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A uniform bound on the smallest surjective prime of an elliptic curve

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.02345 v2 pith:BRT34MTK submitted 2025-01-04 math.NT

Tyler Genao , Jacob Mayle , Jeremy Rouse This is my paper · ORCID
classification math.NT MSC 11G0511F80
keywords non-CMellipticcurvesGaloisrepresentationssurjectiveprimesSerreuniformitymodularrationalpointsChabauty'smethodℓ-adicimages
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 2, Lemma 5; used in Section 3] 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.
  2. [Section 3, rational-point computations] 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.
minor comments (5)
  1. [Section 3, third tree and Table 5] 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).
  2. [Section 1 and repository footnote] 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.
  3. [Section 2, Table 1] 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.
  4. [Tables 2–6] Rows with j-invariant ∞ are cusps; labeling them explicitly as 'cusp' in the j-invariant column would make the tables easier to read.
  5. [Section 2, proof of Lemma 5] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central rational-point computation is self-contained and the self-citations are to public, independently developed databases.

full rationale

The paper's main derivation is a finite rational-point search over fiber products of modular curves; Theorem 2's dichotomy (j in J vs. smallest surjective prime at most 5) is read off the computed rational places, not imposed by the definition of J. The exceptional set J is discovered by the computation, and the claim that those six j-invariants have smallest surjective prime exactly 7 is verified against the RSZB database entries for specific curves. That verification is a use of a public, parameter-free database, not a fitted input; [23] and [22] are self-citations only insofar as Rouse is a coauthor, but they are independently developed and machine-checked in the broader computational number theory community, so they do not make the argument circular. The rational-point computations themselves (ranks, Chabauty0, local solubility) are new code in a public repository. One non-circular weakness exists: Lemma 5's proof conflates Q-isomorphism with quadratic twisting and does not correctly prove the twist-invariance needed to pass from rational places on fiber products to all curves with a given j-invariant. This is a correctness or completeness gap, not a reduction of the theorem to its own assumptions, and it can in principle be repaired by a group-theoretic argument; it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

All entries are background assumptions pulled from cited literature or from standard computational number theory. The central computation (rational points on fiber products) is new, but its preconditions include the completeness of the maximal subgroup enumeration, the correctness of published j-maps, the reliability of the Magma implementations, and the truth of twist-invariance. None of these are derived in this paper.

assumptions (8)
  • standard math Serre's open image theorem ensures finiteness of nonsurjective primes (Theorem 1)
    Invoked in the introduction to frame the problem; does not enter the finite computation.
  • domain assumption RSZB classification: Table 1 lists all maximal closed subgroups of GL2(Zℓ) with surjective determinant for ℓ=2,3,5
    The proof enumerates 54 fiber products using these maximal subgroups; a missing subgroup would invalidate the classification.
  • domain assumption Sutherland-Zywina j-map formulas for XH to P1 for the subgroups in Table 1
    All fiber product models are constructed by equating these j-maps (Example 4); any error propagates through the rational point computations.
  • domain assumption Correctness of the ell-adic-galois-images database [22] for the six exceptional j-invariants
    The paper states that the database shows nonsurjectivity at 2,3,5 and surjectivity at 7 for each j in J; this is a finite check but is not independently reproduced.
  • domain assumption Baran's theorem on rational points of X+ns(4) times X+ns(5) [3]
    Used to rule out non-CM points on several fiber products involving X+ns(4) times X+ns(5).
  • domain assumption Curious group results of Daniels-Gonzalez-Jimenez [9] and Chiloyan [8]
    Used to reduce rational points on X+ns(3) times X+ns(4) times XS4(5) and related curves to previously analyzed cases.
  • domain assumption Correctness of Magma implementations (Chabauty0, IsLocallySolvable, birational maps, rank determination)
    The rational point computations on genus 1 and genus 2 curves are delegated to Magma; no independent certificates are provided.
  • standard math Twist-invariance: quadratic twists do not change ℓ-adic surjectivity for non-CM elliptic curves
    Needed to classify by j-invariant rather than by isomorphism class. The paper's Lemma 5 attempts to prove this but states it incorrectly; the underlying fact is standard and true.

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Pith. "Pith review of A uniform bound on the smallest surjective prime of an elliptic curve." pith.science (2026). https://pith.science/paper/BRT34MTK

@misc{pith2026250102345,
  author       = {Pith},
  title        = {Pith review of: A uniform bound on the smallest surjective prime of an elliptic curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRT34MTK}},
  note         = {Machine review of arXiv:2501.02345}
}
abstract

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the $\ell$-adic Galois representation $\rho_{E,\ell^\infty}$ is surjective for all but finitely many prime numbers $\ell$. Considerable work has gone into bounding the largest possible nonsurjective prime; a uniform bound of $37$ has been proposed but is yet unproven. We consider an opposing direction, proving that the smallest prime $\ell$ such that $\rho_{E,\ell^\infty}$ is surjective is at most $7$. Moreover, we completely classify all elliptic curves $E/\mathbb{Q}$ for which the smallest surjective prime is exactly $7$.

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