Dedekind zeta functions of non-Galois torsion fields of elliptic curves

Robert Pollack; Tom Weston

arxiv: 2604.08776 · v1 · submitted 2026-04-09 · 🧮 math.NT

Dedekind zeta functions of non-Galois torsion fields of elliptic curves

Robert Pollack , Tom Weston This is my paper

Pith reviewed 2026-05-10 16:54 UTC · model grok-4.3

classification 🧮 math.NT

keywords Dedekind zeta functionelliptic curvetorsion pointprime factorizationnumber fieldnon-Galois extensionalgorithm

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The pith

An algorithm determines factorization types of primes in non-Galois number fields generated by odd-order torsion points on elliptic curves and computes the corresponding Dedekind zeta coefficients.

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

The paper develops an algorithm that classifies how rational primes factor in the number fields obtained by adjoining one point of odd order from an elliptic curve. These extensions are non-Galois, so standard tools from Galois theory do not apply in the usual way. Once the factorization types are known, the coefficients of the Dedekind zeta function of the field can be calculated directly from the Euler product. A reader might care because the zeta function encodes the prime-splitting data of the field, turning an abstract arithmetic object into something that can be computed explicitly.

Core claim

We give an algorithm to determine factorization types of primes in the number fields generated by a single point of odd order on an elliptic curve. We apply this to compute coefficients of the Dedekind zeta function of the field.

What carries the argument

The algorithm that determines the factorization types of primes in the non-Galois extensions generated by odd-order torsion points on elliptic curves.

Load-bearing premise

The torsion point has odd order and the extension it generates is non-Galois.

What would settle it

An explicit odd-order torsion point on some elliptic curve for which the algorithm returns an incorrect factorization type for at least one prime.

Figures

Figures reproduced from arXiv: 2604.08776 by Robert Pollack, Tom Weston.

**Figure 1.** Figure 1: The Newton polygon of t 1 18(α, x) with vα = 2 and p = 3 Assume now that oα¯ | k. We have c1 = 2(α k − 1) k 2 α k−2m − k 2α 2k−2m p µ . Since v(α k − 1) = vα + v(k), we see that v(c0) = 2vα + 2v(k) v(c1) = µ + min{vα + v(k) + v(k) + v(k − 1), 2v(k)} = µ + 2v(k). If p ∤ k, then v(c1) = µ lies on the line of slope µ, so that as above we see that the Newton polygon has vertices (0, 2vα + 2v(k)), (1, µ)… view at source ↗

read the original abstract

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

The paper supplies a concrete algorithm for prime factorization types in non-Galois odd-order torsion fields and uses it for zeta coefficients, but offers no general proof of correctness or termination.

read the letter

The main thing to know is that Pollack and Weston give a step-by-step procedure to find how primes factor in the field generated by a single odd-order torsion point on an elliptic curve when the extension is non-Galois, then apply it to compute some Dedekind zeta coefficients. This targets a narrow case that earlier work on Galois or even-order torsion fields left open. They lay out the algorithm clearly enough that it can be implemented and run on examples, which is the practical payoff here. That part is useful for anyone who needs explicit splitting data in these fields to get zeta coefficients without building the full Galois closure each time. The description is direct and the examples show it producing output, so the computational side is grounded in actual runs rather than just theory. The soft spot is exactly what the stress test flags: the paper describes the steps but does not prove that the output always matches the true factorization type in the field, for instance by comparing to the action of Frobenius in the closure or to the splitting of the minimal polynomial. There is also no argument that the procedure terminates for every odd-order point and every prime. This matters because the zeta application needs reliable types for arbitrarily many primes, and without that justification the results rest on the tested cases alone. The work is aimed at number theorists who do explicit computations with elliptic curve torsion fields and their zeta functions. A reader who wants a tool for this specific class of fields will find something they can use right away. It deserves a serious referee because the algorithmic contribution is new for this setting and the application is concrete, even though the missing correctness argument will need attention in review. I would send it to peer review rather than desk reject.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to give an algorithm that determines the factorization types of primes in the non-Galois number fields K = Q(P) generated by a single point P of odd order on an elliptic curve E, and applies the algorithm to compute coefficients of the Dedekind zeta function of K.

Significance. If the algorithm is shown to be correct and terminating, the work would supply a practical computational procedure for obtaining arithmetic data (prime factorization types and zeta coefficients) in a specific class of non-Galois torsion fields attached to elliptic curves. This could facilitate further study of class numbers, regulators, or Galois representations in these extensions, but the absence of verification currently limits its utility.

major comments (3)

[Algorithm description] The description of the algorithm (presumably in the main algorithmic section following the introduction) supplies a sequence of steps but contains no proof that the output factorization type coincides with the actual splitting of p in K. No comparison is made to the minimal polynomial of a primitive element of K or to the Frobenius conjugacy class in the Galois closure of K.
[Algorithm description] No argument is given that the procedure terminates for every input point of odd order yielding a non-Galois extension. Termination is required for the subsequent computation of zeta coefficients, which needs reliable data for arbitrarily many primes.
[Application to zeta coefficients] The application to Dedekind zeta coefficients therefore rests on unverified output; without correctness, the reported coefficients cannot be taken as established data for the fields in question.

minor comments (2)

The abstract would be clearer if it briefly indicated the main ingredients of the algorithm (e.g., use of division polynomials, reduction modulo p, or Galois-theoretic criteria).
Notation for the field K and the torsion point P should be introduced consistently in the introduction and reused verbatim in later sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the importance of establishing the correctness and termination of the algorithm. We agree that these aspects require explicit treatment in the manuscript and will revise accordingly to provide the necessary proofs.

read point-by-point responses

Referee: The description of the algorithm (presumably in the main algorithmic section following the introduction) supplies a sequence of steps but contains no proof that the output factorization type coincides with the actual splitting of p in K. No comparison is made to the minimal polynomial of a primitive element of K or to the Frobenius conjugacy class in the Galois closure of K.

Authors: The referee correctly observes that the manuscript lacks a formal proof of the algorithm's correctness. The steps are motivated by the Galois action on the torsion points and the resulting field extensions, but we did not include a verification against the minimal polynomial or Frobenius class. In the revised manuscript, we will add a proof section demonstrating that the factorization type produced matches the actual splitting by establishing a correspondence with the conjugacy class of the Frobenius in the Galois closure of K. revision: yes
Referee: No argument is given that the procedure terminates for every input point of odd order yielding a non-Galois extension. Termination is required for the subsequent computation of zeta coefficients, which needs reliable data for arbitrarily many primes.

Authors: We acknowledge the absence of a termination argument. The algorithm consists of a finite number of steps involving computations over finite fields and checks on the elliptic curve group, which are bounded. We will include a subsection proving termination for all such inputs, ensuring that the procedure always halts and thus supports the computation of zeta coefficients for any number of primes. revision: yes
Referee: The application to Dedekind zeta coefficients therefore rests on unverified output; without correctness, the reported coefficients cannot be taken as established data for the fields in question.

Authors: With the addition of the correctness and termination proofs in the revision, the computed coefficients of the Dedekind zeta functions will be placed on a rigorous foundation. We will revise the application section to explicitly reference these proofs when presenting the coefficients. revision: yes

Circularity Check

0 steps flagged

No significant circularity in algorithmic presentation

full rationale

The paper presents an algorithm for determining prime factorization types in number fields K = Q(P) generated by odd-order torsion points on elliptic curves, then applies the algorithm to compute Dedekind zeta coefficients. No equations, fitted parameters, self-referential definitions, or load-bearing self-citations appear in the provided abstract or description. The contribution is framed as a computational procedure rather than a derivation that reduces to its own inputs by construction. Concerns about termination or general correctness are verification issues, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. The central claim rests on the unstated correctness of the described algorithm and on standard facts about elliptic curves and Dedekind zeta functions.

pith-pipeline@v0.9.0 · 5313 in / 1066 out tokens · 52816 ms · 2026-05-10T16:54:49.440094+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Andrew Sutherland,Computing Hilbert class polynomials with the Chinese remainder theo- rem, Mathematics of Computation80(2011), 501–538

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Gaetan Bisson,Computing endomorphism rings of elliptic curves under the GRH, Journal of Mathematical Cryptology5(2012), 101—114

work page 2012

[2] [2]

Gaetan Bisson and Andrew Sutherland,Computing the endomorphism ring of an ordinary elliptic curve of a finite field, Journal of Number Theory131(2011), 815—831

work page 2011

[3] [3]

Tommaso Centeleghe,https://mathoverflow.net/questions/90370/conjugation -in-gln-p-adic-setting

work page

[4] [4]

William Duke and ´Arp´ ad T´ oth,The splitting of primes in division Fields of elliptic curves, Experimental Mathematics11(2002), 555 – 565

work page 2002

[5] [5]

Benedict Gross,A tameness criterion for Galois representations associated to modualr forms (modp), Duke Mathematical Journal61(1990), 445–517

work page 1990

[6] [6]

Farshid Hajir and Siman Wong,Specializations of one-parameter families of polynomials, Annales de l’Institut Fourier56(2006), 1127–1163

work page 2006

[7] [7]

[8]The L-functions and modular forms database,https://www.lmfdb.org

David Kohel,Endomorphism rings of elliptic curves over finite fields, Doctoral Dissertation, University of California, Berkeley (1992). [8]The L-functions and modular forms database,https://www.lmfdb.org

work page 1992

[8] [8]

Bernard McDonald,Similarity of matrices over artinian principal ideal rings, Linear Algebra and its Applications21(1978), 153–162

work page 1978

[9] [9]

Ren´ e Schoof,Elliptic curves over finite fields and the computation of square roots modp, Mathematics of Computation44(1985), 483–494

work page 1985

[10] [10]

Ren´ e Schoof,Counting points on elliptic curves over finite fields, Journal de Th´ eorie des Nombres de Bordeaux7(1995), 219–254

work page 1995

[11] [11]

Andrew Sutherland,Computing Hilbert class polynomials with the Chinese remainder theo- rem, Mathematics of Computation80(2011), 501–538

work page 2011

[12] [12]

Andrew Sutherland,Computing images of Galois representations attached to elliptic curves, Forum of Mathematics, Sigma4(2016), 79 pp

work page 2016

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Melanie Matchett Wood,How to determine the splitting type of a prime, https://people.math.harvard.edu/~mmwood/Splitting.pdf

work page