REVIEW 2 major objections 5 minor 30 references
Modular Chabauty: Effective S-Integral Point Computation On Curves with Elliptic Fibrations
T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read The paper presents a practical, unconditional algorithm that computes the full set of $S$-integral points on any elliptic moduli problem over $\mathbb{Z}[1/S]$.
desk verdict A promising method with a wrong conductor bound in the core completeness step; fix that and the paper becomes a solid contribution. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the modular period map $\Phi_M\colon \mathcal{Y}\to \mathcal{M}_{1,1}$, the morphism that sends a point of an elliptic moduli problem to the isomorphism class of the fibre of its elliptic fibration; it plays the role ordinarily filled by a $p$-adic period map in Chabauty-type methods. Two mechanisms carry the computation. The conductor bound $N(S)=c_2(S)c_3(S)\prod_{p\in S}p^2$ makes the Step-1 search finite, since every relevant conductor divides it. The fibre step then either composes $\Phi_M$ with the $j$-invariant to obtain a quasi-finite morphism $j_{\mathcal{Y}}$ whose fibre over $E$ is the zero set of one univariate polynomial, or, for $Y_1(N)$, identifies the fibre with exact order-$N$ rational torsion points on $E$, computed with division polynomials whose roots give torsion $x$-coordinates.
What would settle it
For a fixed $S$, say $S=\{2,3,5\}$ with $N(S)=21600$, independently enumerate all rational elliptic curves of conductor dividing $21600$ without using the same table, and compare with the Step-1 list; any missing or extra curve refutes completeness. A smaller check is to recompute one table entry, such as $\#Y_1(8)(\mathbb{Z}[1/\{2,3,5\}])=8$, by an independent $S$-integral point algorithm and compare counts.
Extended reading notes
Core claim
The paper's central claim is that for any elliptic moduli problem — a geometrically connected curve $\mathcal{Y}/\mathbb{Z}[1/S]$ equipped with a non-isotrivial elliptic fibration, equivalently a non-constant modular period map $\Phi_M\colon \mathcal{Y}\to \mathcal{M}_{1,1}$ — Algorithm 2.1 computes the finite set $\mathcal{Y}(\mathbb{Z}[1/S])$. Step 1 lists all rational elliptic curves with good reduction outside $S$: the Modularity Theorem identifies each isogeny class with a weight-2 newform, and the conductor bound $N(S)=c_2(S)c_3(S)\prod_{p\in S}p^2$ makes the list finite and checkable against standard elliptic-curve tables. Step 2 computes, for each listed curve $E$, the fibre $\Phi_M^{-1}(E)(\mathbb{Z}[1/S])$: in general by solving a univariate polynomial obtained from the $j$-map and checking $S$-integrality, and for $Y_1(N)$ by identifying the fibre with the exact order-$N$ rational torsion points of $E$ and computing those points with division polynomials. The implementation verifies the method on the thrice-punctured line and on $Y_1(N)$ for $4\le N\le 10$ and $N=12$, for all $S$ with $\prod_{p\in S}p^2\le 5\cdot 10^5$, with total running time under 3.5 seconds.
Load-bearing premise
The algorithm's completeness rests on the elliptic-curve data table used in Step 1 being complete and correct for every conductor up to 500,000; if a rational isogeny class with conductor dividing $N(S)$ is missing from that table, the corresponding $S$-integral points will be silently omitted from the output.
Editorial extensions
If this is right
- For every input elliptic moduli problem in the supported range, Algorithm 2.1 returns the exact finite set $\mathcal{Y}(\mathbb{Z}[1/S])$, not an approximation or a bound.
- The $S$-unit equation is solved completely for every set $S$ with $\prod_{p\in S}p^2\le 5\cdot 10^5$, including all listed triples, in a few seconds on a standard computer.
- The tables give the exact counts $\#Y_1(N)(\mathbb{Z}[1/S])$ for $N=4,\ldots,10,12$; zero entries are genuine non-existence statements for those ranges.
- The moduli-theoretic description of the fibre gives a model-independent way to prove the presence or absence of $S$-integral points on modular curves.
- Step 2 is logically independent of Step 1, so any improvement in enumerating elliptic curves by conductor extends the method to larger sets $S$ without changing the fibre computation.
Reading between the lines
- A dynamic enumeration of elliptic curves by conductor, generating only the needed levels, could push the method to sets like the first sixteen primes, where it might compete with sieve-based $S$-unit solvers.
- The same two-step scheme should apply to other level structures, such as full level-$N$ or $\Gamma_0(N)$ structures, and to elliptic moduli problems over number fields once an effective Shafarevich input is available, since the fibre step already works over any $S$-integer ring.
- The zero counts in the $Y_1(N)$ tables can be read as unconditional torsion non-existence data; comparing them with the classification of rational torsion points could turn the implementation into a computational check of that classification.
- The running times suggest the method is fast enough to serve as a first computational probe in explorations of effective Mordell and Siegel problems, with the conductor-table bound rather than the fibre method as the current bottleneck.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a two-step algorithm (Algorithm 2.1) for computing the S-integral points of an elliptic moduli problem Y/Z[1/S], i.e., a curve equipped with a non-isotrivial elliptic fibration. The first step, the Effective Shafarevich step, enumerates all rational elliptic curves with good reduction outside S by combining the Modularity Theorem with Cremona's algorithms; the second step, the Fibre Computation step, computes the S-integral fibre of the modular period map above each listed curve, either universally via the j-map or more explicitly for modular curves Y_1(N) via torsion-point computations. The paper reports implementations and tables for the thrice-punctured line and for Y_1(N), N=4,5,6,7,8,9,10,12, for sets S with product of p^2 bounded by 5·10^5, with runtimes of a few seconds.
Significance. If the correctness issue identified below is resolved, this would be a useful and practical contribution: it offers a uniform, conceptually simple algorithm for a class of Diophantine finiteness problems, complementing p-adic Chabauty–Kim and Lawrence–Venkatesh methods. The fibre computation via the j-map and division polynomials is elementary and works in practice; the paper also contains reproducible code links and concrete numerical tables. However, the central completeness claim currently rests on an incorrect conductor bound, so the advertised ranges and exactness of the tables are not established.
major comments (2)
- [Section 2.1, Fact 2.3] The conductor exponent bounds in Fact 2.3 are incorrect for primes 2 and 3. For elliptic curves over Q, the conductor exponent at p=2 can be as large as 8 (for example, the curve y^2=x^3-8x has conductor 2^8) and at p=3 can be as large as 5 (e.g., curves of conductor 3^5). The proof of Fact 2.3 cites bounds f_2≤5 and f_3≤3, which are false. Consequently, the value N(S)=c_2(S)c_3(S)∏_{p∈S}p^2 with c_2=8 and c_3=3 is not a multiple of the conductor of every elliptic curve with good reduction outside S. For instance, E: y^2=x^3-8x has good reduction outside {2} but its conductor 2^8 does not divide N({2})=2^5. Since Algorithm 2.1 Step 1 lists only curves whose conductor divides N(S), such a curve would be omitted, so the algorithm's output is not guaranteed to be the exact set Y(Z[1/S]). This directly invalidates the completeness claim of Algorithm 2.1 and the exactness claims of the tables. The correct constants should be c_2=2^{8-2}=64 and c_3=3^{5-2}=27.
- [Section 3, Tables 1-5 and Abstract] Because Fact 2.3 is the justification for the conductor range in the Effective Shafarevich step, the numerical results in Section 3 are not reliable for the stated parameter range. With the corrected constants, for S={2,3,5} one has N(S)=64·27·2^2·3^2·5^2 = 1,555,200, which exceeds the 5·10^5 conductor limit of Cremona's tables used in the implementation. Thus the advertised range "∏_{p∈S} p^2 ≤ 5·10^5" is incompatible with the completeness of the enumeration. The tables for |S|=3 (e.g., Table 2 and the three-prime rows of Tables 3-5) therefore cannot be accepted as exact until the enumeration is redone with correct conductor bounds and a matching conductor limit.
minor comments (5)
- [Section 2.2.2, Fact 2.17] The statement of Fact 2.17 needs a more careful discussion of stack-theoretic issues for full level-2 structures. The moduli stack Y(2) is a μ_2-gerbe over its coarse space (the thrice-punctured line), and the claimed bijection between coarse-space S-integral points and isomorphism classes of pairs (E,α) with the indicated equivalence relation is plausible but not proved. Since the universal fibre computation of Section 2.2.1 does not rely on this fact, the gap does not affect the algorithm's correctness, but the exposition should be clarified.
- [Section 2.2.2, Proposition 2.15] Proposition 2.15 asserts a bijection between Y_1(N)(O_{K,S}) and pairs (E,P) over K with E having good reduction outside S and P of exact order N. This is correct when N is invertible in O_{K,S}, but the paper does not state that condition explicitly. Since all tables for Y_1(N) include the primes dividing N in S, the applications are safe, but the proposition should mention the required invertibility of N.
- [Section 2.1, reference [10]] The completeness of the Effective Shafarevich step depends on Cremona's database [10], but the citation is to a live GitHub repository with a visit date and no frozen version. For reproducibility and for the claim of an unconditional algorithm, the paper should either cite a precise completeness statement for the database up to the stated bound, or provide the specific conductor data used in the implementation as part of the supplementary material.
- [Section 3, Tables] The abstract states the condition "∏_{p∈S} p^2 ≤ 5·10^5" while the text uses "≲ 5·10^5". This should be made precise, and the statement of the computational range should be reconciled with the corrected conductor bound.
- [General] There are several minor typos and formatting issues, such as the missing space in "EffectiveS-Integral" in the title line of the arXiv listing, and inconsistent use of "≲" versus "≤". These should be cleaned up in revision.
Circularity Check
No significant circularity: the fibre decomposition is definitional but not circular, and self-citations are background only.
full rationale
The central derivation is Algorithm 2.1: enumerate all E/Q with good reduction outside S via the Effective Shafarevich step, then compute the S-integral fibre over each E via the Fibre Computation step. The identity Y(Z[1/S]) = union over E of Phi_M^{-1}(E)(Z[1/S]) is exactly what it means for Phi_M to be the period map of an elliptic moduli problem; it is a decomposition, not a target set smuggled in as an input. Step 1 rests on the Modularity Theorem and Cremona's independently maintained enumeration, not on the point set being predicted. Step 2 solves jY(t)=jE or computes E[N](K) via division polynomials, both of which are independent computations. The self-citations, [8] and [30], are a related-theory pointer and the implementation link, respectively; neither carries the completeness or correctness argument, and the Corwin-Zehhavi citation is not used to justify Algorithm 2.1. Fact 2.3 and Corollary 2.4 cite von Kaenel's prior work, not the author's own. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors to force a choice, and no ansatz is smuggled in through a self-citation. The reader-supplied skeptic note about the conductor exponent bounds in Fact 2.3 (for example, curves with f_2 = 8 being omitted from N(S) when 2 in S) points to a possible correctness or completeness defect in the Effective Shafarevich bound, but that is an external error in a cited lemma, not circularity: a false or too-weak bound does not make the algorithm's output equivalent to its inputs by construction. I therefore find no significant circularity and assign score 0.
Assumptions & free parameters
assumptions (7)
- standard math Modularity Theorem for elliptic curves over Q (every rational elliptic curve is modular)
- domain assumption Completeness and correctness of Cremona's elliptic curve database up to conductor 5*10^5
- standard math Local conductor exponent bounds f_p <= 5 (p=2), <=3 (p=3), <=2 (p>=5)
- standard math Mazur's Rational Isogeny Theorem and Torsion Theorem
- standard math Neron model property: a rational point of order N on an elliptic curve with good reduction outside S extends to a section of exact order N over Z[1/S]
- domain assumption The coarse moduli space description of P^1 minus three points via full level-2 structures (Fact 2.17)
- domain assumption Representability of Y1(N) as a fine moduli space over Z[1/S] for N in {4,...,10,12} with primes dividing N inverted
Cite this review
Pith. "Pith review of Modular Chabauty: Effective S-Integral Point Computation On Curves with Elliptic Fibrations." pith.science (2026). https://pith.science/paper/Z32SHF52
@misc{pith2026250512947,
author = {Pith},
title = {Pith review of: Modular Chabauty: Effective S-Integral Point Computation On Curves with Elliptic Fibrations},
year = {2026},
howpublished = {\url{https://pith.science/paper/Z32SHF52}},
note = {Machine review of arXiv:2505.12947}
}
abstract
We present a practical, unconditional algorithm for determining the $S$-integral points on any elliptic moduli problem $\mathcal{Y}/\mathbb{Z}[1/S]$ -- that is, on any geometrically connected curve carrying a non-isotrivial elliptic fibration $\mathcal{E} \to \mathcal{Y}$. The associated map $\Phi_M\colon \mathcal{Y} \to \mathcal{M}_{1,1}$ (the modular period map) plays the role ordinarily filled by a $p$-adic period map in Chabauty-type methods. Our Modular Chabauty method studies the image and fibres of $\Phi_M$, and proceeds in two steps: an Effective Shafarevich step, in which we combine the modularity theorem with Cremona's enumeration of elliptic curves by conductor and list all rational elliptic curves with good reduction outside $S$; and a Fibre Computation step, in which we compute the $S$-integral points in the corresponding fibre of $\Phi_M$. A Python/Sage implementation computes $\mathcal{Y}(\mathbb{Z}[1/S])$ for $\mathcal{Y}=\mathbb{P}^1\setminus\{0,1,\infty\}$ and for every modular curve $Y_1(N)$ with $4\le N\le 10$ or $N=12$, for all sets $S$ with $\prod_{p\in S} p^{2}\le 5\cdot 10^{5}$, within $3.5$ seconds on a standard computer.
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Reviewed August 15, 2026 · model on record in the stance chip above.
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