4959–4976. 4

    2024

  36. [44]

    Qing Zhang, A local converse theorem for Sp2r, Math. Ann. 372 (2018), no. 1-2, 451–488. 1, 2, 3, 9, 12, 13, 14, 15, 16, 17, 18 34 ALEXANDER HAZELTINE 45. , A local converse theorem for U2r+1, Trans. Amer. Math. Soc. 371 (2019), no. 8, 5631–5654. 1, 2, 3, 9, 12, 13, 14, 15, 16 ...

    2018

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.