{"id":"65f7d3ed-ef5f-44b8-b049-170c39da015c","arxiv_id":"2506.19560","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For prime-power level, rational isolated j-invariants are exactly 15 values on X_1 (13 CM plus two non-CM) and 19 values on X_0 (13 CM plus six non-CM).","lead":"The complete list of rational j-invariants that come from isolated points on the modular curves X_1(l^n) and X_0(l^n) is now known: 15 values for X_1, 19 for X_0. This finishes a classification that had been open for small primes, including a stubborn 7-adic case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The classification rests on the completeness of elladicgens.txt and an unlogged Magma run; if an admissible ℓ-adic image is missing or the script has a bug, Theorems 1 and 2 could miss an exception. Independent verification would settle the claim.","rationale":"I find no mathematical flaw in the architecture of the proof: the reduction to known Galois images, the correction to the Lozano-Robledo error via Smith, the nonsplit Cartan cases via Furio, and the dedicated handling of the 7-adic exceptional group are all coherent, and the final lists match the expected counts (13 CM rational j-invariants plus the stated exceptions). The reader's weakest assumption points to exactly the place I would also look: the finite classification is produced by running custom Magma code over a database supplied by Rouse, with no certificate of completeness or of the computation itself. Because the theorems are unconditional 'if and only if' statements ranging over all prime powers, a missing admissible image or a silent script bug would not be detectable from the paper alone. This is a genuine load-bearing computational premise. The posted code makes the check possible, but it has not yet been independently performed. I therefore recommend CONDITIONAL rather than unconditional acceptance: acceptance should be contingent on an independent verification that elladicgens.txt is complete and that the scripts reproduce the claimed empty/nonempty outputs.","tokens_in":12391,"tokens_out":23367,"duration_ms":232979,"concrete_test":"Independently regenerate the list of admissible ℓ-adic images from the RSZB tables (or from the authors' GitHub data) and compare setwise with elladicgens.txt; then re-run isolated_points_from_image_Gamma1.m and isolated_points_from_image_Gamma0.m with fresh Magma, instrumenting the scripts to print for each image the reduced list of (level, degree) pairs. If any image in the RSZB classification is absent from the database, or any non-exception image yields a nonempty list, the classification fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The pivotal step is in the proofs of Theorems 4 and 5 (§3.1 and §3.2): the authors run a modified version of the algorithms of [7] and [19] on 'all known ℓ-adic images' contained in elladicgens.txt and assert the output is empty except for the listed images. No log, certificate, or independent run is included, and the code is not machine-checked. If an actual ℓ-adic image of a non-CM elliptic curve over Q is absent from elladicgens.txt — especially a non-maximal subgroup of one of the listed groups — the loop would not test that image, and a hypothetical isolated j-invariant could escape the classification. Similarly, a bug in the Magma implementation of the degree computation or of the P1-parameterization elimination could delete a surviving pair and falsely produce the empty output. The 7-adic exceptional case in Section 5 also depends on additional Magma computations that are only referenced without a log. Thus the unconditional finite classification is only as reliable as the completeness of the database and the correctness of the scripts.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12599,"tokens_out":15868,"duration_ms":155652,"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":[{"comment":"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":"Section 3.1-3.2, Theorems 4 and 5"},{"comment":"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.","section":"Section 5, Proposition 1/Corollary 3"}],"minor_comments":[{"comment":"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":"Section 4, proof of Theorem 6"},{"comment":"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":"Section 5"},{"comment":"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":"Section 5"},{"comment":"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.","section":"Section 1.2 and 5"}],"recommendation":"major_revision","confidential_remarks":"The mathematical architecture of the paper is sound and the result is significant. The only substantive obstacle to acceptance is the lack of a fully reproducible computational record for Theorems 4, 5, and the 7-adic computations; this is a verifiability issue rather than a detected mathematical error. If the journal's policy accepts computational proofs backed by publicly available, inspectable code, the authors may be able to satisfy the request by supplying the exact scripts, database version, and complete output logs, possibly as ancillary material."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"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.","tokens_in":13175,"tokens_out":761,"would_cite":true,"duration_ms":9236,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","11F80","11G05","14G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an explicit finite classification of the rational isolated $j$-invariants that occur on the modular curves $X_1(\\ell^n)$ and $X_0(\\ell^n)$.","keywords":["isolated points","sporadic points","modular curves","j-invariants","Galois representations","complex multiplication","prime-power level","rational points"],"falsifier":"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 $X_0(7^n)$.","tokens_in":12165,"feed_emoji":"🔢","tokens_out":10731,"duration_ms":99236,"temperature":0.7,"pith_summary":"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.","feed_headline":"Finite list settles isolated j-invariants at prime-power levels","feed_subtitle":"On X1 and X0 curves at prime-power levels, only complex-multiplication values plus two or six exceptions are isolated.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the descent theorem for isolated points and the result that all CM $j$-invariants are $\\Gamma_1$-isolated.","marker":"[4]"},{"why":"Provides the original isolation-detection algorithm for $X_1(n)$ that the authors modify to take an $\\ell$-adic image as input; also supplies the genus and parameterization criteria.","marker":"[7]"},{"why":"Provides the analogous algorithm and criteria for $X_0(n)$, used for the $\\Gamma_0$ classification.","marker":"[19]"},{"why":"Supplies the classification database of $\\ell$-adic images that anchors Theorems 4 and 5.","marker":"[26]"},{"why":"Gives the prior partial classification for primes $\\ell>7$, whose proof is corrected here using newer ramification bounds.","marker":"[13]"},{"why":"Provides the effective classification of full $\\ell$-adic images in the normalizer-of-nonsplit-Cartan case, used in Section 4.","marker":"[16]"},{"why":"Establishes the absence of non-CM rational points on a quotient curve used to rule out one possible 7-adic subgroup.","marker":"[23]"}],"fun_headline_variants":["Exactly 15 on X1, 19 on X0: rational isolated j-invariants","Complete list of rational isolated j-invariants at prime-power levels","Isolated j-invariants on X1 and X0: finite sets for prime-power levels","All rational isolated j-invariants at prime-power levels now known","Rational isolated j-invariants at prime-power levels fully classified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exactly 15 on X1, 19 on X0: rational isolated j-invariants","Complete list of rational isolated j-invariants at prime-power levels","Isolated j-invariants on X1 and X0: finite sets for prime-power levels","All rational isolated j-invariants at prime-power levels now known","Rational isolated j-invariants at prime-power levels fully classified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001178,"raw_usage":{"total_tokens":4893,"prompt_tokens":997,"completion_tokens":3896,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":3794}},"tokens_in":613,"tokens_out":3896,"duration_ms":26824,"temperature":1.0,"reasoning_tokens":3794,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:35:59.648481+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $X_0(7^n)$.","supporting_citations":[{"cited_title":"Math.357(2019), 106824, 33","cited_arxiv_id":null,"evidence_quote":"Supplies the descent theorem for isolated points and the result that all CM $j$-invariants are $\\Gamma_1$-isolated."},{"cited_title":"Comp.94(2025), no","cited_arxiv_id":null,"evidence_quote":"Provides the original isolation-detection algorithm for $X_1(n)$ that the authors modify to take an $\\ell$-adic image as input; also supplies the genus and parameterization criteria."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the analogous algorithm and criteria for $X_0(n)$, used for the $\\Gamma_0$ classification."},{"cited_title":"Sutherland, and David Zureick-Brown,ℓ-adic images of Galois for elliptic curves overQ(and an appendix with John Voight), Forum Math","cited_arxiv_id":null,"evidence_quote":"Supplies the classification database of $\\ell$-adic images that anchors Theorems 4 and 5."},{"cited_title":"Number Theory8(2022), no","cited_arxiv_id":null,"evidence_quote":"Gives the prior partial classification for primes $\\ell>7$, whose proof is corrected here using newer ramification bounds."},{"cited_title":"Effective bounds for adelic Galois representations attached to elliptic curves over the rationals","cited_arxiv_id":"2412.10340","evidence_quote":"Provides the effective classification of full $\\ell$-adic images in the normalizer-of-nonsplit-Cartan case, used in Section 4."},{"cited_title":"J.165(2002), 159–178","cited_arxiv_id":null,"evidence_quote":"Establishes the absence of non-CM rational points on a quotient curve used to rule out one possible 7-adic subgroup."}],"review_version":2}