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arxiv: 2407.14322 · v2 · submitted 2024-07-19 · 🧮 math.NT

Minimal torsion curves in geometric isogeny classes

Abbey Bourdon , Nina Ryalls , Lori D. Watson This is my paper

Pith reviewed 2026-05-23 22:42 UTC · model grok-4.3

classification 🧮 math.NT
keywords elliptic curvesisogeny classesmodular curvestorsion pointscomplex multiplicationrational j-invariantsminimal degreesprime power levels
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The pith

In rational or CM geometric isogeny classes of elliptic curves, the minimal degree of a point on X1(N) is completely characterized for prime-power N under odd-degree restriction or when the class is CM.

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

The paper defines minimal torsion curves as those achieving the smallest degree point on X1(N) within a fixed geometric isogeny class of elliptic curves over the algebraic closure of Q. It aims to find this minimal degree for given N, focusing on classes that contain a curve with rational j-invariant or that consist entirely of CM curves. For N a power of a single prime, complete characterizations are provided when considering only odd-degree points and separately when the class has CM. This classification matters for understanding the possible torsion structures that elliptic curves can acquire over number fields of small degree within an isogeny class.

Core claim

The authors establish that for a geometric isogeny class that is rational or CM, the least degree of a point of order N on X1(N) associated to a curve in the class admits a complete description when N is a prime power and one restricts to odd degrees, as well as in the full CM case for any N.

What carries the argument

The minimal degree of a point on the modular curve X1(N) corresponding to an elliptic curve in the given geometric isogeny class, which captures the smallest extension degree needed for N-torsion in that class.

If this is right

  • The minimal degree for odd-degree points on X1(ℓ^k) is explicitly determined for rational isogeny classes.
  • CM isogeny classes have their minimal N-torsion degrees fully described for all N.
  • Partial results extend the characterization to some non-prime-power levels in rational and CM classes.
  • These characterizations identify which curve in the class achieves the minimal degree.

Where Pith is reading between the lines

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

  • The method of comparing degrees across isogenous curves could apply to other properties preserved or related under isogeny.
  • General levels might be handled by factoring N into prime powers and using the characterizations for each.
  • Such minimal degrees provide a way to bound the possible torsion over low-degree fields uniformly within an isogeny class.

Load-bearing premise

The isogeny class must either contain an elliptic curve with rational j-invariant or consist only of CM curves, and for prime-power N the points considered must have odd degree.

What would settle it

An explicit example of a rational isogeny class or a CM class together with a prime power N where the computed minimal odd degree for a point of order N on X1(N) contradicts the characterization given in the paper.

read the original abstract

In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{\mathbb{Q}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in \mathbb{Z}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. In the present work, we consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational $j$-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterization upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting.

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

0 major / 3 minor

Summary. The paper introduces the study of minimal torsion curves in a fixed geometric isogeny class E of elliptic curves over Q-bar. For N positive integer, it determines the least degree of a point on X_1(N) associated to any E in the class. Complete characterizations are given when E is rational (contains a curve with rational j-invariant) or consists entirely of CM curves; for N = ℓ^k a prime power, full results hold upon restricting to odd-degree points on X_1(N) or when E is CM, with various partial results in the general setting.

Significance. If the derivations hold, the work provides a systematic treatment of minimal degrees of torsion points on modular curves within isogeny classes, extending standard modular curve theory to this geometric setting. The explicit scoping to rational/CM classes and odd-degree points for prime powers avoids overclaiming and allows clean statements; the partial results outside these cases indicate where further work is needed. No machine-checked proofs or parameter-free derivations are present, but the results are falsifiable via explicit computation of points on X_1(N).

minor comments (3)
  1. §1: the definition of 'minimal torsion curve' should be stated as a numbered definition or equation for easy reference in later sections.
  2. Table 1 (or equivalent summary table for prime-power cases): clarify whether the listed degrees are achieved only for odd-degree points or in full generality.
  3. The transition from the rational case to the CM case in §4 could include a short comparison paragraph highlighting where the proofs diverge.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The report accurately captures the scope and contributions of the paper.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces and characterizes minimal torsion curves in geometric isogeny classes of elliptic curves, restricting to rational or CM cases and (for prime-power N) to odd-degree points on X1(N). The abstract and description contain no equations, fitted parameters, predictions, or self-referential definitions. Results rest on standard modular-curve theory with explicitly stated scoping conditions; no load-bearing step reduces by construction to its inputs or to a self-citation chain. This is a self-contained mathematical derivation with no visible circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard theory of modular curves X1(N), isogenies of elliptic curves over Q-bar, and the distinction between rational and CM isogeny classes; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • standard math Standard properties of the modular curve X1(N) and the action of Galois on torsion points of elliptic curves.
    Invoked when relating degrees of points on X1(N) to field extensions generated by torsion points.
  • standard math Existence and basic arithmetic of isogenies between elliptic curves over the algebraic closure of Q.
    Used to define geometric isogeny classes.

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Reference graph

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