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arxiv: 2606.24532 · v1 · pith:GNMTWPERnew · submitted 2026-06-23 · 🧮 math.NT · math.AG· math.KT

Elements in K₄ and regulator maps of Fermat curves

Fran\c{c}ois Brunault , David T.-B. G. Lilienfeldt , Yusuke Nemoto This is my paper

Pith reviewed 2026-06-25 22:51 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.KT
keywords Fermat curvesK_4 groupsBeilinson regulatorpolylogarithmic complexesZagier trilogarithmhypergeometric functionsBeilinson conjectures
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The pith

Fermat curves x^N + y^N = 1 admit explicit non-trivial elements in K_4^{(3)} detected by non-zero images under Beilinson's regulator that equal special values of Zagier's trilogarithm.

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

The paper constructs explicit elements in the group K_4^{(3)} of the Fermat curves x^N + y^N =1 for all N≥3. The construction, which is uniform in N, uses polylogarithmic complexes and a map of de Jeu to K-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to 3/2 ζ(3) N² as N→+∞.

Core claim

We construct explicit elements in the group K_4^{(3)} of the Fermat curves x^N + y^N =1 for all N≥3. The construction, which is uniform in N, uses polylogarithmic complexes and a map of de Jeu to K-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to 3/2 ζ(3) N² as N→+∞. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for K_2 groups of Ferm

What carries the argument

Polylogarithmic complexes combined with de Jeu's map to K-theory, which produce the uniform explicit classes in K_4^{(3)}, detected as non-zero by their images under Beilinson's regulator map whose integrals reduce to Zagier's trilogarithm values.

If this is right

  • The regulator integrals are asymptotic to (3/2) ζ(3) N² as N → +∞.
  • The regulators admit explicit formulas in terms of hypergeometric functions that generalize Otsubo's earlier results for the K_2 groups of the same curves.
  • Numerical checks confirm portions of Beilinson's conjectures relating these regulators to special values of L-functions at s=3 for the cases N=3,4,6.

Where Pith is reading between the lines

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

  • The uniform-in-N construction suggests the same polylogarithmic complexes could be applied to other algebraic curves equipped with similar de Jeu maps to produce regulator formulas.
  • The hypergeometric expressions for the regulators may allow closed-form evaluations for additional small N beyond the cases already checked numerically.
  • The observed quadratic growth in N supplies a concrete growth rate that could be compared against independent predictions for the rank of K_4^{(3)} on these curves.

Load-bearing premise

The polylogarithmic complexes together with de Jeu's map produce well-defined classes inside the correct K_4^{(3)} group for every N, and the subsequent regulator integrals are computed without hidden normalization choices or curve-specific adjustments that would affect the claimed non-vanishing and asymptotic statements.

What would settle it

An explicit computation for N=5 in which the constructed class maps to zero under the regulator integral, or in which the integral fails to match the stated hypergeometric or trilogarithm formula, would falsify the non-triviality and explicit-formula claims.

read the original abstract

We construct explicit elements in the group $K_4^{(3)}$ of the Fermat curves $x^N+y^N=1$ for all $N\geq 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\frac{3}{2}\zeta(3)N^2$ as $N\to +\infty$. Moreover, we derive formulas for the regulators of our elements in terms of hypergeometric functions, generalizing results of Otsubo for $K_2$ groups of Fermat curves. Finally, we numerically verify some cases of Beilinson's conjectures on special values of $L$-functions at $s=3$ for $N\in \{ 3,4,6 \}$.

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

2 major / 2 minor

Summary. The paper constructs explicit elements in the group K_4^{(3)} of the Fermat curves X_N : x^N + y^N =1 for all N≥3. The construction is uniform in N and proceeds by mapping polylogarithmic complexes through de Jeu's map into K-theory. Non-triviality is established by showing that the images under Beilinson's regulator are non-zero, with explicit formulas for the regulator integrals in terms of Zagier's trilogarithm. As a corollary an asymptotic (3/2)ζ(3)N² is derived as N→∞. Additional results include hypergeometric expressions for the regulators (generalizing Otsubo) and numerical verification of Beilinson's conjectures at s=3 for N=3,4,6.

Significance. If the central identification holds, the work supplies the first uniform, explicit family of non-trivial classes in K_4^{(3)}(X_N) together with closed-form regulator expressions. The asymptotic and the numerical checks for small N would then give concrete support for Beilinson's conjectures on special values of L-functions of Fermat curves. The generalization of Otsubo's K_2 formulas is a further positive feature.

major comments (2)
  1. [construction of the elements via de Jeu's map] The application of de Jeu's map (construction section, around the definition of the classes c_N): the manuscript must explicitly confirm that the image lies in the Adams eigenspace K_4^{(3)} rather than a different filtration or requiring an N-dependent scalar multiple. The regulator formulas, the claimed non-vanishing, and the asymptotic (3/2)ζ(3)N² all presuppose this exact landing; without a verification (e.g., via Adams operations or a cited theorem guaranteeing the eigenspace for this polylog input) the central claims rest on an unverified intermediate step.
  2. [regulator integrals] Regulator computation (section deriving the explicit trilog formulas): the passage from the de Jeu image to the integral involving Zagier's trilogarithm L_3(·) should include a check that no hidden normalization constants depending on N appear when identifying the class inside K_4^{(3)}; any such factor would rescale the asymptotic and invalidate the numerical checks for N=3,4,6.
minor comments (2)
  1. Notation for the polylogarithmic complexes should be made uniform across sections; currently the same symbol is reused for slightly different complexes in the construction and in the regulator paragraphs.
  2. The numerical tables for N=3,4,6 would benefit from an additional column showing the raw regulator integral before any normalization, to allow independent verification of the non-vanishing claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the significance of our results. We address each major comment below and will revise the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: [construction of the elements via de Jeu's map] The application of de Jeu's map (construction section, around the definition of the classes c_N): the manuscript must explicitly confirm that the image lies in the Adams eigenspace K_4^{(3)} rather than a different filtration or requiring an N-dependent scalar multiple. The regulator formulas, the claimed non-vanishing, and the asymptotic (3/2)ζ(3)N² all presuppose this exact landing; without a verification (e.g., via Adams operations or a cited theorem guaranteeing the eigenspace for this polylog input) the central claims rest on an unverified intermediate step.

    Authors: We agree that an explicit confirmation strengthens the paper. The de Jeu map is functorial and preserves the Adams filtration when applied to the polylogarithmic complexes in the appropriate degree; this is guaranteed by the construction in de Jeu's original work on the map from polylog complexes to K-theory. In the revised manuscript we will add a short paragraph (in the construction section) citing the relevant result from de Jeu that ensures the image of our classes c_N lands precisely in K_4^{(3)} with no N-dependent scalar factor. This will also make clear that the subsequent regulator formulas and the asymptotic remain unchanged. revision: yes

  2. Referee: [regulator integrals] Regulator computation (section deriving the explicit trilog formulas): the passage from the de Jeu image to the integral involving Zagier's trilogarithm L_3(·) should include a check that no hidden normalization constants depending on N appear when identifying the class inside K_4^{(3)}; any such factor would rescale the asymptotic and invalidate the numerical checks for N=3,4,6.

    Authors: The regulator is computed directly from the image class under the normalized Beilinson regulator map, and the explicit trilogarithm expressions are obtained without introducing any N-dependent normalization constants; the uniformity of the construction ensures the constants are absolute. Nevertheless, to make this transparent we will insert a brief verification paragraph tracing the normalizations through de Jeu's map and the regulator, confirming that no rescaling occurs. This will also reinforce the validity of the asymptotic and the numerical checks. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external maps (de Jeu, Beilinson regulator) and independent special values (Zagier trilog) without self-referential reduction.

full rationale

The paper constructs classes in K_4^{(3)} via polylogarithmic complexes and de Jeu's map (external reference), then applies Beilinson's regulator to obtain explicit integrals involving Zagier's trilogarithm. These are shown non-zero for all N≥3, yielding the (3/2)ζ(3)N² asymptotic and hypergeometric formulas as corollaries. No step equates a claimed prediction or class to a fitted parameter or self-citation by construction; the landing in the correct Adams eigenspace and the regulator formulas are derived from the cited maps rather than redefined in terms of the output. Numerical checks for N=3,4,6 are verifications against independent L-value conjectures, not inputs. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract the central claim rests on standard background results in algebraic K-theory and regulator theory; no free parameters, no new postulated entities, and no ad-hoc axioms are visible.

axioms (1)
  • standard math Standard properties of algebraic K-theory groups, polylogarithmic complexes, de Jeu's map, and Beilinson's regulator map hold for Fermat curves.
    The construction and non-vanishing proof invoke these established tools without re-deriving them.

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discussion (0)

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Reference graph

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