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arxiv: 2604.09954 · v1 · submitted 2026-04-10 · 🧮 math.KT

Recognition: unknown

The K-theory of finite Tambara fields: away from p

Noah Wisdom

Pith reviewed 2026-05-10 16:46 UTC · model grok-4.3

classification 🧮 math.KT
keywords algebraic K-theoryTambara fieldstorsion groupsp-inversionfinite fieldscyclic Galois groupsconstant fieldsodd primes
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The pith

The algebraic K-theory groups of any constant C_{p^n}-Tambara field valued in a finite field of characteristic p are torsion groups and are fully determined after inverting p.

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

This paper proves that the algebraic K-theory groups of constant C_{p^n}-Tambara fields with values in finite fields of characteristic p consist entirely of torsion elements. It gives a complete description of these groups after inverting p, where the away-from-p part follows a simple pattern already seen in the prime-2 case. The result uses a new computational approach that succeeds for odd primes, where earlier methods stopped working. A computer check shows that the p-power torsion is nontrivial in general. This provides explicit information about K-theory for these structures beyond the earlier C_2 computation.

Core claim

In previous work the author and Chan computed the algebraic K-theory of the constant C_2-Tambara field valued in the field with two elements using a method that fails at odd primes. The present work uses a completely new idea to make progress on the odd-primary cases. It shows that the K-theory groups of any constant C_{p^n}-Tambara field with value a characteristic-p finite field are torsion, and determines these groups completely after inverting p. The away-from-p torsion satisfies a simple pattern predicted by previous work, while a computer-aided computation shows that the p-power torsion is nontrivial in general.

What carries the argument

Constant C_{p^n}-Tambara field with finite characteristic-p value, analyzed by a new computational idea that isolates the away-from-p part of its K-theory.

If this is right

  • The K-theory groups are torsion in all degrees.
  • After inverting p the groups are given by a simple explicit pattern.
  • The p-power torsion is present and nontrivial for general choices of p and n.
  • The result extends the earlier C_2 computation to all odd primes using a method that avoids the previous obstruction.
  • The groups become completely known once p is inverted.

Where Pith is reading between the lines

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

  • The same torsion phenomenon may appear in K-theory computations for non-constant Tambara fields or for other Galois actions.
  • The explicit pattern away from p could be used to simplify spectral-sequence arguments that involve these K-theory groups.
  • The computer method for detecting p-torsion might be applied to larger values of p or n to map out the full torsion structure.
  • The separation between away-from-p and p-power parts suggests that the K-theory splits in a controlled way when p is inverted.

Load-bearing premise

The new computational idea correctly isolates and determines the away-from-p K-theory for odd primes, and the computer-aided check accurately detects the presence of p-power torsion without calculation errors.

What would settle it

An explicit calculation of the K-theory groups for p equal to 3 and n equal to 1 that either shows a different pattern after inverting p or finds no p-power torsion at all would disprove the main claims.

read the original abstract

In previous work, the author and Chan computed the algebraic $K$-theory of the constant $C_2$-Tambara field with value the field with two elements, using a method which fails at odd primes. Herein we make progress towards the corresponding odd primary computations using a completely new idea. Particularly, we show that the $K$-theory groups of any constant $C_{p^n}$-Tambara field with value a characteristic $p$ finite field are torsion, and we completely determine these groups after inverting $p$. The away-from-$p$-torsion satisfies a simple pattern predicted by previous work, and a computer-aided computation shows that the $p$-power torsion is nontrivial in general.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces a new method to compute the algebraic K-theory of constant C_{p^n}-Tambara fields with values in finite fields of characteristic p, extending prior work limited to the C_2 case at p=2. It proves that these K-theory groups are torsion, completely determines the groups after inverting p, shows that the away-from-p torsion follows a simple pattern from previous results, and uses a computer-aided computation to establish that the p-power torsion is nontrivial in general.

Significance. If the results hold, this constitutes meaningful progress on K-theory computations for Tambara fields at odd primes, where earlier techniques failed. The explicit determination after p-inversion and the identification of torsion provide concrete data points that may guide conjectures in algebraic K-theory and equivariant homotopy. The computer-assisted evidence for nontrivial p-torsion supplies empirical support for the general claim, though its scope requires clarification.

major comments (1)
  1. The section describing the computer-aided computation for p-power torsion does not specify the range of p and n examined, the exact algorithm or implementation details, or how 'nontrivial in general' is formalized and verified. This is load-bearing for the central claim of nontrivial p-power torsion, as limitations to small cases or undetected errors would leave the general statement unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying a point that requires clarification in the presentation of our computational results. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The section describing the computer-aided computation for p-power torsion does not specify the range of p and n examined, the exact algorithm or implementation details, or how 'nontrivial in general' is formalized and verified. This is load-bearing for the central claim of nontrivial p-power torsion, as limitations to small cases or undetected errors would leave the general statement unsupported.

    Authors: We agree that the current description of the computer-aided computation is insufficiently detailed to fully substantiate the claim. In the revised manuscript we will expand the relevant section to specify the precise range of primes p and exponents n that were examined, provide a complete description of the algorithm together with its implementation (including the software environment and verification procedures used to confirm the output), and clarify the sense in which the p-power torsion is shown to be nontrivial in general—namely, that it is observed to be nontrivial for every pair (p,n) in the computed range. These additions will make the supporting evidence explicit while leaving the mathematical statements of the paper unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new idea and computations are independent of prior predictions

full rationale

The paper explicitly introduces a completely new idea for odd-primary computations that replaces the method which failed at odd primes. It derives that the K-theory groups are torsion and determines the groups after inverting p, while separately noting that the away-from-p torsion matches a pattern from prior work. The p-power torsion is established via computer-aided computation. No load-bearing step reduces by definition, fitted input, or self-citation chain to the paper's own inputs; the central claims rest on the new approach and explicit verification rather than re-deriving or renaming prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on a new computational idea introduced in the paper, whose details are not available in the abstract, along with standard background in algebraic K-theory and Tambara functors. No free parameters or invented entities are explicitly mentioned.

axioms (2)
  • standard math Algebraic K-theory satisfies standard properties such as being a functor and having certain exact sequences.
    The computations rely on established properties of K-theory.
  • domain assumption Tambara functors have the expected structure for constant ones over fields.
    Assumed from the definition in the context of the paper.

pith-pipeline@v0.9.0 · 5406 in / 1440 out tokens · 72369 ms · 2026-05-10T16:46:12.325217+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

  1. [1]

    Nemo/Hecke: Computer algebra and number theory packages for the Julia programming language

    1, 6 [FHHJ17] Claus Fieker, William Hart, Tommy Hofmann, and Fredrik Johansson. Nemo/Hecke: Computer algebra and number theory packages for the Julia programming language. In Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation , ISSAC ’17, page 157–164, New York, NY, USA,

    2017

  2. [2]

    The structure of Mackey functors.Trans

    1, 5 [TW95] Jacques Th´ evenaz and Peter Webb. The structure of Mackey functors.Trans. Amer. Math. Soc., 347(6):1865– 1961,

    1961

  3. [3]

    4 [Wis25a] Noah Wisdom

    An introduction to algebraic K-theory. 4 [Wis25a] Noah Wisdom. Clarification and coinduction of Tambara functors. https://arxiv.org/abs/2505.08066,

    work page arXiv