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arxiv: 2604.04124 · v1 · submitted 2026-04-05 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Relation between Anderson Generating Functions and Weil Pairing

Chuangqiang Hu , Yixuan Ou-Yang
Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:00 UTC · model grok-4.3

classification 🧮 math.NT
keywords Drinfeld modulesWeil pairingAnderson generating functionsMoore determinantfunction field arithmetictorsion modulestensor products
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The pith

The rank-r Weil pairing equals a specific coefficient in the Moore determinant of Anderson generating functions.

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

The paper connects Anderson generating functions to the Weil pairing on Drinfeld modules of arbitrary rank. It examines the Weil operator and shows that this operator corresponds to the remainder polynomial of the generating functions taken modulo a fixed polynomial f. From this correspondence the authors extract the pairing value as one coefficient inside the Moore determinant built from those functions. A sympathetic reader would care because the construction supplies an explicit and elementary formula that avoids Anderson motives entirely.

Core claim

The authors investigate the Weil operator, establish its connection with the remainder polynomial of Anderson generating functions modulo a fixed polynomial f, and finally derive an extremely simple interpretation: the value of the rank-r Weil pairing is essentially the specific coefficient in the Moore determinant of certain Anderson generating functions.

What carries the argument

The Moore determinant arising from Hamahata's tensor product of Drinfeld modules, applied to torsion bases, which isolates the Weil pairing value as a coefficient in the Anderson generating functions after reduction modulo f.

Load-bearing premise

The Weil operator connects directly to the remainder polynomial of Anderson generating functions modulo f through the Moore determinant coming from the tensor product and torsion bases.

What would settle it

Explicit computation of the Moore determinant coefficient for a rank-3 Drinfeld module and a concrete choice of f, compared against the independently known Weil pairing value, would show mismatch if the claimed relation fails.

Figures

Figures reproduced from arXiv: 2604.04124 by Chuangqiang Hu, Yixuan Ou-Yang.

Figure 1
Figure 1. Figure 1: Graphs Gγi According to Proposition 2.8, we get O (2) f (α, γj ) O(2) f (γj , γj+1) ≡ O (2) f (α, γj+1) O(2) f (γj , γj+1) Mod f(∗) [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
read the original abstract

The existence of the Weil pairing for Drinfeld modules was proved by van~der~Heiden using the Anderson $t$-motive. Papikian's note provided the explicit formula for the rank-two Weil pairing that avoids Anderson motives. Following this approach, Katen extended the formula to higher ranks. As Papikian observed, this method is more elementary than the approach using Anderson motives, but it is less conceptual. This paper is devoted to a new insight into Katen's formula motivated by the Moore determinant coming from Hamahata's tensor product of Drinfeld modules and the basis of torsion modules found by Maurischat and Perkins. We investigate the Weil operator, establish its connection with the remainder polynomial of Anderson generating functions modulo a fixed polynomial $\f$, and finally derive an extremely simple interpretation: the value of the rank-$r$ Weil pairing is essentially the specific coefficient in the Moore determinant of certain Anderson generating functions.

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 claims that the rank-r Weil pairing on Drinfeld modules admits an extremely simple algebraic interpretation: its value is essentially a specific coefficient extracted from the Moore determinant built from certain Anderson generating functions. The argument proceeds by defining a Weil operator, relating it to the remainder polynomial of an Anderson generating function modulo a fixed polynomial f, and then invoking Hamahata’s tensor-product construction together with the Maurischat–Perkins torsion bases to produce the Moore determinant; this recovers Katen’s explicit formula in a more conceptual manner than the earlier elementary approaches of Papikian and Katen.

Significance. If the identification is correct, the work supplies a direct algebraic bridge between Anderson generating functions and the Weil pairing that avoids explicit appeal to Anderson motives while still yielding a coordinate-free description. The use of the Moore determinant arising from Hamahata’s tensor product is a genuine conceptual gain and may facilitate explicit calculations and generalizations to other pairings or higher-rank settings.

major comments (2)
  1. [§3] §3, after the definition of the Weil operator: the precise statement that the operator applied to the remainder polynomial yields the desired coefficient in the Moore determinant must be stated as a theorem with an explicit formula; without it the central claim remains an outline rather than a verified identity.
  2. [§4] §4, construction of the Moore determinant: verify that the torsion bases of Maurischat–Perkins are used exactly once and that no additional normalization constants appear; any hidden scalar would contradict the claim that the pairing value is “essentially” the coefficient.
minor comments (2)
  1. The fixed polynomial f is introduced without an explicit definition or reference to its degree; add a sentence in the introduction clarifying that f is the minimal polynomial of the chosen torsion point.
  2. Notation for Anderson generating functions is introduced in §2 but reused in §4 with a different subscript; adopt a uniform subscript convention throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments help clarify the central identification between the Weil pairing and the Moore determinant coefficient. We address each major point below and will revise the manuscript to make the arguments fully explicit while preserving the conceptual approach via Hamahata’s tensor product and Maurischat–Perkins bases.

read point-by-point responses

  1. Referee: [§3] §3, after the definition of the Weil operator: the precise statement that the operator applied to the remainder polynomial yields the desired coefficient in the Moore determinant must be stated as a theorem with an explicit formula; without it the central claim remains an outline rather than a verified identity.

    Authors: We agree that the link must be stated as a precise theorem. In the revised manuscript we will insert Theorem 3.1 immediately after the definition of the Weil operator: Let W denote the Weil operator and let R_f be the remainder of the Anderson generating function modulo f. Then the rank-r Weil pairing equals the coefficient of x^{r-1} in the Moore determinant of the r-tuple (W(R_f(ω_1)), …, W(R_f(ω_r))). The proof is obtained by combining the relation between W and the remainder polynomial (already derived in §3) with the explicit matrix representation coming from Hamahata’s construction; we will write out the short verification in full. revision: yes

  2. Referee: [§4] §4, construction of the Moore determinant: verify that the torsion bases of Maurischat–Perkins are used exactly once and that no additional normalization constants appear; any hidden scalar would contradict the claim that the pairing value is “essentially” the coefficient.

    Authors: The construction in §4 applies the Maurischat–Perkins torsion bases exactly once, when assembling the matrix whose Moore determinant is taken; the bases enter canonically with no extra scalar factors. This is already implicit in the appeal to Hamahata’s tensor product, but we acknowledge the need for explicit confirmation. We will add a one-paragraph remark in §4 stating that the chosen bases are the standard Maurischat–Perkins bases (no rescaling) and that the resulting coefficient therefore matches the Weil pairing value without hidden constants, thereby removing any ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives an interpretive link between the rank-r Weil pairing and a specific coefficient in the Moore determinant of Anderson generating functions by connecting the Weil operator to the remainder polynomial modulo a fixed polynomial f, using Hamahata's tensor product and Maurischat-Perkins torsion bases. This builds directly on independent external results from van der Heiden, Papikian, Katen, Hamahata, Maurischat, and Perkins without any self-citations by the current authors, self-definitional reductions, fitted parameters renamed as predictions, or ansatzes smuggled via prior work. The central claim recovers Katen's formula conceptually rather than by construction from its own inputs, so the derivation chain remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard properties of Drinfeld modules, Anderson generating functions, and the Moore determinant from tensor products, all drawn from the cited prior literature.

axioms (1)
  • domain assumption Standard properties of Drinfeld modules, their torsion modules, and Anderson generating functions as established in prior literature.
    The paper invokes these as background without re-deriving them.

pith-pipeline@v0.9.0 · 5453 in / 1238 out tokens · 60507 ms · 2026-05-13T17:00:08.375881+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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