arxiv: 2604.21601 · v1 · submitted 2026-04-23 · 🧮 math.NT

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The Smallest Invariant Factor of Elliptic Curves, and Coincidences

Alexander Milner , Jack Shotton

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Pith reviewed 2026-05-08 14:14 UTC · model grok-4.3

classification 🧮 math.NT

keywords elliptic curvesGalois representationsinvariant factorsdivision fieldsprime densitiescomplex multiplicationabelian extensions

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The pith

For non-CM elliptic curves over Q, the positivity of the density constant C_E,j is determined by the image of the adelic Galois representation.

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

Cojocaru showed that an explicit constant C_E,j gives the density, assuming GRH, of primes p of good reduction such that the smallest invariant factor of the group E(F_p) equals a fixed natural number j. For elliptic curves E without complex multiplication, the paper investigates precisely when this constant is positive, which is a necessary condition for infinitely many such primes to exist and, under GRH, also sufficient. The authors replace earlier approaches with group-theoretic arguments that read the answer directly off the image of the adelic Galois representation attached to E. They strengthen a prior result of Kim in this way and observe experimentally that C_E,j vanishes only when the division fields of E exhibit coincidences, documenting several infinite families of such coincidences that arise from abelian extensions.

Core claim

For a non-CM elliptic curve E over Q the constant C_E,j is positive exactly when the image of the adelic Galois representation rho_E(Gal(Qbar/Q)) inside GL_2(Zhat) meets a certain collection of conjugacy classes that permit the smallest invariant factor of the reduction to be j; the positivity question is thereby reduced to a question about this image group. Experimentally the constant vanishes only in the presence of coincidences among the division fields of E, and several infinite families of curves exhibiting such coincidences are constructed from abelian division fields.

What carries the argument

The image of the adelic Galois representation of E inside GL_2(Zhat), which encodes the possible Frobenius conjugacy classes and thereby controls the possible group structures of E(F_p).

If this is right

Whenever the adelic Galois image meets the required classes, C_E,j is positive and therefore there are infinitely many primes p with smallest invariant factor j in E(F_p), assuming GRH.
The set of all j for which C_E,j > 0 is completely determined by the adelic image group of E.
Coincidences between division fields cause C_E,j to vanish for certain j, and this accounts for all observed zeros.
Infinite families of curves arising from abelian division fields exhibit systematic vanishing of C_E,j.

Where Pith is reading between the lines

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

If the experimental pattern is general, then every case of vanishing density is explained by equality of certain division fields rather than by other obstructions in the Galois image.
The documented families suggest that abelianness of a division field forces the adelic image to be smaller than expected, thereby producing zero densities.
The group-theoretic criteria could be used to compute the full set of attainable smallest invariant factors for a given E by inspecting only its Galois image.

Load-bearing premise

The positivity of C_E,j is completely determined by the image of the adelic Galois representation, and experimental vanishing occurs only at division-field coincidences.

What would settle it

A non-CM elliptic curve E together with a j for which the adelic Galois image satisfies the group-theoretic positivity condition yet direct computation of C_E,j yields zero, or a curve with no division-field coincidence yet C_E,j vanishes.

read the original abstract

For an elliptic curve E over Q and a natural number j, Cojocaru has shown that there is an explicit constant C_E,j giving (under GRH) the density of primes p of good reduction such that the smallest invariant factor of E(F_p) is j. For E without complex multiplication, we study the question of when C_E,j is positive (a necessary and, on GRH, sufficient condition for there to be infinitely many such p), strengthening a result by Kim. Our arguments are group-theoretic using the image of the adelic Galois representation of E. Experimentally, C_E,j appears to vanish only when there is a coincidence of division fields; we document a number of families of such coincidences arising from abelian division fields.

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

This paper gives explicit group-theoretic criteria on the adelic Galois image for positivity of C_E,j on non-CM curves and classifies some new abelian division-field coincidence families.

read the letter

The main point is that for elliptic curves without complex multiplication, whether the constant C_E,j is positive reduces to a check on the image of the adelic Galois representation. The authors make this reduction explicit using conjugacy classes and strengthen Kim's earlier density result. They also list several families of abelian division fields that coincide and cause the constant to vanish, which had not been classified before. Experimentally they suggest these coincidences are the only places where vanishing happens. That part is useful for people who care about explicit examples in this area. The group-theoretic approach is straightforward and relies on standard facts like Serre openness, so the core logic looks clean on the surface. They give credit to Cojocaru and Kim where it is due and do not overclaim unconditional results. The soft spots are modest. The abstract supplies no proofs or sample computations, so the details of how the conjugacy-class conditions translate into the invariant-factor count still need checking in the full text. The sufficiency direction stays conditional on GRH, which is honest but limits the reach. The classification is restricted to abelian cases, so it may not be exhaustive. Overall this is aimed at specialists in elliptic curves and Galois representations who already know the background on reduction densities. A reader looking for concrete criteria or new families of coincidences will get something out of it. The work shows clear thinking and direct engagement with the literature, so it deserves a serious referee even if the proofs need polishing.

Referee Report

2 major / 2 minor

Summary. The paper claims that for non-CM elliptic curves E over Q, the positivity of Cojocaru's constant C_{E,j} (which under GRH gives the density of primes p of good reduction such that the smallest invariant factor of E(F_p) equals j) is completely determined by the image of the adelic Galois representation of E. It strengthens a result of Kim via group-theoretic arguments on this image and experimentally observes that C_{E,j} vanishes only in the presence of division-field coincidences, documenting several families of such coincidences arising from abelian division fields.

Significance. If the group-theoretic reduction holds, the work supplies a concrete criterion, in terms of a standard object (the adelic image), for when infinitely many primes exist with a given smallest invariant factor. This clarifies the link between Galois images and invariant-factor statistics and supplies explicit families of division-field coincidences that can serve as test cases for future work on when such coincidences occur. The experimental component, while not a proof, provides falsifiable predictions about vanishing loci.

major comments (2)

[Main theorem / group-theoretic section] The central reduction of positivity of C_{E,j} to a property of the adelic image (presumably the main theorem in the group-theoretic section) should include an explicit lemma or proposition showing how the relevant conjugacy-class condition on the image translates into the invariant-factor condition without additional assumptions beyond Serre openness.
[Experimental section / families of coincidences] The experimental claim that vanishing occurs only at division-field coincidences is supported only by computation; the paper should state the precise range of curves, primes, and j values tested, together with any error analysis or sampling method, so that the strength of the observation can be assessed.

minor comments (2)

[Introduction] The introduction should explicitly contrast the new group-theoretic criterion with Kim's original statement, citing the precise result being strengthened.
[Throughout] Notation for the adelic image and for the constant C_{E,j} should be fixed early and used consistently; a short table summarizing the families of coincidences would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, agreeing to incorporate clarifications and documentation in the revised version.

read point-by-point responses

Referee: [Main theorem / group-theoretic section] The central reduction of positivity of C_{E,j} to a property of the adelic image (presumably the main theorem in the group-theoretic section) should include an explicit lemma or proposition showing how the relevant conjugacy-class condition on the image translates into the invariant-factor condition without additional assumptions beyond Serre openness.

Authors: We agree that an explicit bridge between the conjugacy-class condition and the invariant-factor condition would improve readability. In the revised manuscript we will add a short lemma in the group-theoretic section that derives the positivity criterion for C_{E,j} directly from the relevant conjugacy classes in the adelic image, invoking only Serre openness and the definition of the smallest invariant factor; no further arithmetic assumptions will be required. revision: yes
Referee: [Experimental section / families of coincidences] The experimental claim that vanishing occurs only at division-field coincidences is supported only by computation; the paper should state the precise range of curves, primes, and j values tested, together with any error analysis or sampling method, so that the strength of the observation can be assessed.

Authors: We will expand the experimental section to record the precise computational range (curves, primes, and j values examined), the sampling procedure employed, and any error or precision analysis. This addition will make the empirical support for the observed link between vanishing and division-field coincidences fully transparent and assessable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external Galois theory independently

full rationale

The paper takes the explicit constant C_E,j from Cojocaru's prior work and studies its positivity for non-CM curves via the adelic Galois representation image, invoking the standard Serre openness theorem (external). The group-theoretic criterion for when C_E,j > 0 is derived directly from conjugacy class conditions in the image group, without reducing to fitted parameters, self-defined quantities, or a load-bearing self-citation chain. Experimental vanishing cases are documented as separate observations on division-field coincidences, not used to force the main result. The strengthening of Kim's result consists of making the group-theoretic reduction explicit, which remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard theory of adelic Galois representations for non-CM elliptic curves (Serre's openness theorem) and the generalized Riemann hypothesis for the density-to-infinitely-many implication.

axioms (2)

domain assumption The image of the adelic Galois representation of a non-CM elliptic curve is open in the adelic group (Serre's theorem).
Invoked to translate positivity of C_E,j into a group-theoretic condition on the image.
domain assumption GRH
Used to make positivity of C_E,j sufficient for infinitely many primes p.

pith-pipeline@v0.9.0 · 5422 in / 1247 out tokens · 80038 ms · 2026-05-08T14:14:38.555584+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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