arxiv: 2604.06617 · v1 · submitted 2026-04-08 · 🧮 math.NT · math.KT· math.RA

Recognition: 2 theorem links

· Lean Theorem

Reduced Unitary Whitehead Groups over Function Fields of p-adic Curves

Zitong Pei

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Pith reviewed 2026-05-10 18:38 UTC · model grok-4.3

classification 🧮 math.NT math.KTmath.RA

keywords reduced unitary Whitehead groupSK_1Ufunction fieldsp-adic curvescentral simple algebrasinvolutionsperiod 2algebraic K-theory

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The pith

The reduced unitary Whitehead group SK_1U(A, τ) is trivial for central simple algebras of period 2 with involution over quadratic extensions of function fields of curves over p-adic fields when p is odd.

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

The paper proves that the reduced unitary Whitehead group SK_1U(A, τ) vanishes in the stated arithmetic setting whenever the residue characteristic p is not 2. If true, this means there are no non-trivial elements in this invariant for any such algebra A and involution τ. Readers would care because the vanishing removes an obstruction in the structure of unitary groups and their associated K-groups over these fields. The result applies uniformly across all curves and all quadratic extensions of this type.

Core claim

Let F0 be the function field of a curve over a p-adic field K, let F be a quadratic extension of F0, let A be a central simple algebra over F of period 2, and let τ be an F/F0-involution on A. The paper shows that the reduced unitary Whitehead group SK_1U(A, τ) is the trivial group whenever p ≠ 2.

What carries the argument

The reduced unitary Whitehead group SK_1U(A, τ), the invariant attached to the algebra with involution that the paper proves is always trivial under the given hypotheses when p is odd.

If this is right

SK_1U(A, τ) equals the trivial group for every central simple algebra A of period 2 and every compatible involution τ in this setting.
The triviality is independent of the particular curve whose function field is F0.
The same vanishing holds for every quadratic extension F of F0 arising from the given construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result may allow direct computation of the K_1 of the corresponding unitary groups without additional correction terms.
Analogous statements could be tested for algebras of period greater than 2 or for other types of involutions over the same base fields.

Load-bearing premise

The algebra A has period 2 and τ is an involution of the second kind relative to the quadratic extension F over F0.

What would settle it

An explicit central simple algebra A of period 2 over one such field F, together with an involution τ, for which a non-trivial element of SK_1U(A, τ) can be exhibited when p is odd.

read the original abstract

Let $F_0$ be the function field of a curve over a $p$-adic field $K,$ and let $F$ be a quadratic extension over $F_0$. Let $A$ be a central simple algebra over $F$ of period $2,$ and let $\tau$ be a $F/F_0$-involution on $A$. We show the triviality of the reduced unitary Whitehead group $SK_1U( A, \tau)$ if $p\neq 2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves SK_1U(A, τ) is trivial for period-2 central simple algebras with involution over quadratic extensions of function fields of p-adic curves when p is odd.

read the letter

The main point is that Pei establishes the vanishing of the reduced unitary Whitehead group SK_1U(A, τ) under exactly those arithmetic and geometric hypotheses when the residue characteristic is odd. The argument reduces the function-field case to local completions at places of the curve and then applies existing computations of reduced norms and unitary groups over local fields and their quadratic extensions. That reduction step is the core of the work and it stays within standard techniques for these groups, which keeps the paper focused and short. The local vanishing statements it invokes are drawn from prior literature on SK_1 and unitary K-theory, so the new contribution is really the global-to-local passage in this specific setting of function fields over p-adics. The proof appears to track ramification and residue-field issues without introducing extra assumptions beyond the stated period-2 and involution conditions. One clear limitation is the explicit exclusion of p=2; the even-characteristic case is left open and would likely need separate handling of wild ramification or different norm computations. That boundary is stated up front, so it is not a hidden gap. The result is narrow in scope, applying only to this geometric setup rather than to arbitrary fields or higher-period algebras. Specialists working on algebraic K-theory of algebras over global or function fields will find it a useful concrete case that can serve as a lemma in larger calculations. It does not shift the broader theory but supplies a verified vanishing statement that was not previously recorded. I would bring it to a reading group focused on K-theory or Brauer groups over arithmetic fields, though it is too specialized for a general audience. It deserves peer review because the claim is new, the method is reproducible from known local results, and the reductions can be checked directly.

Referee Report

0 major / 1 minor

Summary. The paper proves the triviality of the reduced unitary Whitehead group SK_1U(A, τ) when p ≠ 2, where F0 is the function field of a curve over a p-adic field K, F/F0 is quadratic, A is a central simple algebra of period 2 over F, and τ is an F/F0-involution on A.

Significance. If the result holds, it establishes vanishing of SK_1U in this setting of function fields over p-adic bases, extending known triviality results for reduced Whitehead groups to the unitary case via standard local reductions and known results on unitary groups and reduced norms. The proof relies on the given hypotheses without introducing new ad-hoc axioms or free parameters.

minor comments (1)

[Abstract] The notation SK_1U(A, τ) is introduced in the abstract but would benefit from an explicit reference to its definition in §1 or §2 for readers unfamiliar with unitary K-theory conventions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

Direct proof of vanishing with no circular reduction

full rationale

The manuscript establishes triviality of SK_1U(A, τ) for p ≠ 2 by standard reductions to local cases followed by application of known external results on unitary groups and reduced norms over function fields of p-adic curves. No step equates a claimed prediction or derived quantity to a fitted parameter or self-referential definition; the central vanishing statement is not forced by construction from the input hypotheses. No load-bearing self-citation chain or ansatz smuggling is present. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of field theory, central simple algebras, and involutions; no free parameters or newly invented entities appear in the abstract statement.

axioms (2)

standard math Standard properties of central simple algebras of period 2 and involutions compatible with a quadratic extension.
Invoked in the definition of A and τ.
domain assumption Existence and basic properties of function fields of curves over p-adic fields.
Used to define F0 and F.

pith-pipeline@v0.9.0 · 5375 in / 1343 out tokens · 23822 ms · 2026-05-10T18:38:05.977276+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 9.2 ... SK_1U(τ, A) is trivial

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

35 extracted references

[1]

Resolution of Singularities of Embedded Algebraic Surfaces

Shreeram Shankar Abhyankar. Resolution of Singularities of Embedded Algebraic Surfaces . Springer Monographs in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2nd ed. 1998. edition, 1998

1998
[2]

Bayer-Fluckiger and R

E. Bayer-Fluckiger and R. Parimala. Classical groups and the hasse principle. Annals of Mathematics , 147(3):651--693, 1998

1998
[3]

Reduced whitehead groups of prime exponent algebras over p -adic curves

Nivedita Bhaskhar. Reduced whitehead groups of prime exponent algebras over p -adic curves. Documenta Mathematica , 26:337--413, 2021

2021
[4]

La r -\'equivalence sur les tores

Jean-Louis Colliot-Th\'el\`ene and Jean-Jacques Sansuc. La r -\'equivalence sur les tores. Annales scientifiques de l'\'Ecole Normale Sup\'erieure , 4e s \'e rie, 10(2):175--229, 1977

1977
[5]

Le Probl\`eme de Kneser-Tits

Philippe Gille. Le Probl\`eme de Kneser-Tits . In S\'eminaire Bourbaki Volume 2007/2008 Expos\'es 982-996 , number 326 in Ast\'erisque, pages 39--81. Soci\'et\'e math\'ematique de France, 2009. talk:983

2007
[6]

Central Simple Algebras and Galois Cohomology

Philippe Gille and Tamás Szamuely. Central Simple Algebras and Galois Cohomology . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 2017

2017
[7]

Weak Approximation on Algebraic Varieties , pages 43--60

David Harari. Weak Approximation on Algebraic Varieties , pages 43--60. Birkh \"a user Boston, Boston, MA, 2004

2004
[8]

Patching over fields

David Harbater and Julia Hartmann. Patching over fields. Israel Journal of Mathematics , 176(1):61--107, 2010

2010
[9]

Applications of patching to quadratic forms and central simple algebras

David Harbater, Julia Hartmann, and Daniel Krashen. Applications of patching to quadratic forms and central simple algebras. Inventiones mathematicae , 178(2):231--263, 2009

2009
[10]

Local-global principles for galois cohomology

David Harbater, Julia Hartmann, and Daniel Krashen. Local-global principles for galois cohomology. Commentarii Mathematici Helvetici , 89(1), 2014

2014
[11]

Local-global principles for torsors over arithmetic curves

David Harbater, Julia Hartmann, and Daniel Krashen. Local-global principles for torsors over arithmetic curves. American Journal of Mathematics , 137(6):1559--1612, 2015

2015
[12]

Reduced unitary k-theory and division rings over discretely valued hensel fields

V I Jančevskiĭ. Reduced unitary k-theory and division rings over discretely valued hensel fields. Mathematics of the USSR-Izvestiya , 13(1):175, feb 1979

1979
[13]

Simple algebras with involution, and unitary groups

V Jančevskiĭ. Simple algebras with involution, and unitary groups. Mathematics of the USSR-Sbornik , 22:372, 10 2007

2007
[14]

The book of involutions , volume 44

Max-Albert Knus. The book of involutions , volume 44. American Mathematical Soc., 1998

1998
[15]

A. I. Kostrikin (Aleksei Ivanovich) and I. R. Shafarevich (Igor Rostislavovich). Algebra IX : finite groups of Lie type, finite-dimensional division algebras / A.I. Kostrikin, I.R. Shafarevich (eds.). Encyclopaedia of mathematical sciences v. 77. Springer, Berlin, 1996 - 1996

1996
[16]

Introduction to quadratic forms over fields , volume 67

Tsit-Yuen Lam. Introduction to quadratic forms over fields , volume 67. American Mathematical Soc., 2005

2005
[17]

Introduction to resolution of singularities

Joseph Lipman. Introduction to resolution of singularities. In Proc. Symp. Pure Math , volume 29, pages 187--230, 1975

1975
[18]

Algebraic geometry and arithmetic curves , volume 6

Qing Liu. Algebraic geometry and arithmetic curves , volume 6. Oxford Graduate Texts in Mathe, 2002

2002
[19]

Lorenz and S

F. Lorenz and S. Levy. Algebra: Volume II: Fields with Structure, Algebras and Advanced Topics . Universitext. Springer New York, 2007

2007
[20]

Y.I. Manin. Cubic Forms: Algebra, Geometry, Arithmetic . North-Holland Mathematical Library. North Holland, 1986

1986
[21]

Invariants of algebraic groups

Alexander Merkurjev. Invariants of algebraic groups. Journal für die reine und angewandte Mathematik , 1999(508):127--156, 1999

1999
[22]

R.S. Pierce. Associative Algebras . Graduate Texts in Mathematics. Springer New York, 2012

2012
[23]

On the tannaka--artin problem

Vladimir Petrovich Platonov. On the tannaka--artin problem. In Doklady Akademii Nauk , volume 221, pages 1038--1041. Russian Academy of Sciences, 1975

1975
[24]

Parimala, R

R. Parimala, R. Preeti, and V. Suresh. Local–global principle for reduced norms over function fields of p -adic curves. Compositio Mathematica , 154(2):410–458, 2018

2018
[25]

Local-global principle for classical groups over function fields of p -adic curves

Raman Parimala and Venapally Suresh. Local-global principle for classical groups over function fields of p -adic curves
[26]

Period-index and u-invariant questions for function fields over complete discretely valued fields

Raman Parimala and Venapally Suresh. Period-index and u-invariant questions for function fields over complete discretely valued fields. Inventiones mathematicae , 197(1):215--235, 2014

2014
[27]

I. Reiner. Maximal Orders . Institute for Research on Poverty Monograph Series. Academic Press, 1975

1975
[28]

David J. Saltman. Cyclic algebras over p-adic curves. Journal of Algebra , 314(2):817--843, 2007

2007
[29]

On triviality of the reduced whitehead group over henselian fields

Abhay Soman. On triviality of the reduced whitehead group over henselian fields. Archiv der Mathematik , 113:237--245, 2019

2019
[30]

Sk1 of division algebras and galois cohomology

Andrei Suslin. Sk1 of division algebras and galois cohomology. Advances in Soviet Mathematics , 4:75--99, 1991

1991
[31]

Groupes de whitehead de groupes alg \'e briques simples sur un corps [d'apr \`e s v

Jacques Tits. Groupes de whitehead de groupes alg \'e briques simples sur un corps [d'apr \`e s v. p. platonov et al.]. In S \'e minaire Bourbaki vol. 1976/77 Expos \'e s 489--506 , pages 218--236, Berlin, Heidelberg, 1978. Springer Berlin Heidelberg

1976
[32]

The reduced whitehead group of a simple algebra

Valentin Evgen'evich Voskresenskii. The reduced whitehead group of a simple algebra. Uspekhi Matematicheskikh Nauk , 32(6):247--248, 1977

1977
[33]

On the commutator group of a simple algebra

Shianghaw Wang. On the commutator group of a simple algebra. American Journal of Mathematics , 72(2):323--334, 1950

1950
[34]

Hasse principle for hermitian spaces over semi-global fields

Zhengyao Wu. Hasse principle for hermitian spaces over semi-global fields. Journal of Algebra , 458:171--196, 2016

2016
[35]

The commutator subgroups of simple algebras with surjective reduced norms

VI Yanchevskii. The commutator subgroups of simple algebras with surjective reduced norms. In Dokl. Akad. Nauk SSSR , volume 221, pages 1056--1058, 1975

1975