Heavenly elliptic curves over quadratic fields

Cam McLeman; Christopher Rasmussen

arxiv: 2410.18389 · v5 · pith:XR556WOLnew · submitted 2024-10-24 · 🧮 math.NT

Heavenly elliptic curves over quadratic fields

Cam McLeman , Christopher Rasmussen This is my paper

Pith reviewed 2026-05-23 19:33 UTC · model grok-4.3

classification 🧮 math.NT

keywords elliptic curvesheavenly elliptic curvesquadratic fieldscomplex multiplicationfinitenessdivision fieldsFrobenius traces

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The pith

For any fixed prime ℓ at least 7, only finitely many heavenly elliptic curves exist over quadratic fields up to K-isomorphism.

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

The paper proves a finiteness theorem for heavenly elliptic curves over quadratic fields when the prime is held fixed. An elliptic curve is heavenly at ℓ when the extension obtained by adjoining all ℓ-power torsion points over the cyclotomic extension is pro-ℓ and unramified outside ℓ. This result complements the known finiteness for a fixed quadratic field as ℓ varies, by using a trace comparison modulo ℓ with complex-multiplication curves to handle the case of varying base fields. The authors also give the complete list of heavenly curves that have complex multiplication and irrational j-invariant.

Core claim

For a fixed prime ℓ≥7, there are only finitely many K-isomorphism classes of heavenly elliptic curves, even running over all quadratic fields K. The argument proceeds from the known finiteness result for fixed K by comparing Frobenius traces modulo ℓ with those of CM curves and thereby controlling the varying-K case.

What carries the argument

The heavenly condition that K(A[ℓ^∞])/K(μ_ℓ^∞) is pro-ℓ and unramified away from ℓ, which permits trace-mod-ℓ comparison with CM curves to bound the collection over all quadratic K.

If this is right

The complete list of heavenly elliptic curves with complex multiplication and irrational j-invariant over quadratic fields is determined up to isomorphism.
Frobenius traces modulo ℓ of heavenly curves behave like those of CM curves.
Finiteness statements extend in limited ways to abelian varieties of higher dimension and to base fields of higher degree.

Where Pith is reading between the lines

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

The finiteness question when both the quadratic field K and the prime ℓ are allowed to vary simultaneously remains open.
The trace comparison technique may apply to other constrained division-field conditions on elliptic curves.
Explicit computation of heavenly curves for small fixed ℓ could produce concrete lists that test the finiteness bound.

Load-bearing premise

The heavenly definition together with the known finiteness result for fixed K is sufficient to control the varying-K case via the trace-mod-ℓ comparison with CM curves.

What would settle it

Exhibiting infinitely many pairwise non-isomorphic heavenly elliptic curves over distinct quadratic fields for some fixed ℓ≥7 would disprove the claim.

Figures

Figures reproduced from arXiv: 2410.18389 by Cam McLeman, Christopher Rasmussen.

**Figure 1.** Figure 1: The set R inside N × F 1.3. Outline. In §2 we review previous results used in the rest of the paper. In §3 we prove Theorem 1.3 using Levin’s generalization of Siegel’s theorem on S-integral points on curves. In §4, 5, we demonstrate that (uniformly and dependent only on the dimension g and the degree of K), abelian varieties over K which are heavenly at a prime ℓ ≫ 0 possess an additional property: the st… view at source ↗

read the original abstract

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism classes of heavenly elliptic curves is finite, even running over all primes $\ell$. We prove a complementary result, that for a fixed prime $\ell\geq 7$, there are only finitely many such classes, even running over all quadratic fields. This naturally raises the question of whether to expect a finiteness result when both $K$ and $\ell$ are allowed to vary. We demonstrate similarities in the behavior of heavenly elliptic curves and elliptic curves with complex multiplication, in terms of their Frobenius traces modulo $\ell$. We determine the complete list of heavenly elliptic curves defined over quadratic fields with complex multiplication and with irrational $j$-invariant (up to isomorphism). We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible.

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 proves a new finiteness result for heavenly elliptic curves with fixed ℓ over all quadratic K, plus an explicit CM list, but the trace-mod-ℓ reduction needs checking to confirm it bounds discriminants without circularity.

read the letter

The main takeaway is that for fixed ℓ at least 7, only finitely many K-isomorphism classes of heavenly elliptic curves exist even as K runs over all quadratic fields. This complements the known result for fixed K and varying ℓ. The paper also gives the complete list of CM heavenly examples with irrational j-invariant and notes that their Frobenius traces mod ℓ behave like those of CM curves in general. These are the concrete contributions. The definition of heavenly is clear, and the statement of the theorem is direct. The CM list is useful as a reference point for anyone working in this corner of Galois representations. The trace comparison is presented as evidence of similarity, which is a reasonable observation to record. The potential issue is whether matching traces mod ℓ is enough by itself to force a bound on the discriminant of K when K varies. There are infinitely many quadratic fields that admit curves with any given trace mod ℓ, so the argument must contain an extra step that uses the heavenly condition to restrict the possible K uniformly in terms of ℓ alone before invoking the fixed-K finiteness. The abstract does not show the derivation, so the full text needs to be read to see if that step is present and non-circular. If it is, the result stands; if the comparison is used without first producing the uniform bound, the reduction would be incomplete. This work sits in a narrow subfield of arithmetic geometry. Specialists on elliptic curves over number fields and their ℓ-adic representations will find the finiteness statement and the CM list worth consulting. It is not broad enough to interest a general number theorist. The paper deserves a serious referee because it states a clean complementary theorem and supplies explicit data. The referee can check the key reduction step against the stress-test concern and confirm whether the proof controls the varying-K case properly.

Referee Report

2 major / 2 minor

Summary. The paper defines an abelian variety A/K as heavenly at ℓ if K(A[ℓ^∞])/K(μ_ℓ^∞) is pro-ℓ and unramified away from ℓ. It proves that for fixed ℓ ≥ 7 there are finitely many K-isomorphism classes of heavenly elliptic curves over quadratic fields K (varying over all such K), complements the known fixed-K finiteness result, establishes that heavenly curves have Frobenius traces mod ℓ matching those of CM curves, gives the complete list of heavenly CM elliptic curves over quadratic fields with irrational j-invariant, and extends some results to higher-degree fields and higher-dimensional abelian varieties.

Significance. If the main finiteness theorem holds, it supplies a uniform bound across varying quadratic fields for fixed ℓ, which is a natural complement to the fixed-K case and strengthens the arithmetic control on these Galois representations. The explicit classification of the CM heavenly cases and the trace-mod-ℓ comparison provide concrete data that may be useful for further study of non-CM heavenly curves or for generalizations.

major comments (2)

[Abstract / main theorem proof] Abstract and the proof of the main finiteness theorem (likely §3): the reduction of the varying-K case to the known fixed-K finiteness via trace-mod-ℓ matching with CM curves does not a priori bound disc(K), since infinitely many quadratic fields admit elliptic curves with any prescribed trace mod ℓ. The manuscript must explicitly produce (or cite) a uniform bound on disc(K) in terms of ℓ alone that follows from the heavenly condition plus the trace equality; without this step the argument risks being incomplete.
[§4] §4 (classification of CM cases): the list of heavenly CM elliptic curves with irrational j-invariant is stated to be complete, but the proof should verify that the pro-ℓ unramified condition is checked for each candidate and that no additional CM curves over quadratic fields satisfy the heavenly property at ℓ ≥ 7.

minor comments (2)

[§2] Notation for the extension K(A[ℓ^∞])/K(μ_ℓ^∞) is introduced in the abstract but should be restated with a numbered definition in §2 for clarity.
[final section] The extensions to higher-degree fields and higher-dimensional abelian varieties are mentioned only briefly; a short subsection summarizing which statements survive would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the exposition of the main finiteness argument and the CM classification can be strengthened. Both concerns are addressable by adding explicit steps that were implicit in the original manuscript; we outline the revisions below.

read point-by-point responses

Referee: [Abstract / main theorem proof] Abstract and the proof of the main finiteness theorem (likely §3): the reduction of the varying-K case to the known fixed-K finiteness via trace-mod-ℓ matching with CM curves does not a priori bound disc(K), since infinitely many quadratic fields admit elliptic curves with any prescribed trace mod ℓ. The manuscript must explicitly produce (or cite) a uniform bound on disc(K) in terms of ℓ alone that follows from the heavenly condition plus the trace equality; without this step the argument risks being incomplete.

Authors: We agree that the reduction step requires an explicit uniform bound on |disc(K)| in terms of ℓ. The heavenly condition (that K(E[ℓ^∞])/K(μ_ℓ^∞) is pro-ℓ and unramified outside ℓ) together with the trace-mod-ℓ equality to a CM curve implies that the conductor of the associated Galois representation is a power of ℓ only; for quadratic K this forces |disc(K)| to be bounded by a constant depending only on ℓ (via the conductor-discriminant formula and the fact that the representation factors through a pro-ℓ extension of bounded ramification). We will add a short lemma in §3 making this bound explicit (or citing the relevant result from the theory of ℓ-adic representations of elliptic curves over number fields) before invoking the fixed-K finiteness theorem. This renders the argument complete. revision: yes
Referee: [§4] §4 (classification of CM cases): the list of heavenly CM elliptic curves with irrational j-invariant is stated to be complete, but the proof should verify that the pro-ℓ unramified condition is checked for each candidate and that no additional CM curves over quadratic fields satisfy the heavenly property at ℓ ≥ 7.

Authors: We accept the suggestion to make the verification fully explicit. The classification in §4 proceeds by enumerating all CM j-invariants over quadratic fields with irrational j and checking the heavenly condition at each ℓ ≥ 7; we will expand the proof to include, for every listed curve, a direct verification that the extension is pro-ℓ and unramified away from ℓ, together with a short argument that the list of possible CM discriminants is exhaustive for quadratic fields (using the known classification of CM elliptic curves over number fields of bounded degree). No additional curves satisfy the condition. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain.

full rationale

The central claim (finiteness for fixed ℓ over varying quadratic K) rests on an external known finiteness result for fixed K (cited as 'it is known') plus trace-mod-ℓ comparison to CM curves. No self-citation is load-bearing, no parameter is fitted then renamed as prediction, and no equation reduces to its input by definition. The derivation is self-contained against external benchmarks.

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 definition of heavenly and the comparison with CM curves are the structural ingredients, but their status cannot be audited without the manuscript.

pith-pipeline@v0.9.0 · 5714 in / 1120 out tokens · 20235 ms · 2026-05-23T19:33:56.926929+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 2002, pp. 275–284 (↑ 5). [Sie29] C. L. Siegel.Über einege Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys. Math.1 (1929), pp. 41–69 (↑ 8). [Sil09] J. H. Silverman.The arithmetic of elliptic curves. Second. Vol

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[1] [1]

Anderson and Y

[AI88] G. Anderson and Y. Ihara.Pro-l branched coverings ofP1 and higher circularl-units. Ann. of Math. (2)128.2 (1988), pp. 271–293 (↑ 5). [AM14] K. Arai and F. Momose.Algebraic points on Shimura curves ofΓ0(p)-type. J. Reine Angew. Math. 690 (2014), pp. 179–202 (↑ 7). [BCS17] A. Bourdon, P. L. Clark, and J. Stankewicz.Torsion points on CM elliptic curve...

work page 1988

[2] [2]

Finiteness theorems for abelian varieties over number fields

CMS Conf. Proc. Amer. Math. Soc., Providence, RI, 1995, pp. 39–133 (↑ 8). [DKP99] C. David, H. Kisilevsky, and F. Pappalardi.Galois representations with non-surjective traces. Canad. J. Math.51.5 (1999), pp. 936–951 (↑ 16). [DL15] H. B. Daniels and Á. Lozano-Robledo.On the number of isomorphism classes of CM elliptic curves defined over a number field. J....

work page 1995

[3] [3]

Quad./Monogr. Ed. Norm., Pisa, 2014, pp. 1–80 (↑ 8). [Iha86] Y. Ihara.Profinite braid groups, Galois representations and complex multiplications. Ann. of Math. (2) 123.1 (1986), pp. 43–106 (↑ 5). [Kar19] R. Karpisz.Conditional bounds on heavenly elliptic curves over quadratic number fields. MA thesis: Wesleyan University, May 2019 (↑ 20). [Lan87] S. Lang....

work page 2014

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With an appendix by J

Graduate Texts in Mathematics. With an appendix by J. Tate. Springer-Verlag, New York, 1987, pp. xii+326 (↑ 14). [Lev09] A. Levin. Generalizations of Siegel’s and Picard’s theorems. Ann. of Math. (2) 170.2 (2009), pp. 609–655 (↑ 8). [Lev16] A. Levin.Integral points of bounded degree on affine curves. Compos. Math.152.4 (2016), pp. 754– 768 (↑ 8). [LMFDB] ...

work page 1987

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[Oze13] Y

arXiv: 2505.17474 [math.NT] (↑ 3). [Oze13] Y. Ozeki.Non-existence of certain CM abelian varieties with prime power torsion. Tohoku Math. J. (2)65.3 (2013), pp. 357–371 (↑ 7). [RS09] K. Rubin and A. Silverberg.Point counting on reductions of CM elliptic curves. J. Number Theory 129.12 (2009), pp. 2903–2923 (↑ 14). 28 REFERENCES [RT08] C. Rasmussen and A. T...

work page arXiv 2013

[6] [6]

Grothendieck

Sèminaire de Gèomètrie Algèbrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigè par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. Springer-Verlag, Berlin-New York, 1972, pp. viii+523 (↑ 4). [Sha02] R. T. Sharifi.Relationships between conjectures on the structure of pro-p Galois groups unramified outside p. Arithmetic fundamental groups an...

work page 1967

[7] [7]

Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 2002, pp. 275–284 (↑ 5). [Sie29] C. L. Siegel.Über einege Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. Phys. Math.1 (1929), pp. 41–69 (↑ 8). [Sil09] J. H. Silverman.The arithmetic of elliptic curves. Second. Vol

work page 2002

[8] [8]

Springer, Dordrecht, 2009, pp

Graduate Texts in Mathematics. Springer, Dordrecht, 2009, pp. xx+513 (↑ 15). [Sil94] J. H. Silverman.Advanced topics in the arithmetic of elliptic curves. Vol

work page 2009

[9] [9]

Springer-Verlag, New York, 1994 (↑ 14, 17)

Graduate Texts in Mathematics. Springer-Verlag, New York, 1994 (↑ 14, 17). [SM] The Sage Developers.SageMath, the Sage Mathematics Software System (Version 9.8). Available at https://www.sagemath.org. 2024 (↑ 13, 18). [ST68] J.-P. Serre and J. Tate.Good reduction of abelian varieties. Ann. of Math. (2)88 (1968), pp. 492– 517 (↑ 3, 6). [Sup63] D. A. Suprun...

work page 1994