{"id":"7f225953-2614-47b1-a30a-aee173bad510","arxiv_id":"2501.02345","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every non-CM elliptic curve over Q, one of its 2-, 3-, or 5-adic Galois representations is surjective unless its j-invariant is one of six listed values, in which case 7 is the minimal surjective prime.","lead":"Elliptic curves over the rationals without complex multiplication always have a Galois representation that becomes surjective at a small prime: at most 7. The paper proves this uniform bound and lists the six exceptional j-invariants where the smallest such prime is exactly 7.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's Lemma 5 does not establish the quadratic-twist invariance that the j-invariant classification relies on; without a correct proof, the claim that all twists of the six exceptional j-invariants have smallest surjective prime 7 is unsupported.","rationale":"Good-faith reading: the paper's strategy is sound and the computational infrastructure is a real strength. The rational-point enumeration reduces the claim to finitely many modular curves, and the public Magma code makes the computer-assisted part checkable. I do not doubt the underlying theorem on the basis of this review. The single most load-bearing weakness is the logical step from Part (i) to Part (ii). The tables show that for each j∈J there exists a twist with the required nonsurjective 2/3/5 and surjective ≥7 behavior, but the statement of Theorem 2 is about every elliptic curve with that j-invariant. The paper's Lemma 5 is not a correct proof of the needed quadratic-twist invariance: it proves only the trivial invariance under Q-isomorphism and its proof's first sentence conflates Q-isomorphic with quadratic twists. The reader identified this as a concrete issue; I agree. I did not choose the completeness of Table 1 or the Magma computations as the primary concern, because those are external classifications or reproducible code and are less likely to be the weak point; they are also explicitly highlighted by the authors. The verdict should remain CONDITIONAL: the main theorem is plausible and likely true, but the manuscript should be revised to state and prove the twist-invariance lemma correctly, or to check all twists of the six exceptional j-invariants.","tokens_in":13879,"tokens_out":31773,"duration_ms":328699,"concrete_test":"Write a correct lemma: for every quadratic twist E_d of E over Q and every ℓ, im ρ_{E_d,ℓ∞} is full GL2(Z_ℓ) whenever im ρ_{E,ℓ∞} is full. The essential check is group-theoretic: prove that no subgroup G≤GL2(Z_ℓ) with G/{±I}=PGL2(Z_ℓ) omits -I, i.e., that the central extension 1→{±I}→GL2(Z_ℓ)→PGL2(Z_ℓ)→1 does not split for ℓ=2,3,5,7; then apply this to G=im(χ_d⊗ρ_E). If this proof succeeds, the j-classification goes through; if it fails for any ℓ, the six exceptional j-invariants must be rechecked twist by twist, for example by computing the 7-adic images of all quadratic twists of 50.a1 in the RSZB/LMFDB data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem has two parts: (i) if E has nonsurjective 2-, 3-, and 5-adic images then j(E) is one of the six values in J; (ii) for these six j-invariants the smallest surjective prime is exactly 7. Part (i) is established by the rational-point search. Part (ii) is not established by that search: a rational point on a fiber product gives a j-invariant, but infinitely many quadratic twists share that j-invariant, and the tables only analyze one representative twist for each point. To conclude that every E with j(E) in J has nonsurjective 3-adic (etc.) and surjective 7-adic image, the paper needs the fact that ℓ-adic surjectivity is invariant under quadratic twisting. Lemma 5 is supposed to supply this, but as written it states only the trivial Q-isomorphism invariance, its proof conflates Q-isomorphic curves with quadratic twists, and it even asserts the false claim that Q-isomorphism class is determined by j-invariant. The invariance statement is true and can be proved group-theoretically, but it does not appear correctly in the manuscript. Because the six exceptional j-invariants are precisely the sharp case of the theorem, this gap is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves Theorem 2: every non-CM elliptic curve E/Q has a surjective ℓ-adic representation for some prime ℓ ≤ 7, and if j(E) is not one of six explicitly listed j-invariants, then there is even a surjective prime ℓ ≤ 5. The proof assumes that ρ_{E,2∞}, ρ_{E,3∞}, and ρ_{E,5∞} are all nonsurjective and studies rational points on fiber products of modular curves X_{H_2} × X_{H_3} × X_{H_5} attached to the maximal subgroups in Table 1. The rational-point computations reduce the possibilities to the six j-invariants in J, and for those j-invariants the authors verify nonsurjectivity at 2, 3, 5 and surjectivity at all primes ≥ 7. The argument combines the Sutherland–Zywina/RSZB classification, low-genus rational-point techniques, Chabauty0, local solubility, and Magma computations, with the code available in a public repository.","tokens_in":13978,"tokens_out":13220,"duration_ms":133650,"significance":"If the computational claims are correct, this completely solves the problem of uniformly bounding the smallest surjective prime for elliptic curves over Q and is sharp: infinitely many curves have minimal surjective prime 5, while the six exceptional j-invariants have minimal surjective prime exactly 7. The result is a natural complement to the better-studied largest-nonsurjective-prime problem. The proof is parameter-free and the computation is publicly available, which are substantial strengths. The main flaw in the current write-up is the incorrect proof of the quadratic-twist invariance lemma; because the needed statement is true and the overall strategy is sound, the theorem is very likely salvageable.","major_comments":[{"comment":"Lemma 5 as stated and proved does not support the argument. The statement is about Q-isomorphic curves and is trivially true, but the proof actually tries to establish invariance under quadratic twisting. The preamble that 'the Q-isomorphism class of an elliptic curve is determined by its j-invariant' is false: the j-invariant determines the isomorphism class over Qbar, not over Q, and nontrivial quadratic twists give counterexamples. Likewise, the assertion that Q-isomorphic curves 'must be quadratic twists' is only true with the trivial character. The application in Section 3 ('Lemma 5 implies ... invariant on Q-isomorphism classes') therefore does not justify the crucial inference from the rational places in Tables 2 and 3 to the claim that every elliptic curve with one of the six j-invariants in J has nonsurjective 2-, 3-, and 5-adic images and surjective 7-adic image. Because Theorem 2 classifies by j-invariant and the six j-invariants are exactly the sharp case, this gap is load-bearing. The needed fact is true: one should prove, or cite a correct proof, that for a quadratic twist E' of E, ρ_{E',ℓ∞} is surjective if and only if ρ_{E,ℓ∞} is surjective for every prime ℓ. The standard group-theoretic argument should replace the current Lemma 5.","section":"Section 2, Lemma 5; used in Section 3"},{"comment":"The exhaustive rational-point searches are the core of the proof, but several are described only as 'handled in the same way' or 'a similar analysis reveals', for example X+ns(3)×Xns(2), X+ns(3)×4.2.0.1, 9.27.0.1×8.2.0.1, 9.27.0.1×8.2.0.2, and the rank-0 elliptic-curve cases in the two trees. For these cases the text does not give the birational model, the computed point set, or the output verifying that every rational place is cuspidal, CM, or listed in the tables. The linked Magma repository is a good start, but the printed proof should either include these data in an appendix or provide a verification table keyed to the exact scripts and outputs in the repository. Without this, the exhaustion claim on which the main theorem depends cannot be checked from the paper alone.","section":"Section 3, rational-point computations"}],"minor_comments":[{"comment":"The notation X+sp(5) is used without definition; please define it as the modular curve attached to the normalizer of a split Cartan subgroup of GL2(F5).","section":"Section 3, third tree and Table 5"},{"comment":"Please state the exact Magma version and the commit hash of the GitHub repository used for the computations, so that the output can be reproduced independently.","section":"Section 1 and repository footnote"},{"comment":"The table caption says 'all maximal closed subgroups ...'; please add a precise citation to the classification theorem in [30] and to [23] for the RSZB labels, so that the exhaustiveness step is explicitly supported by the literature.","section":"Section 2, Table 1"},{"comment":"Rows with j-invariant ∞ are cusps; labeling them explicitly as 'cusp' in the j-invariant column would make the tables easier to read.","section":"Tables 2–6"},{"comment":"The Magma computation for index-2 subgroups of GL2(Z/8Z) should be written out or explicitly included in the repository, since the current sentence is hard to check by hand.","section":"Section 2, proof of Lemma 5"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper solves a clean and natural problem, and the computational strategy is convincing. The main issue is the incorrect proof of Lemma 5, which conflates Q-isomorphism with quadratic twisting; since the needed twist-invariance statement is true, this is repairable rather than fatal. I would be willing to accept after the authors supply a correct proof of twist-invariance and expand the computational verification details for the load-bearing rational-point cases."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my quick take. The headline result is real: a uniform bound of 7 on the smallest surjective prime for non-CM curves over Q, plus a complete list of the six j-invariants where 7 is minimal. That's genuinely new and removes the ineffectiveness in Larson-Vaintrob. The proof strategy is also sound: assume 2, 3, 5 all fail, reduce to finitely many fiber products of modular curves, and determine rational points by rank-0 Jacobian/Chabauty, local solubility, and a few clever 'curious group' reductions. The Magma code is public, and Table 1 is based on the Sutherland-Zywina j-maps and the RSZB classification. This is a serious computational paper.\n\nThe main soft spot is Lemma 5. As written it states that the Q-isomorphism class of an elliptic curve is determined by its j-invariant, which is false over Q, and the proof conflates Q-isomorphic curves with quadratic twists. The paper needs exactly the twist-invariance of ℓ-adic surjectivity to move from the one twist of each exceptional j-invariant that the rational point computation produces to all twists with that j. The invariance is true in the cases that matter (essentially because the maximal subgroups in play contain -I), but the lemma as stated does not prove it. This is a load-bearing gap in exposition, not in the underlying mathematics. A referee should ask for a correct statement and proof.\n\nThere are also several places where rational point claims are just 'a similar analysis' or are left to Magma commands without enough information to check by hand. That is a transparency issue, not an error. The reliance on the authors' own ell-adic-galois-images database is fine: it is public and well tested.\n\nBottom line: the theorem is likely right, and the flaws are repairable. The paper is for anyone working on Serre uniformity or Galois images of elliptic curves; it is directly relevant and should be cited. This deserves a serious referee and probably acceptance after revision. I'd bring it to reading group.","headline":"Sharp uniform bound on the smallest surjective prime, correctly proved in outline; the quadratic-twist lemma is misstated and needs a fix before the classification is fully supported.","tokens_in":14641,"tokens_out":4840,"would_cite":true,"duration_ms":50661,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11F80"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every non-CM elliptic curve over the rationals, the smallest surjective prime is at most 7; apart from six explicit j-invariants it is at most 5.","keywords":["non-CM elliptic curves","Galois representations","surjective primes","Serre uniformity","modular curves","rational points","Chabauty's method","ℓ-adic images"],"falsifier":"Search the paper's own electronic data for a non-CM curve whose 2-, 3-, and 5-adic images are all nonsurjective and whose j-invariant is not one of the six values in J; even one such curve would disprove the classification. Alternatively, compute the 7-adic image of a curve with j-invariant in J: if it is nonsurjective, or if a smaller prime turns out to be surjective, the claim that 7 is the smallest surjective prime for those curves fails.","tokens_in":13532,"feed_emoji":"🔢","tokens_out":13495,"duration_ms":122908,"temperature":0.7,"pith_summary":"Serre's open image theorem guarantees that, for a non-CM elliptic curve over the rationals, the ℓ-adic Galois representation is onto GL2(Zℓ) for all but finitely many primes, and the hard remaining problem is to bound the largest prime where it can fail. This paper attacks the opposite end: how small can the first surjective prime be? It proves that the answer is uniformly at most 7, and that unless the j-invariant is one of six explicit values, the first surjective prime is at most 5. The six exceptional j-invariants are completely classified: for each of them the 2-, 3-, and 5-adic representations all fail to be surjective while the 7-adic representation is surjective, so 7 is truly the smallest. This settles the small-prime analogue of Serre's uniformity question and gives a sharp, completely explicit bound.","feed_headline":"Smallest surjective prime of an elliptic curve is at most 7","feed_subtitle":"A complete classification shows only six j-invariants require the prime 7; all others are covered by 2, 3, or 5.","key_machinery":"The central object is the modular curve X_H attached to a subgroup H ⊆ GL2(Z/NZ) containing −I: its rational points classify elliptic curves whose mod-N Galois image lies in H, via the j-map X_H → $P^{1}$. To force all three small primes to fail, the paper forms the fiber product X_{H2} × X_{H3} × X_{H5} over the j-line, where each Hℓ runs through the maximal closed subgroups of GL2(Zℓ) with surjective determinant for ℓ = 2, 3, 5 (six, three, and three possibilities respectively, listed with their j-maps in Table 1). A curve with all three small images nonsurjective gives a rational point on one of these 54 fiber products. The proof then determines all rational points on the relevant fiber products, using local solubility obstructions, elliptic curves of rank zero, and Chabauty's method for genus-2 curves with rank-zero Jacobians, together with previously known classifications of rational points on some of the curves.","core_discovery":"The paper proves Theorem 2, stated as follows. If E/Q is a non-CM elliptic curve, then the smallest prime ℓ such that ρ_{E,ℓ∞} is surjective is at most 7. Moreover, if j(E) is not one of the six values $$\\{-$2^{{-3}}$\\cdot $5^{{2}}$\\cdot 2413,\\; -$2^{{4}}$\\cdot $3^{{2}}$\\cdot $13^{{3}}$,\\; -$2^{{-5}}$\\cdot 5\\cdot $29^{{3}}$,\\; -$2^{{-1}}$\\cdot $5^{{2}}$,\\; $2^{{4}}$\\cdot $3^{{3}}$,\\; $2^{{-15}}$\\cdot 5\\cdot $211^{{3}}$\\},$$ then the smallest surjective prime is at most 5. For each of the six listed j-invariants, the 2-, 3-, and 5-adic representations are nonsurjective and the 7-adic representation is surjective, so these curves have smallest surjective prime exactly 7 and the uniform bound is sharp. The proof establishes the finite classification by showing that any curve whose 2-, 3-, and 5-adic images are all nonsurjective must have one of these six j-invariants.","pith_inferences":["One could independently verify the classification by scanning tables of ℓ-adic Galois images: every non-CM curve outside the six j-invariants should have at least one of its 2-, 3-, or 5-adic images surjective.","The same small-prime-first strategy may transfer to abelian surfaces or Jacobians of genus-2 curves, where the smallest surjective prime is currently bounded only in terms of the conductor; carrying this over would require an analogous finite enumeration of maximal subgroups at the smallest levels.","Because the six exceptional curves form a finite set, one expects that in most families of elliptic curves the smallest surjective prime is 2; computing the actual density of curves with smallest surjective prime 2, 3, or 5 would require a finer moduli analysis than this paper gives."],"forward_implications":["Every non-CM elliptic curve over the rationals has a surjective prime among 2, 3, 5, and 7, and the six listed j-invariants are the only curves for which 7 is needed.","For every non-CM curve, at least one of the mod-2, mod-3, or mod-5 Galois representations is surjective.","There are infinitely many j-invariants for which the smallest surjective prime is exactly 5, so the improved bound 5 for the generic case cannot be lowered uniformly to 3.","For each exceptional j-invariant, all primes ℓ ≥ 7 are surjective, so the simultaneous failure at 2, 3, and 5 is an isolated small-prime phenomenon rather than part of a chain of large nonsurjective primes.","Any curve with all three small-prime images nonsurjective must appear among the six exceptional j-invariants, giving a finite, checkable obstruction to nonsurjectivity at 2, 3, and 5 simultaneously."],"supporting_citations":[{"why":"Supplies the complete list of maximal closed subgroups of GL2(Zℓ) with surjective determinant for ℓ = 2, 3, 5 and the j-maps of their modular curves (Table 1), which is the enumeration every fiber product starts from.","marker":"[30]"},{"why":"Provides the ℓ-adic image labels used to identify the images of the six exceptional curves and to check that all primes ℓ ≥ 7 are surjective.","marker":"[23]"},{"why":"Computes the ℓ-adic Galois images of specific elliptic curves, giving the nonsurjectivity at 2, 3, 5 and surjectivity at 7 for the exceptional j-invariants.","marker":"[22]"},{"why":"Introduces the curious-groups phenomenon and supplies rational-point results on several fiber products that let the proof discard or reduce cases.","marker":"[9]"},{"why":"Classifies curious Galois groups as direct products, used to reduce one fiber product case to a curve whose rational points are already known.","marker":"[8]"},{"why":"Shows all rational points on a key non-split Cartan fiber product are CM, dismissing one remaining case.","marker":"[3]"},{"why":"Provides the model-construction and point-finding routines used for fiber products that need better models than the direct j-map equation.","marker":"[35]"},{"why":"Yields infinitely many j-invariants with smallest surjective prime exactly 5, establishing sharpness of the generic bound.","marker":"[21]"}],"fun_headline_variants":["Smallest surjective prime for elliptic curves: ≤ 7","Proven: elliptic curve surjective prime is at most 7","Surjective prime bound for elliptic curves: ≤7, sharp","Smallest surjective prime: max 7 (six exceptions)","Elliptic curve surjective prime is ≤ 7; six need 7"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the published enumeration of all maximal possible ℓ-adic image groups for ℓ = 2, 3, and 5 is complete, that the computer rational-point computations are error-free, and that quadratic twists do not change which primes are surjective; if any of these fails, a curve whose smallest surjective prime exceeds 7 could be missed.","fun_headline_variants_meta":{"raw":{"variants":["Smallest surjective prime for elliptic curves: ≤ 7","Proven: elliptic curve surjective prime is at most 7","Surjective prime bound for elliptic curves: ≤7, sharp","Smallest surjective prime: max 7 (six exceptions)","Elliptic curve surjective prime is ≤ 7; six need 7"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000758,"raw_usage":{"total_tokens":3359,"prompt_tokens":928,"completion_tokens":2431,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":2343}},"tokens_in":544,"tokens_out":2431,"duration_ms":16426,"temperature":1.0,"reasoning_tokens":2343,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:16:20.006444+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search the paper's own electronic data for a non-CM curve whose 2-, 3-, and 5-adic images are all nonsurjective and whose j-invariant is not one of the six values in J; even one such curve would disprove the classification. Alternatively, compute the 7-adic image of a curve with j-invariant in J: if it is nonsurjective, or if a smaller prime turns out to be surjective, the claim that 7 is the smallest surjective prime for those curves fails.","supporting_citations":[{"cited_title":"Sutherland and David Zywina, Modular curves of prime-power level with infinitely many rat ional points, Algebra Number Theory 11 (2017), no","cited_arxiv_id":null,"evidence_quote":"Supplies the complete list of maximal closed subgroups of GL2(Zℓ) with surjective determinant for ℓ = 2, 3, 5 and the j-maps of their modular curves (Table 1), which is the enumeration every fiber product starts from."},{"cited_title":"Sutherland, and David Zureick-B rown, ℓ-adic images of Galois for elliptic curves over Q (and an appendix with John Voight) , Forum Math","cited_arxiv_id":null,"evidence_quote":"Provides the ℓ-adic image labels used to identify the images of the six exceptional curves and to check that all primes ℓ ≥ 7 are surjective."},{"cited_title":"Sutherland, and David Zureick-B rown, ell-adic-galois-images, https://github.com/AndrewVSutherland/ell-adic-galois-images , 2021, GitHub repository","cited_arxiv_id":null,"evidence_quote":"Computes the ℓ-adic Galois images of specific elliptic curves, giving the nonsurjectivity at 2, 3, 5 and surjectivity at 7 for the exceptional j-invariants."},{"cited_title":"Daniels and Enrique Gonz´ alez-Jim´ enez, Serre’s constant of elliptic curves over the rationals , Exp","cited_arxiv_id":null,"evidence_quote":"Introduces the curious-groups phenomenon and supplies rational-point results on several fiber products that let the proof discard or reduce cases."},{"cited_title":"A classification of curious Galois groups as direct products","cited_arxiv_id":"2310.19987","evidence_quote":"Classifies curious Galois groups as direct products, used to reduce one fiber product case to a curve whose rational points are already known."},{"cited_title":"Number Theory 130 (2010), no","cited_arxiv_id":null,"evidence_quote":"Shows all rational points on a key non-split Cartan fiber product are CM, dismissing one remaining case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the model-construction and point-finding routines used for fiber products that need better models than the direct j-map equation."},{"cited_title":"Morrow, Composite images of Galois for elliptic curves over Q and entanglement fields , Math","cited_arxiv_id":null,"evidence_quote":"Yields infinitely many j-invariants with smallest surjective prime exactly 5, establishing sharpness of the generic bound."}],"review_version":1}