Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$

Abbey Bourdon; \"Ozlem Ejder

REVIEW 2 major objections 4 minor 35 references

Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves an explicit finite classification of the rational isolated $j$-invariants that occur on the modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$.

desk verdict A complete, credible classification of rational isolated j-invariants for prime-power levels, with the main caveat being the unlogged Magma runs over the RSZB database. read the letter →

arxiv 2506.19560 v1 pith:YMJHDV3Y submitted 2025-06-24 math.NT

Abbey Bourdon , Özlem Ejder This is my paper · ORCID

classification math.NT MSC 11G1811F8011G0514G05

keywords isolatedpointssporadicmodularcurvesj-invariantsGaloisrepresentationscomplexmultiplicationprime-powerlevelrational

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

A point on a curve is isolated when it is not part of an infinite family of points of the same degree. This paper proves that, over all prime-power levels, the rational $j$-invariants that occur as isolated points on $X_1(\ell^n)$ and $X_0(\ell^n)$ are exactly the complex-multiplication (CM) $j$-invariants together with, respectively, two and six explicit non-CM values. For $X_1(\ell^n)$ the exceptions are $-7\cdot 11^3$ and $-7\cdot 137^3\cdot 2083^3$, both realized only at level $\ell=37$; for $X_0(\ell^n)$ there are six exceptions, with $\ell\in\{11,17,37\}$. This settles an unconditional finite classification for an infinite family of modular curves, the kind of result that underpins questions about torsion subgroups of elliptic curves over number fields. The proof combines a modified isolation-detection algorithm run over a database of known $\ell$-adic Galois images with case analyses for the remaining possible images, so the classification is exactly as strong as that database and the accompanying script.

What carries the argument

The load-bearing object is a finite database of $\ell$-adic Galois images attached to non-CM elliptic curves over $\mathbb{Q}$, together with a modified isolation algorithm that accepts one such image as input and outputs a finite list of pairs $(\ell^{a_i}, d_i)$ for which an isolated point could exist. The algorithm works by using the image to compute degrees of points on $X_1(\ell^k)$ and $X_0(\ell^k)$, then applying a descent principle: an isolated point that maps with full degree forces its image to be isolated. Pairs with degree exceeding the genus are discarded by the Riemann-Roch theorem, and genus-zero cases are discarded as parameterized, leaving only a short list of exceptions. To cover images not already in the database, the proof adds a case analysis for images contained in the normalizer of a non-split Cartan subgroup and a separate argument showing the only remaining 7-adic possibility is the group labeled $49.196.9.1$, which the algorithm rules out.

What would settle it

An independent reimplementation of the search, run on an independently compiled complete list of admissible $\ell$-adic images, should reproduce exactly the empty output for every image except those named in Theorems 4 and 5; any non-empty output for an image not on the lists, or a constructed rational $j$-invariant lying on an isolated point at prime-power level but absent from the lists, would disprove the classification. Concretely, the script's claimed empty output for the exceptional 7-adic image 49.196.9.1 can be checked by exhibiting or disproving an elliptic curve with that image and an isolated point on $X_1(7^n)$ or $X_0(7^n)$.

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Extended reading notes

Core claim

The paper's central claim is that, over the full family of prime-power levels, there are only finitely many rational $j$-invariants that can belong to an isolated point, and the list is explicit. For $X_1(\ell^n)$, $j\in\mathbb{Q}$ is the image of an isolated point exactly when $j$ is a CM $j$-invariant or one of $-7\cdot 11^3$ and $-7\cdot 137^3\cdot 2083^3$, the last two occurring only for $\ell=37$. For $X_0(\ell^n)$ the non-CM exceptions are $-11\cdot 131^3$, $-11^2$, $-17^2\cdot 101^3/2$, $-17\cdot 373^3/2^{17}$, $-7\cdot 11^3$, and $-7\cdot 137^3\cdot 2083^3$, each realized by a rational point on $X_0(\ell)$ for $\ell=11,17,37$; all other rational isolated values are CM. In the authors' phrasing, these are the rational $\Gamma_1$-isolated and $\Gamma_0$-isolated $j$-invariants at prime-power level.

Load-bearing premise

The classification depends on the database of possible Galois actions on prime-power torsion points of non-CM rational elliptic curves being complete and on the accompanying search script being correct; if either fails, the claimed finite lists could miss a rational isolated $j$-invariant.

Editorial extensions

If this is right

The union over all prime-power levels of rational $\Gamma_1$-isolated $j$-invariants is exactly the CM values plus the two level-37 numbers $-7\cdot 11^3$ and $-7\cdot 137^3\cdot 2083^3$.
The corresponding union for $\Gamma_0$ is exactly the CM values plus six explicit numbers, each witnessed by a rational point on $X_0(\ell)$ already present at level $\ell$.
Because the classification is unconditional, it places a finite bound on the prime-power contribution to the conjectured list of rational $\Gamma_1$-isolated $j$-invariants; the remaining work is confined to composite levels.
Every non-CM $\Gamma_1$-isolated value is also $\Gamma_0$-isolated at prime-power level, but the reverse containment is strict and fails globally, since $351/4$ is $\Gamma_1$-isolated but not $\Gamma_0$-isolated.

Reading between the lines

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

Inference: the same image-input algorithm could be rerun on composite levels to classify rational isolated $j$-invariants for all $X_1(n)$ and $X_0(n)$; the main obstacle would be the absence of an equally complete database over composite levels.
Inference: rerunning the script after any update to the underlying image database is a natural verification step; the architecture of the proof means an error would most plausibly surface as a nonempty output for some image currently asserted to be empty.
Inference: for the conjectured full classification of rational $\Gamma_1$-isolated $j$-invariants, the prime-power case now contributes exactly two non-CM values, so any further non-CM examples must occur at composite level and would be genuinely new phenomena.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper closes the prime-power case for X_1(l^n) and X_0(l^n), and it does so with a sensible strategy: reduce to known or partially classified Galois images, repair the known error in Ejder's earlier work by swapping in Smith's ramification bounds, handle the nonsplit Cartan cases with Furio's classification plus a Riemann-Roch degree argument, and then isolate the one remaining exceptional 7-adic image. The statement of the classification itself is clean: 15 rational j-invariants for X_1, 19 for X_0, with the non-CM exceptions only at l = 11, 17, 37 (and 7 for X_0). If correct, the union over all prime-power levels is exactly these finite lists, which sharpens the conjectural picture from the 17-element list for X_1.

The genuinely new work is in the small primes, especially the 7-adic exceptional case (Section 5). There the authors show, via Magma computations plus a reduction to known rational points on X_0^+(74), that the only possible 7-adic image is the full group 49.196.9.1. That argument is coherent, and the correction to Ejder's proof using Smith's ramification bounds is a real service to the literature.

The soft spot is the same one the stress-test note flagged, and the paper does not fully answer it. Theorems 4 and 5 depend on running a modified algorithm over all known l-adic images in elladicgens.txt, with no log, certificate, or independent verification. A missing non-maximal subgroup or a bug in the degree computation could let an isolated exception escape. The authors post code, which is good and helps reproducibility, but the classification is only as solid as the completeness of that database and the correctness of the Magma script. This is a load-bearing computational premise, not a minor caveat. That said, the same kind of database dependency underlies all the prior work in this area (RSZB, Sutherland, Zywina), and the authors do cite those sources directly; the reader's circularity score of 2.0 seems fair. The self-citation is not a red flag here—the prior algorithms are genuinely theirs and are being adapted, not recycled to produce the same output.

The math I can check by hand holds up: the degree computations from the Galois group orders, the Riemann-Roch dimension arguments, and the reduction trick using Theorem 3 are all sound. The use of [23, Theorem 3.14] for X_0^+(74) is a standard external input, fine for this context.

Who this is for: anyone working on isolated points, torsion of elliptic curves, or Galois images of elliptic curves over Q. It deserves a serious referee, and I would read the revised version.

Recommendation: send to peer review. Ask the referees to run the Magma scripts (or at least spot-check the algorithm on a few known small cases) and to confirm the completeness of elladicgens.txt for the relevant levels. If the computation checks out, the classification stands.

Referee Report

2 major / 4 minor

Summary. The paper proves an unconditional classification of rational j-invariants arising from isolated points on the modular curves X_1(ℓ^n) and X_0(ℓ^n) for prime powers ℓ^n. Theorem 1 states that the rational Γ_1-isolated j-invariants are exactly the 13 rational CM j-invariants together with −7·11^3 and −7·137^3·2083^3, and that the non-CM exceptions occur only at ℓ=37. Theorem 2 gives the analogous Γ_0-isolated list, adding −11·131^3, −11^2, −17^2·101^3/2, and −17·373^3/2^17. The proof combines a corrected version of Ejder's earlier argument (replacing the erroneous theorem of Lozano-Robledo with Smith's work), a new treatment of nonsplit Cartan cases using Furio's classification, and a computational elimination of all known ℓ-adic images using modified versions of the algorithms from [7] and [19]. A separate argument handles the remaining 7-adic exceptional image 49.196.9.1.

Significance. If the result is correct, it completes the classification of rational isolated j-invariants for prime-power level, sharpening the conjectured finite list from [7] in this important case. The paper also repairs a known error in the literature and demonstrates the effectiveness of combining recent adelic Galois image classifications (RSZB, Zywina, Smith, Furio) with algorithmic elimination. The authors provide their Magma code in a public repository, which is a useful step toward reproducibility. The main theorems give crisp, falsifiable statements: exactly 15 and 19 rational j-invariants respectively, with non-CM exceptions confined to ℓ=11, 17, and 37.

major comments (2)

[Section 3.1-3.2, Theorems 4 and 5] The proofs of Theorems 4 and 5 rest on the assertion that running the modified Magma algorithms on elladicgens.txt produces the empty set except for the listed images, but no output log, certificate, or complete trace is included in the paper. Since these theorems are load-bearing for the unconditional classifications in Theorems 1 and 2, the authors should provide a fully reproducible computational record: the exact scripts, a version or checksum of the database, and the complete output (or a certificate listing every pair ⟨ℓ^{a_i}, d_i⟩ considered and the reason for its elimination). Without this, the classification cannot be independently verified from the paper alone.
[Section 5, Proposition 1/Corollary 3] The 7-adic exceptional case is excluded using two unlogged Magma computations: the statement that imρ_{E,49} is either all of 49.196.9.1 or the unique index-49 conjugacy class, and the claim that only the full preimage of 49.196.9.1 reduces to 49.196.9.1 modulo 49. These computations are load-bearing because they are needed to conclude imρ_{E,7^∞} = 49.196.9.1. Please include the relevant scripts and their output, or replace them with an explicit mathematical description of the finite checks performed, so that this step is reproducible.

minor comments (4)

[Section 4, proof of Theorem 6] In the first bullet, the displayed equality should read deg(x) = deg(f(x))·deg(f) rather than deg(x) = deg(f(x))·deg(x).
[Section 5] The notation X_s^+(72), X_0^+(74), and X_s(72) appears to have lost superscripts: these should presumably be X_s^+(7^2), X_0^+(7^4), and X_s(7^2), respectively. As printed, the equations are confusing.
[Section 5] The equality [Q(P):Q] = ℓ^{2d−2}(ℓ^2−1) in the first bullet of Theorem 6 would benefit from an explicit justification of the upper bound; the bound follows from the fact that a fiber over a rational point has at most deg(f) closed-point degrees summing to deg(f), but this is not stated.
[Section 1.2 and 5] The GitHub link is a welcome resource, but it would help to include a versioned commit hash or archive snapshot, since the manuscript does not specify which version of the code was used for the computational claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the classification reduces to external classifications and published theorems; the computational runs are verification, not fitted predictions.

full rationale

The derivation is self-contained in the relevant sense. The target j-invariants in Theorems 1 and 2 are not used as inputs to any computation; instead, the proofs enumerate admissible ℓ-adic images from the external classifications of Rouse–Zureick-Brown and Rouse–Sutherland–Zureick-Brown (elladicgens.txt) and run algorithms that, given an ℓ-adic image, output candidate level-degree pairs. The nonempty outputs for the exceptional j-invariants are then checked separately against published gonality and isolation results. The reduction criterion in Theorem 3 and the CM isolation result [4, Theorem 7.1] are cited from the authors' own earlier work, but they are peer-reviewed theorems with stated hypotheses that do not assume the present classification; they are therefore independent support rather than self-citation load-bearing. The corrected proof of Proposition 4 replaces an erroneous cited result with Smith's theorem and is proved in the paper. The main residual concern is reproducibility: Theorems 4 and 5 rely on an unlogged Magma run over all known ℓ-adic images, so a missing entry in elladicgens.txt or a bug in the implementation could affect the lists. That is a correctness and verification issue, not circularity, because the computation does not reduce by construction to the claimed output and no parameter is fitted to the published list. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The classification is built on a stack of external computational classifications and theorems: the RSZB l-adic image database, Zywina's mod 7 classification, Furio's nonsplit-Cartan analysis, Smith's ramification bounds, and Momose-Shimura's result on X_0^+(74). None of these is proved in the paper, and all are load-bearing for the reduction. There are no free parameters or invented entities; the only inputs are the external databases and the Magma scripts.

assumptions (7)

domain assumption The RSZB database elladicgens.txt lists every l-adic image that can occur for a non-CM elliptic curve over Q, except for the nonsplit-Cartan cases and the group 49.196.9.1.
Theorems 4 and 5 and Section 5 reduce to a scan of this database; a missing image would invalidate the classification. The paper states the 2-adic classification is complete and uses [26, Theorem 1.6] to reduce the odd-prime case.
domain assumption Zywina's mod 7 classification [35, Theorem 1.5] correctly identifies when 7Ns.2.1 and 7Ns.3.1 occur.
Section 5 uses it to conclude these images occur only for j=3^3*5*7^5/27 and that the 7-adic image is then 7.112.1.2.
domain assumption Furio's Theorem 1.9 [16] correctly characterizes l-adic images whose mod l image lies in the normalizer of a nonsplit Cartan.
Theorem 6 is built on this; the paper explicitly notes the classification of this case is otherwise incomplete.
domain assumption Smith's ramification bound [29, Theorem 1.1] applies to the potential supersingular reduction setting.
Proposition 4 uses it to repair the flawed argument via [21, Theorem 1.2].
domain assumption The modified algorithms from [7] and [19] correctly output the finite candidate list for isolated points from an l-adic image.
Theorems 4 and 5 are asserted by running these algorithms; the paper gives only an outline of steps (1)-(3) and no machine-readable log.
domain assumption Momose-Shimura [23, Theorem 3.14]: X_0^+(74)(Q) has no non-CM, non-cuspidal rational points.
Section 5 rules out the index-49 conjugate subgroup using this result.
standard math Riemann-Roch: a degree d point with d greater than the genus is P1-parameterized.
Used in Theorem 6 and Corollary 2 to show points of degree exceeding the genus are not isolated.

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Pith. "Pith review of Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$." pith.science (2026). https://pith.science/paper/YMJHDV3Y

@misc{pith2026250619560,
  author       = {Pith},
  title        = {Pith review of: Rational isolated $j$-invariants from $X_1(\ell^n)$ and $X_0(\ell^n)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMJHDV3Y}},
  note         = {Machine review of arXiv:2506.19560}
}

abstract

Let $\ell$ and $n$ be positive integers with $\ell$ prime. The modular curves $X_1(\ell^n)$ and $X_0(\ell^n)$ are algebraic curves over $\mathbb{Q}$ whose non-cuspidal points parameterize elliptic curves with a distinguished point of order $\ell^n$ or a distinguished cyclic subgroup of order $\ell^n$, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 $j$-invariants in $\mathbb{Q}$ which arise as the image of an isolated point $x\in X_1(\ell^n)$ under the natural map $j:X_1(\ell^n) \rightarrow X_1(1)$. This completes a prior partial classification of Ejder. We also identify the 19 rational $j$-invariants which correspond to isolated points on $X_0(\ell^n)$.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [7]

Comp.94(2025), no

Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry-Duda, Travis Morrison, Filip Najman, and Himanshu Shukla,Towards a classification of isolatedj-invariants, Math. Comp.94(2025), no. 351, 447–473, With an appendix by Maarten Derickx and Mark van Hoeij. MR 4807817

work page 2025

[2] [19]

Meghan Lee,Isolatedj-invariants arisign from the modular curveX 0(n), Master’s thesis, Wake Forest University, Winston- Salem, NC, May 2025, available athttps://users.wfu.edu/bourdoam/MeghanLeeThesis.pdf

work page 2025

[3] [1]

Number Theory76(1999), no

Francesc Bars,Bielliptic modular curves, J. Number Theory76(1999), no. 1, 154–165. MR 1688168

work page 1999

[4] [2]

Yuri Bilu, Pierre Parent, and Marusia Rebolledo,Rational points onX + 0 (pr), Ann. Inst. Fourier (Grenoble)63(2013), no. 3, 957–984. MR 3137477

work page 2013

[5] [3]

Wieb Bosma, John Cannon, and Catherine Playoust,The Magma algebra system. I. The user language, vol. 24, 1997, Computational algebra and number theory (London, 1993), pp. 235–265. MR 1484478

work page 1997

[6] [4]

Math.357(2019), 106824, 33

Abbey Bourdon, ¨Ozlem Ejder, Yuan Liu, Frances Odumodu, and Bianca Viray,On the level of modular curves that give rise to isolatedj-invariants, Adv. Math.357(2019), 106824, 33

work page 2019

[7] [5]

Abbey Bourdon and Tyler Genao,Uniform polynomial bounds on torsion from rational geometric isogeny classes, available at arxiv.org:2409.08214

work page Pith review arXiv

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Gill, Jeremy Rouse, and Lori D

Abbey Bourdon, David R. Gill, Jeremy Rouse, and Lori D. Watson,Odd degree isolated points onX 1(N)with rational j-invariant, Res. Number Theory10(2024), no. 1, Paper No. 5, 32. MR 4678892

work page 2024