Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$

Kenji Terao

REVIEW 2 major objections 5 minor 40 references

Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper classifies all degrees of closed points with rational $j$-invariant on the modular curves $X_0(n)$ and $X_1(n)$.

desk verdict New H-closure framework, but the unconditional classification depends on an unpinned LMFDB completeness claim that needs a snapshot or independent enumeration. read the letter →

arxiv 2507.13199 v1 pith:XIDOBY6W submitted 2025-07-17 math.NT

Kenji Terao This is my paper · ORCID

classification math.NT MSC 11G1811F8014G0511G05

keywords modularcurvesrationalj-invariantGaloisrepresentationsH-closuresdegreesofpointsisolatedX0(n)X1(n)

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

The paper determines, for every $n$, the set of degrees $d$ for which $X_0(n)$ or $X_1(n)$ has closed points of degree $d$ with rational $j$-invariant. For degrees occurring for infinitely many such points, the classification is unconditional and is expressed as a union over divisors $m$ of $n$ of sets $d \cdot \deg(X(n)\to X(m))$ with $d$ in explicit finite lists. For degrees occurring only finitely often, the paper gives an analogous classification of all degrees of non-cuspidal, non-CM points, conditional on a broad conjecture stated in the paper. The engine is a new group-theoretic invariant, the $H$-closure of a Galois image, which exactly encodes the orbit sizes that determine these degrees. If correct, this yields a complete computable answer to a natural arithmetic-geometry question and also settles the conjectural list of rational $j$-invariants of isolated points on $X_1(n)$.

What carries the argument

The central object is the $H$-closure of a subgroup $G$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$: the unique maximal overgroup of $G$ whose right action on the coset space $H\backslash \operatorname{GL}_2(\widehat{\mathbb{Z}})$ produces the same orbit decomposition as the action of $G$ itself. A related finer notion, the $H$-closure, records the full orbit decomposition rather than only the orbit of the identity coset. The size of such an orbit is exactly the degree of a closed point on the modular curve $X_H$ with a given rational $j$-invariant, so computing $B_0(n)$- and $B_1(n)$-closures of adelic Galois images attached to elliptic curves over $\mathbb{Q}$ computes the degrees of rational-$j$ points and fibers on $X_0(n)$ and $X_1(n)$. The paper computes these closures for the relevant Galois images: unconditionally for closures occurring infinitely often, and conditionally on Conjecture 5.5 for those occurring only finitely often, with the rational-point computation on 160 auxiliary modular curves as the final step.

What would settle it

Exhibit a conjugacy class $G$ of open subgroups of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of level at most 70 with full determinant, genus at most 1, with $G$ equal to its own $B_0(n)$- or $B_1(n)$-closure for some $n$, and with $X_G(\mathbb{Q})$ infinite, that is not listed in Table 1 or Table 2; equivalently, exhibit a rational non-CM $j$ and an $n$ whose $B_0(n)$- or $B_1(n)$-closure is not among the classes listed in Tables 1/7 or 2/8. A single such example would disprove the corresponding theorem.

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Extended reading notes

Core claim

The paper's central claim is that the degrees of points with rational $j$-invariant on $X_0(n)$ and $X_1(n)$ are exactly the union over divisors $m$ of $n$ of the multiples $d \cdot \deg(X_0(n)\to X_0(m))$ and $d \cdot \deg(X_1(n)\to X_1(m))$, with $d$ ranging over explicit finite sets: unconditionally for degrees occurring infinitely often, and conditionally on Conjecture 5.5 for degrees occurring finitely often, excluding CM $j$-invariants. The classification is finer than a statement about individual point degrees: it gives the full multiset of degrees in each rational fiber of the $j$-map, and the point-degree theorems are deduced from this fiber-level computation. The same machinery also yields that, assuming Conjecture 5.5, the only rational $j$-invariants of non-cuspidal, non-CM isolated closed points on any $X_1(n)$ are the four numbers listed in Theorem 1.5.

Load-bearing premise

The unconditional classification assumes that the online database of conjugacy classes of open subgroups of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$ of level at most 70 with full determinant, together with its genus and rational-point data, is complete; the finite classification additionally assumes Conjecture 5.5.

Editorial extensions

If this is right

For every $n$, the full set of degrees of rational-$j$ points on $X_0(n)$ and $X_1(n)$ can be computed by a finite algorithm without case-by-case analysis of elliptic curves.
The infinite-degree classification is unconditional, so the explicit sets $D^\infty_0(m)$ and $D^\infty_1(m)$ give exact answers for infinitely occurring degrees for all $n$.
If Conjecture 5.5 is proved, the finite-degree lists $D_0(m)$ and $D_1(m)$ become unconditional, completing the classification for non-CM points.
The fiber-level computation refines the point-degree classification by recording how the degrees split above each rational $j$-invariant, not merely which degrees occur.
The isolated-point result reduces the possible rational $j$-invariants of non-cuspidal, non-CM isolated points on $X_1(n)$ to four explicit numbers, matching the conjecture stated in the paper's reference [3].

Reading between the lines

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

The $H$-closure framework is not tied to the specific subgroups $B_0(n)$ and $B_1(n)$; the same orbit-size dictionary should work for any open subgroup $H$ of $\operatorname{GL}_2(\widehat{\mathbb{Z}})$, giving a general method for degree classifications on arbitrary modular curves.
The observation that all genus-1 candidates in Tables 1 and 2 have analytic rank zero suggests a hidden structural constraint relating $B_0(n)$- and $B_1(n)$-closed subgroups to modular ranks, which could simplify the finite classification.
A direct testable extension is to re-run the enumeration with more complete database coverage beyond level 70; any newly appearing conjugacy class would force a revision of the tables in the paper.
The conditional finite classification could be converted into an unconditional statement for each fixed $n$ by proving the underlying conjecture for only the finitely many $j$-invariants that actually appear, a potentially more tractable task than the full conjecture.

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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

Kenji Terao's paper gives the first full classification of the degrees of rational-j points on X0(n) and X1(n), split into infinite (unconditional) and finite (conditional on Zywina's conjecture). The main new tool is the H-closure framework, which cleanly translates the orbit structure of Galois images into degree lists. The reduction in Corollaries 2.2 and 3.3 is solid, and the application to isolated j-invariants (Theorem 1.5) is a genuine bonus.

The infinite-degree theorems (1.1 and 1.2) are the core contribution. The logic from H-closures to the finite list of candidates is coherent, and the orbit computations are straightforward to redo. The conditional theorems are clearly labeled and the reliance on Zywina's conjecture is explicit, so no issue there.

The soft spot is exactly where the stress-test note points: Theorem 4.2, which feeds the unconditional theorems, is obtained by querying the LMFDB for all full-determinant conjugacy classes of level at most 70. The paper asserts that the database contains all such classes 'at the time of writing' but gives no snapshot, no version, and no independent enumeration. The rational-point data is also incomplete, with only a claim that it is complete for the subgroups needed. Since Theorems 1.1 and 1.2 are exhaustive classifications, a single missing conjugacy class with full determinant, level in S, and infinite rational points would change the D∞ sets. This is a correctness risk, not just a reproducibility nicety. The code is on GitHub but not pinned to a commit.

The author is aware of the frailty—Remark 4.3 notes the surprising fact that all the relevant genus-1 curves have rank 0, which smells like a selection effect but is not itself a flaw.

Overall: the mathematical architecture is sound and the conditional results are trustworthy conditional on the conjecture. The unconditional results need a verifiable input: an LMFDB snapshot or an independent enumeration of the relevant subgroups. This is fixable, and the paper deserves a serious referee. I'd send it to review with a request to add a snapshot or an independent verification of the completeness claim.

Referee Report

2 major / 5 minor

Summary. The paper introduces the notions of H-equivalence and H-closures for subgroups of GL2(Ẑ) and uses them to translate the problem of degrees of points with rational j-invariant on X0(n) and X1(n) into finite orbit computations for adelic Galois images. Theorems 1.1 and 1.2 give an unconditional classification of the degrees that occur infinitely often, via the sets D∞0(m) and D∞1(m). Theorems 1.3 and 1.4 give the analogous classification of all degrees of non-cuspidal, non-CM points with rational j-invariant, conditional on Zywina's Conjecture 5.5. Theorem 1.5 applies the same methods to isolate four possible j-invariants of isolated points on X1(n).

Significance. If the computational inputs are correct, this is a substantial and useful classification: the H-closure formalism cleanly separates the Galois-theoretic core from the modular-curve computations, and the orbit-to-degree translation in Corollaries 2.2 and 3.3 is elegant and sound. The Hilbert irreducibility argument for infinitely occurring closures (Theorem 4.4 and Corollary 4.6) is a nice contribution, and Section 6.5 contains a self-contained Chabauty computation for one exceptional genus-3 curve. The paper also ships Magma code, which is a strength. However, the unconditional theorems depend on a completeness assertion about the LMFDB that is not accompanied by a version, snapshot, or certificate, and the finite-part computations are summarized rather than independently verifiable from the manuscript alone.

major comments (2)

[§4.2, Theorem 4.2] The unconditional Theorems 1.1 and 1.2 rest on the assertion that the LMFDB contains all conjugacy classes of open subgroups of GL2(Ẑ) of level at most 70 with full determinant, and that the associated rational-point data are complete for the subgroups 'required in our application.' No LMFDB version, frozen snapshot, or independent certificate is supplied. Because Theorems 1.1 and 1.2 are exhaustive classifications, a single omitted conjugacy class of genus 0 or 1 with infinitely many rational points would alter the D∞0(m) or D∞1(m) sets and hence the final degree lists. The phrase 'complete for all subgroups G which are required in our application' is also circular as written, since the set of required subgroups is exactly what the enumeration is supposed to determine. Please supply a versioned database snapshot or an independent enumeration from the Cummins–Pauli data, together with scripts that verify completeness of the list in Tables 1 and 2.
[§5.2, Theorem 5.9 and Table 6] The classification of finitely occurring closures depends on a Magma enumeration of 2651 B1-closed conjugacy classes that is presented only as Figure 1, and on rational-point computations for 160 modular curves summarized in Table 5 by method, without machine-readable output or exact versions of Magma and the LMFDB. This part is conditional on Conjecture 5.5, so it is not a threat to the unconditional theorems, but Tables 6–8 are load-bearing for the finite-degree theorems and for Theorem 1.5. Please include the enumeration data, the exact scripts with version information, or an independent verification of the exceptional j-invariants listed in Table 6.

minor comments (5)

[Title and Abstract] The title contains spacing artifacts: 'RA TIONAL j-INV ARIANT' should read 'RATIONAL j-INVARIANT'. Please correct throughout.
[§4.3, first paragraph] There is a typo: 'we showed that it the problem of determining' should read 'we showed that the problem of determining'.
[§4.2, bullet on B0/B1-closed subgroups] The text says 'By Theorem 3.11', but the relevant statement is Lemma 3.11; please correct the cross-reference.
[§7, Remark 7.3] In the sentence beginning 'Therefore, Theorem 7.1, the j-invariant...', a preposition is missing; it should read 'Therefore, by Theorem 7.1, the j-invariant...'.
[Table 4] Several entries in Table 4 are written as products such as '9 · 12063' and '51 · 78843'; this notation is not explained and the values do not look like the surrounding j-invariants. Please clarify whether these are products or typesetting artifacts.

Circularity Check

0 steps flagged · score 1.0 of 10

Central degree classification is an honest orbit computation against fixed subgroups; no fitted parameter reappears as a prediction. Minor reliance on the author's prior isolated-point paper is not load-bearing circularity.

full rationale

No circular reduction can be exhibited. The paper defines H-closures and proves (Corollary 2.2, Corollary 3.3) that the degree multiset of the fiber over j is literally the orbit-size multiset of B0(n)\GL2(\widehat{Z}) under Gj; the subsequent classification is then an enumeration against the fixed subgroups B0(n)/B1(n), not a fit to the target degrees. Theorem 4.1 is an equivalence of definitions, and Theorem 4.4 (Hilbert irreducibility) supplies infinitely many j realizing each genus-zero closure, so the D-infinity lists are not being read back from the data. The finite/isolated part is explicitly conditional on Zywina's Conjecture 5.5, a stated external assumption, and Section 6 computes rational points on 160 auxiliary curves by standard Chabauty and local-solubility methods; no parameter fitted to Theorems 1.3 and 1.4 is relabelled as a prediction. The isolated-j-invariant theorem invokes the author's earlier [31] and the independent algorithm of [3]; those are outside the derivation chain of the degree classification. The one genuine caveat is non-circular: the enumeration in Section 4.2 relies on a live LMFDB completeness assertion ('at the time of writing, the LMFDB contains a list of all conjugacy classes G ... of level at most 70' and rational-point data 'complete for all subgroups G which are required in our application') with no snapshot, which is a correctness and reproducibility risk rather than a circular reduction, because the completeness claim is not equivalent to any of the theorem statements. Consequently the score is 1, reflecting a minor self-citation [31] in the ancillary isolated-point transfer, not load-bearing circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted constants or invented physical entities. The central results are obtained from finite group actions and external computational inputs; the main assumptions are the database completeness claims and the explicit Zywina conjecture.

assumptions (5)

domain assumption Zywina's Conjecture 5.5 classifies the intersections Gj ∩ SL2(hat Z) for non-CM rational j.
This conjecture, stated in Section 5.1 and taken from Zywina, is the explicit basis for the finite-occurrence classifications in Theorems 1.3, 1.4, 1.5. It packages Serre's uniformity conjecture and a high-genus rational-point conjecture.
domain assumption The LMFDB contains all conjugacy classes of open subgroups of GL2(hat Z) of level at most 70 with full determinant, with complete genus and rational-point data.
Section 4.2 asserts this completeness at the time of writing and uses it to enumerate B0- and B1-closed subgroups with infinitely many rational points. No proof or version pin is supplied.
domain assumption The Cummins-Pauli classification of congruence subgroups of genus at most 24 is complete and yields the SL2-level bound set S in Section 4.2.
The finite bound on levels for genus 0 and 1 modular curves is drawn from this classification, and the transfer to GL2-levels is used to restrict the enumeration.
standard math Faltings's theorem, Hilbert irreducibility, Kolyvagin's theorem, and the Chabauty-Coleman method are valid in the settings where they are invoked.
These results power the reductions in Sections 4 and 6: Faltings reduces infinite rational points to genus at most 1, Hilbert irreducibility produces infinitely many j-invariants, Kolyvagin converts analytic rank zero to algebraic rank zero, and Chabauty computes rational points on several modular curves.
domain assumption The Magma computations reported in the paper and the by-hand Chabauty argument for the curve 15.90.3.c.1 are correct.
The rational-point classification of the 160 maximal curves in Table 5 and the closure tables depend on these computations. Code is provided but no commit hash or environment pin, so this is an assumption for reproduction.

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Pith. "Pith review of Degrees of points with rational $j$-invariant on $X_{0}(n)$ and $X_{1}(n)$." pith.science (2026). https://pith.science/paper/XIDOBY6W

@misc{pith2026250713199,
  author       = {Pith},
  title        = {Pith review of: Degrees of points with rational $j$-invariant on $X_0(n)$ and $X_1(n)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XIDOBY6W}},
  note         = {Machine review of arXiv:2507.13199}
}

abstract

We give a classification of the degrees of the points with rational $j$-invariant on the modular curves $X_{0}(n)$ and $X_{1}(n)$. The degrees which occur infinitely often are computed unconditionally, while those which occur finitely often are determined assuming a conjecture of Zywina. To achieve this, we define the notion of $\mathcal{H}$-closures of subgroups of $\operatorname{GL}_{2}(\widehat{\mathbb{Z}})$, and compute the $\mathcal{B}_{0}(n)$- and $\mathcal{B}_{1}(n)$-closures of images of Galois representations of elliptic curves defined over $\mathbb{Q}$. An application to computing the set of isolated $j$-invariants in $\mathbb{Q}$ is also given.

Figures

Figures reproduced from arXiv: 2507.13199 by the authors.

**Figure 1.** The graph of the B1-closed conjugacy classes of subgroups of GL2(Zb) containing one of the conjugacy classes listed in [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

The Magma algebra system. I. The user language

Wieb Bosma, John Cannon, and Catherine Playoust. “The Magma algebra system. I. The user language”. In: J. Symbolic Comput. 24.3-4 (1997). Com- putational algebra and number theory (London, 1993), pp. 235–265

work page 1997

[2] [2]

On the level of modular curves that give rise to isolated j-invariants

Abbey Bourdon, ¨Ozlem Ejder, Yuan Liu, Frances Odumodu, and Bianca Vi- ray. “On the level of modular curves that give rise to isolated j-invariants”. In: Adv. Math. 357 (2019), pp. 106824, 33

work page 2019

[3] [3]

Towards a classification of isolated j-invariants

Abbey Bourdon, Sachi Hashimoto, Timo Keller, Zev Klagsbrun, David Lowry- Duda, Travis Morrison, Filip Najman, and Himanshu Shukla. “Towards a classification of isolated j-invariants”. In: Math. Comp. 94.351 (2025). With an appendix by Maarten Derickx and Mark van Hoeij, pp. 447–473

work page 2025

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Torsion of rational elliptic curves over quartic Galois number fields

Michael Chou. “Torsion of rational elliptic curves over quartic Galois number fields”. In: J. Number Theory 160 (2016), pp. 603–628

work page 2016

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Torsion points on curves and p-adic abelian integrals

Robert F. Coleman. “Torsion points on curves and p-adic abelian integrals”. In: Ann. of Math. (2) 121.1 (1985), pp. 111–168

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