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Reciprocity and the Kernel of Dedekind Sums

T0 review · 1 major / 0 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Atkin-Lehner operators generate a family of reciprocity formulas for Dedekind sums attached to newforms, supplying symmetries that probe the kernel of these sums.

desk verdict The paper uses Atkin-Lehner operators on newforms to produce a family of reciprocity formulas for the attached Dedekind sums and then applies the symmetries to study the kernel. read the letter →

arxiv 2110.12269 v1 submitted 2021-10-23 math.NT

Alexis LaBelle , Emily Van Bergeyk , Matthew P. Young This is my paper
classification math.NT
keywords DedekindsumsnewformsAtkin-Lehneroperatorsreciprocityformulaskernelmodularforms
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

The paper shows how to apply Atkin-Lehner operators to newforms in order to produce multiple reciprocity relations satisfied by the associated Dedekind sums. These relations create additional symmetries among the sums. The symmetries are then used to study which linear combinations of the sums lie in the kernel. A reader cares because Dedekind sums encode arithmetic information that appears across modular forms and number theory, and clearer control over their kernels can simplify many explicit calculations.

What carries the argument

The action of Atkin-Lehner operators on newforms, which produces a family of reciprocity formulas for the attached Dedekind sums and thereby generates symmetries among them.

What would settle it

An explicit newform for which at least one Atkin-Lehner translate fails to satisfy the expected reciprocity identity with its Dedekind sum.

Watch

Extended reading notes

Core claim

We use the action of Atkin-Lehner operators to generate a family of reciprocity formulas for newform Dedekind sums. This family of reciprocity formulas provides symmetries which we use to investigate the kernel of these Dedekind sums.

Load-bearing premise

Applying Atkin-Lehner operators to newforms produces reciprocity formulas that remain valid when transferred to the associated Dedekind sums.

Editorial extensions

If this is right

  • New reciprocity identities hold for every Dedekind sum attached to a newform.
  • The generated symmetries relate the values of these sums across different Atkin-Lehner orbits.
  • The kernel of the Dedekind-sum map can be described using the linear relations coming from the new identities.
  • Explicit membership tests for the kernel become available once the symmetries are written down.

Reading between the lines

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

  • The same operator technique may produce kernel information for other arithmetic sums attached to newforms.
  • The resulting relations could be combined with existing modular-form identities to obtain closed-form evaluations.
  • Similar symmetry arguments might apply to Dedekind sums of higher level or weight.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript claims that the action of Atkin-Lehner operators on newforms can be used to produce a family of reciprocity formulas satisfied by the associated Dedekind sums; these formulas are then applied to generate symmetries that help describe the kernel of the Dedekind-sum map.

Significance. If the derivations are valid, the work would supply an explicit mechanism for producing reciprocity relations in the newform setting and a systematic way to constrain the kernel, both of which are of interest in the arithmetic theory of modular forms and Dedekind sums.

major comments (1)
  1. [Abstract] Abstract: the central step—that the action of an Atkin-Lehner operator on a newform induces a reciprocity formula for the corresponding Dedekind sum—is asserted without any explicit formula, operator definition, or sample calculation; this step is load-bearing for the entire claim and cannot be checked from the given text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central step—that the action of an Atkin-Lehner operator on a newform induces a reciprocity formula for the corresponding Dedekind sum—is asserted without any explicit formula, operator definition, or sample calculation; this step is load-bearing for the entire claim and cannot be checked from the given text.

    Authors: The abstract is a high-level summary and does not contain the explicit constructions. The action of the Atkin-Lehner operators, the resulting reciprocity formulas for the newform Dedekind sums, the relevant operator definitions, and sample calculations are all supplied in the body of the manuscript. The full text therefore permits verification of the central claim. To improve accessibility, we will revise the abstract to include a brief reference to the main formula and the relevant theorem. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The derivation relies on the standard action of Atkin-Lehner operators on newforms to produce reciprocity formulas for associated Dedekind sums, followed by symmetry analysis of the kernel. This chain uses established modular-forms techniques with no self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce claims to their own inputs. The abstract and described approach are self-contained against external number-theory benchmarks.

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

Abstract supplies no information on free parameters, background axioms, or new entities introduced by the work.

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Cite this review

Pith. "Pith review of Reciprocity and the Kernel of Dedekind Sums." pith.science (2026). https://pith.science/paper/2110.12269

@misc{pith2026211012269,
  author       = {Pith},
  title        = {Pith review of: Reciprocity and the Kernel of Dedekind Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2110.12269}},
  note         = {Machine review of arXiv:2110.12269}
}
read the original abstract

We use the action of Atkin-Lehner operators to generate a family of reciprocity formulas for newform Dedekind sums. This family of reciprocity formulas provides symmetries which we use to investigate the kernel of these Dedekind sums.

Figures

Figures reproduced from arXiv: 2110.12269 by the authors.

Figure 1
Figure 1. Elements of K3,5 with 1 ≤ c ≤ 10q1q2 The following relationship follows from Theorem 1.1: (5) Sχ 0 1 ,χ0 2 (γ) = 0 ⇐⇒ Sχ1,χ2 (γ 0 ) = 0. Technically, the simplified version in Theorem 1.1 gives this relation if γ, γ0 ∈ Γ1(N), but the more general reciprocity formula in Theorem 3.1 implies (5) for γ, γ0 ∈ Γ0(N). Using (5), we derive the following theorem (see Section 4 for the proof). Theorem 1.3. Let q1q2 = N = QR w… view at source ↗
Figure 2
Figure 2. Elements of K7,11 for 1 ≤ c ≤ 10q1q2 Note that in Figures 1 and 2, the only kernel elements present in rows c = N kq2 are those equivalent to (±1, N kq2). We refer the reader to [NRY21] for additional examples of this phenomenon. The following proposition gives some evidence for this sparseness by demonstrating that no element of the form (a, c) = (±1 + N kr, N kq2) with nonzero k is in the kernel. Proposition 4.2. … view at source ↗

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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    Tom M. Apostol. Modular functions and D irichlet series in number theory , volume 41 of Graduate Texts in Mathematics . Springer-Verlag, New York, second edition, 1990

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    Ramirez, and Matthew P

    Evuilynn Nguyen, Juan J. Ramirez, and Matthew P. Young. The kernel of newform D edekind sums. J. Number Theory , 223:53--63, 2021

    work page 2021

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    S ageMath, the S age M athematics S oftware S ystem ( V ersion 9.0) , 2021

    S age D evelopers. S ageMath, the S age M athematics S oftware S ystem ( V ersion 9.0) , 2021. https://www.sagemath.org

    work page 2021

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    Stucker, A

    T. Stucker, A. Vennos, and M. P. Young. Dedekind sums arising from newform E isenstein series. Int. J. Number Theory , 16(10):2129--2139, 2020

    work page 2020

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    Some Results on Classical E isenstein Series and Modular Forms Over Function Fields

    James Weisinger. Some Results on Classical E isenstein Series and Modular Forms Over Function Fields . ProQuest LLC, Ann Arbor, MI, 1977. Thesis (Ph.D.)--Harvard University

    work page 1977

  7. [7]

    Matthew P. Young. Explicit calculations with E isenstein series. J. Number Theory , 199:1--48, 2019

    work page 2019

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Reviewed May 24, 2026 · model on record in the stance chip above.