{"id":"eeb09eff-6fac-4718-bea9-de5d33057eeb","arxiv_id":"2505.12947","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A modular-period-map variant of Chabauty computes all S-integral points on elliptic-fibred curves, including Y1(N) and the S-unit equation, in seconds.","lead":"This paper gives a fast, unconditional algorithm for computing S-integral points on curves that carry an elliptic fibration, using modularity and known tables of elliptic curves to enumerate and then test finitely many candidates. A Sage implementation handles the thrice-punctured line and modular curves Y1(N) for small N, returning complete point sets in seconds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fact 2.3's conductor bound is too small: f_2 can be 8 and f_3 can be 5, so Step 1 silently omits valid curves such as y^2=x^3-8x (conductor 256, good outside {2}).","rationale":"The paper's central claim is exact enumeration of Y(Z[1/S]) via Algorithm 2.1, and completeness of Step 1 is a necessary condition. The reader identified the external Cremona database as the main weak point, but the more immediate obstruction is the paper's own conductor bound. Fact 2.3 restricts the search to conductors dividing a number that is too small: standard local conductor tables allow f_2 up to 8 and f_3 up to 5, and an explicit curve with good reduction outside {2} has conductor 256. Since the algorithm searches only divisors of N(S) (32 when S={2}), it never sees that curve. This is an internal inconsistency rather than a reproducibility issue. Correcting the constants would enlarge the conductor search beyond the 5·10^5 table limit for S={2,3}, so the stated ranges are not usable as written. The fibre-computation ideas may be salvageable after fixing this, but the current central claim is unsupported, so the verdict should move to REJECT.","tokens_in":16844,"tokens_out":46580,"duration_ms":500066,"concrete_test":"Run Tate's algorithm or Sage's EllipticCurve([0,0,0,-8,0]).conductor() on y^2=x^3-8x. If the conductor is 256, then Fact 2.3's N({2})=32 is disproved. Then re-run the full pipeline for S={2} and S={2,3} with the corrected conductor bound N(S)=2^8·3^5·∏_{p∈S} p^2 and check whether any previously omitted curves contribute to the tables in Section 3.","verdict_should_be":"REJECT","load_bearing_attack":"Section 2.1, Fact 2.3 is the load-bearing estimate for Step 1: it claims any E/Q with good reduction outside S has conductor dividing N(S)=c2(S)c3(S)∏_{p∈S} p^2, with c2=8, c3=3, and the proof cites local bounds f_2≤5, f_3≤3. These local bounds are not correct for elliptic curves over Q: the conductor exponent at p=2 can be 8 (e.g., E: y^2=x^3-8x, Cremona label 256a1, conductor 2^8) and at p=3 can be 5 (conductor 243 also occurs in Cremona's data). For S={2}, the paper's N(S) is 8·4=32, but 256∤32, so the Effective Shafarevich step will not list E. Any elliptic moduli problem whose period-map fibre contains this curve, or any curve with f_2=6,7,8 or f_3=4,5, will have its S-integral points omitted, and Algorithm 2.1 will not output the claimed exact set. This is an internal error in the manuscript, independent of database versioning or completeness. Correcting the constants (c2=64, c3=27) raises N(S) well above 5·10^5 even for S={2,3}, so the advertised computational ranges also need re-derivation.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17131,"tokens_out":19573,"duration_ms":193213,"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":[{"comment":"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":"Section 2.1, Fact 2.3"},{"comment":"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.","section":"Section 3, Tables 1-5 and Abstract"}],"minor_comments":[{"comment":"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":"Section 2.2.2, Fact 2.17"},{"comment":"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":"Section 2.2.2, Proposition 2.15"},{"comment":"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":"Section 2.1, reference [10]"},{"comment":"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.","section":"Section 3, Tables"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The key issue is the incorrect conductor exponent bound in Fact 2.3. This is a load-bearing mathematical error, not a stylistic concern: it affects the completeness proof of Algorithm 2.1 and the exactness of the tables. The rest of the framework appears sound, and the error is fixable by using the standard bounds f_2≤8 and f_3≤5 and recomputing the affected ranges. I would encourage the editor to seek a revised version addressing this point rather than a rejection, since the paper's core idea has merit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The core idea is good: replace p-adic period maps with the modular period map, list elliptic curves via Effective Shafarevich, and compute S-integral fibres. The universal fibre computation—solve the j-equation and check S-integrality—is a real improvement over von Känel–Matschke's case-by-case fibre handling. The moduli-theoretic fibre description for Y1(N) is clean, and the runtimes are impressive.\n\nBut there is a serious problem in the load-bearing Step 1. Fact 2.3 states that an elliptic curve over Q with good reduction outside S has conductor dividing N(S) = c2(S)c3(S)∏p^2, with c2=8 and c3=3. That relies on local bounds f_2≤5 and f_3≤3, which are simply wrong. For example, y^2=x^3-8x has conductor 2^8=256 and good reduction outside {2}. For S={2}, N(S)=32, so this curve is omitted from the Shafarevich list. The same issue arises at p=3 with f_3=5. This is an internal error, not a question of Cremona table coverage.\n\nCorrecting the bound to f_2≤8, f_3≤5 means c2=64 and c3=27. Then N(S)=64·27·∏p^2 when 2,3∈S, which exceeds 5·10^5 for some S of size 3 that the paper tabulates. So the advertised computational range and at least some of the tables would need re-checking.\n\nThe secondary weakness is the unfrozen GitHub data with a visit date but no commit hash. By itself that is minor, but compounded with the conductor error it means the exact outputs are not reproducible as stated.\n\nIf the author corrects Fact 2.3, re-derives the ranges, and re-publishes the tables, this becomes a useful paper. As it stands, the central claim of an unconditional algorithm is not established. The method deserves referee attention because it is promising and the error is localised.\n\nI wouldn't cite it until corrected, but I'd bring it to a reading group as an instructive example of an effectivity argument with a hidden non-effectivity.","headline":"A promising method with a wrong conductor bound in the core completeness step; fix that and the paper becomes a solid contribution.","tokens_in":17655,"tokens_out":5495,"would_cite":false,"duration_ms":54181,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G18","11Y50","14G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper presents a practical, unconditional algorithm that computes the full set of $S$-integral points on any elliptic moduli problem over $\\mathbb{Z}[1/S]$.","keywords":["modular Chabauty","S-integral points","elliptic moduli problems","modular period map","effective Shafarevich","S-unit equation","modular curves","division polynomials"],"falsifier":"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.","tokens_in":16614,"feed_emoji":"🧮","tokens_out":13026,"duration_ms":105837,"temperature":0.7,"pith_summary":"The paper aims to show that the classical finiteness of $S$-integral points on hyperbolic curves can be made effective, in a uniform way, for every curve that carries a non-isotrivial elliptic fibration — an elliptic moduli problem over $\\mathbb{Z}[1/S]$. Its algorithm replaces the $p$-adic period map used in Chabauty-type methods with the modular period map, which sends a point of the curve to the elliptic curve above it, and then computes the fibre over every elliptic curve with good reduction outside $S$. The method is implemented and returns exact point sets for the thrice-punctured line (the $S$-unit equation) and for the modular curves $Y_1(N)$ with $4\\le N\\le 10$ or $N=12$, for all sets $S$ with $\\prod_{p\\in S}p^2\\le 5\\cdot 10^5$, in under 3.5 seconds. A sympathetic reader should care because effective computation of $S$-integral points is usually blocked by impractical height bounds, whereas this route is direct, unconditional, and fast enough for routine use in the stated range.","feed_headline":"Two-step algorithm computes S-integral points in seconds","feed_subtitle":"Modular Chabauty turns the period map into a fast point-finder, with exact outputs up to conductor 5e5.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Proves the Modularity Theorem pairing isogeny classes of conductor d with weight-2 newforms; this supplies the Effective Shafarevich step's theoretical foundation.","marker":"[29, 27, 6]"},{"why":"Supplies the algorithms that enumerate isogeny classes and all curves in each class, used as the computational engine of Step 1.","marker":"[9]"},{"why":"Supplies the elliptic-curve data table through conductor 500000 whose completeness the Step-1 search relies on.","marker":"[10]"},{"why":"Derives the conductor divisibility bound N(S) that reduces the Shafarevich search to finitely many conductors.","marker":"[17]"},{"why":"Introduced the modularity-based two-step strategy on specific families; this paper generalizes the second step to arbitrary elliptic moduli problems.","marker":"[18]"},{"why":"Supplies the rational isogeny and torsion restrictions that bound the within-class search and the range of N for Y1(N).","marker":"[23]"},{"why":"Bounds the number of isomorphism classes in each rational isogeny class, guaranteeing the enumeration terminates.","marker":"[20]"},{"why":"Supplies the explicit quotient formula used to construct all curves in an isogeny class.","marker":"[28]"}],"fun_headline_variants":["Modular Chabauty computes S-integral points in seconds","Two-step algorithm finds S-integral points on elliptic curves","Effective modular method computes S-integral points quickly","Fast algorithm for S-integral points on elliptic moduli problems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Modular Chabauty computes S-integral points in seconds","Two-step algorithm finds S-integral points on elliptic curves","Effective modular method computes S-integral points quickly","Fast algorithm for S-integral points on elliptic moduli problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000715,"raw_usage":{"total_tokens":3319,"prompt_tokens":1152,"completion_tokens":2167,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":768,"completion_tokens_details":{"reasoning_tokens":2100}},"tokens_in":768,"tokens_out":2167,"duration_ms":15225,"temperature":1.0,"reasoning_tokens":2100,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:24:05.973379+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Algorithms for Modular Elliptic Curves Full Canadian Binding","cited_arxiv_id":null,"evidence_quote":"Supplies the algorithms that enumerate isogeny classes and all curves in each class, used as the computational engine of Step 1."},{"cited_title":"Elliptic Curve Data.GitHubrepository.2025","cited_arxiv_id":null,"evidence_quote":"Supplies the elliptic-curve data table through conductor 500000 whose completeness the Step-1 search relies on."},{"cited_title":"Modularity and integral points on moduli schemes","cited_arxiv_id":"1310.7263","evidence_quote":"Derives the conductor divisibility bound N(S) that reduces the Shafarevich search to finitely many conductors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the modularity-based two-step strategy on specific families; this paper generalizes the second step to arbitrary elliptic moduli problems."},{"cited_title":"Rational isogenies of prime degree","cited_arxiv_id":null,"evidence_quote":"Supplies the rational isogeny and torsion restrictions that bound the within-class search and the range of N for Y1(N)."},{"cited_title":"On the number of Q-isomorphism classes of elliptic curves in eachQ-isogeny class","cited_arxiv_id":null,"evidence_quote":"Bounds the number of isomorphism classes in each rational isogeny class, guaranteeing the enumeration terminates."},{"cited_title":"Isogénies entre courbes elliptiques","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit quotient formula used to construct all curves in an isogeny class."}],"review_version":1}