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Divisibility Biases in the Orders of Elliptic Curve Reductions

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Pith's one-line read For Serre curves, the density of primes where m divides the order of an elliptic curve reduction is explicitly determined, strictly exceeds 1/m for every m≥2, and matches the average density apart from small correction terms.

desk verdict Solid, genuinely new explicit formulas for Serre-curve divisibility densities; the main caveat is a load-bearing but legitimate citation to Jones's 2-adic identity, and the Magma checks skip the exceptional cases. read the letter →

arxiv 2606.25067 v2 pith:JYYLUR7Z submitted 2026-06-23 math.NT

Sung Min Lee , Nara Sheen This is my paper · ORCID
classification math.NT MSC 11G0511F8011N05
keywords ellipticcurvereductionsm-divisibilitydensitySerrecurvesGaloisrepresentationsChebotarevtheoremaverage2-adiccharactersgroupordersmoduloprimes
topics The Twin Prime Conjecture
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

This paper gives an explicit formula for the natural density of primes p for which m divides the number of F_p-points on a fixed elliptic curve over the rationals, in the generic case of a Serre curve. It shows this density is approximately 1/φ(m) and is always strictly larger than the naive random-integer prediction of 1/m, revealing a systematic bias toward divisibility. The formula expresses the density as the average density C^{m-div} plus a correction term that vanishes except when the adelic level divides m and the 2-adic part of m lies in a narrow exceptional range. It also proves that averaging these densities over large boxes of Weierstrass equations reproduces the average density C^{m-div}, confirming that the bias persists on average. The picture that emerges is that the order of an elliptic curve reduction behaves more like the multiplicative group of a finite field than like a random integer.

What carries the argument

The m-divisibility condition m | #E_p(F_p) is equivalent, for good primes p, to the matrix condition det(I − ρ_{E,m}(Frob_p)) ≡ 0 (mod m), so the density is |G_E(m) ∩ Ψ(m)| / |G_E(m)|, where Ψ(m) is the set of matrices with determinant of I − M equal to 0 mod m. For Serre curves, G_E(m) is either the full group GL_2(Z/mZ) or the index-2 subgroup H_E(m) = ker ψ_m, where ψ_m is an explicit product of quadratic characters — the Legendre symbol of the determinant at odd primes, the sign of the permutation action on the three 2-torsion roots, and the characters χ_4 and χ_8 at the prime 2 — depending on the congruence class of the squarefree discriminant Δ'_E modulo 4 and 8. The proof reduces to c

What would settle it

For a specific Serre curve with Δ'_E ≡ 3 (mod 4) and v_2(m) = 2, or with Δ'_E ≡ 2 (mod 4) and v_2(m) ∈ {3, 4}, compute the mod-16 (or mod-8) Galois image directly from the defining Weierstrass equation by listing the Frobenius matrices that occur, and compare this image with the kernel of the character ψ_m defined in equation (15). If any matrix in the kernel is not attained by a Frobenius element (or vice versa), the identity H_E(m) = ker ψ_m fails for that curve and the corresponding exceptional-case formula in Theorem 3 collapses; the paper cites this identity but does not prove it.

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Extended reading notes

Core claim

For a Serre curve — an elliptic curve over Q whose adelic Galois image is as large as possible, namely the index-2 subgroup forced by the quadratic field Q(√Δ_E) — Theorem 3 provides a closed formula for C^{m-div}_E, the natural density of good primes p with m | #E_p(F_p). Writing m = m_1 m_2 with m_1 = gcd(m, m_E^∞), the density equals the average density C^{m-div} when m_1 is outside a small exceptional 2-adic range; otherwise it is (C^{m1-div} + ∏_{ℓ^α∥m1} −ℓ^{2α−1}/|GL_2(Z/ℓ^αZ)|) · C^{m2-div}. This formula yields Corollary 4: C^{m-div}_E > 1/m for every m ≥ 2. Theorem 5 then shows that the k-th power average of |C^{m-div}_E − C^{m-div}| over the box family tends to zero as the box grows

Load-bearing premise

The entire computation of the correction terms rests on a cited, unproved description of the index-2 Galois subgroup H_E(m) as the kernel of an explicit quadratic character for levels between the adelic level and its powers; if the 2-adic part of that description is wrong for certain discriminants and 2-adic valuations, the exceptional cases in Theorem 3 would require different corrections.

Editorial extensions

If this is right

  • For every Serre curve and every m ≥ 2, the m-divisibility density C^{m-div}_E is strictly larger than 1/m, confirming globally a bias previously seen in local weighted models over finite fields.
  • The average density C^{m-div} equals the density one would obtain from a hypothetical elliptic curve with surjective adelic Galois image, so the generic Galois image governs all averaged statistics of this kind.
  • For m odd or m ∈ {2, 4, 8, 16}, every Serre curve satisfies C^{m-div}_E = C^{m-div}: the correction term vanishes, making the average density exact for these moduli.
  • The explicit formulas enable direct numerical computation of divisibility densities for individual curves; for example, a Serre curve of adelic level 6 has C^{6-div}_E = 5/16 ≈ 0.3125, roughly twice the naive 1/6 prediction and about 7% above the average value.
  • The maximal upward deviation from the average occurs at m = 6 for curves with Δ'_E = −3, and the maximal downward deviation at m = 30 and m = 32, giving quantitative benchmarks for how far an individual curve can depart from the average.

Reading between the lines

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

  • The same ratio-counting technique should extend to the distribution of #E_p(F_p) modulo m in other residue classes: if the class 0 is systematically overweighted, some other classes must be underweighted, suggesting a universal bias pattern that could be made precise by the methods of this paper.
  • The reliance on a cited, unproved 2-adic description of H_E(m) means the exceptional cases are the most fragile part; an independent verification of H_E(m) = ker ψ_m for the specific 2-adic levels would fully de-risk the formula.
  • The quantitative bounds in Theorem 5 hint that the decay rate of the average discrepancy is controlled by the count of curves with small discriminant; sharpening that count (e.g., via sieving over squarefree discriminants) would improve the rate and possibly give a stronger almost-sure statement for random curves in the box.
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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

0 major / 6 minor

Summary. The paper studies the natural density C_E^{m-div} of primes p for which m divides #E_p(F_p), where E/Q is an elliptic curve. For Serre curves, it gives an explicit formula (Theorem 3) expressing C_E^{m-div} as the average density C^{m-div} in certain 'exceptional' cases, and otherwise as C^{m-div} plus a correction factor depending on the adelic level m_E. It proves (Corollary 4) that C_E^{m-div} > 1/m for every m ≥ 2, confirming a bias toward m-divisibility predicted by Howe's local model. It also proves (Theorem 5) that the average of C_E^{m-div} over the box family F(A,B) converges to C^{m-div} with explicit error terms, using Jones's result that almost all elliptic curves are Serre curves. The proofs combine Chebotarev density, explicit counting in GL2(Z/mZ) via Haar measure and character sums, and detailed 2-adic/local computations.

Significance. If correct, the paper provides a clean, explicit refinement of results of Cojocaru and of Banks--Shparlinski, and gives quantitative confirmation that the m-divisibility density is governed by 1/φ(m) rather than 1/m. The identification in Proposition 25 of the average density C^{m-div} with the full-image density C_E^{m-div} is a valuable independent check connecting the local and average viewpoints. The local counting lemmas (notably Lemmas 20, 43, 44 and the 2-adic Lemmas 32--38) are detailed and internally consistent; spot checks for ℓ=2,3 and small α match. The paper also includes numerical examples for m=6,30,32 that are consistent with the theoretical formulas. The main limitation is the reliance on the quoted identity (16) from Jones [13] for the structure of HE(m), especially in the exceptional cases of Theorem 3; however, this is a citation rather than an internal inconsistency, and the surrounding derivations are coherent.

minor comments (6)
  1. [§2.2, Eq. (16)] Theorem 3's exceptional cases depend on the identity HE(m)=ker ψ_m, quoted from Jones [13] without proof. The numerical examples in §4 are all nonexceptional (m=6,30,32), so the cases where C_E^{m-div}=C^{m-div} in Theorem 3 are not independently checked. I do not regard this as a correctness gap, but the authors should state (16) as a named lemma with a precise pointer to [13, Section 4], and ideally add one computational check in an exceptional case (e.g., m=8 with Δ'_E≡2 mod 4, or m=12 with Δ'_E≡3 mod 4).
  2. [Proof of Corollary 4] The displayed formula for H(ℓ^α)-F(ℓ^α)-1 is algebraically incorrect. For ℓ=2, α=2 it gives 1/3, whereas the true value is 5/9. In the subsequent v2(m1)=1 case, the equality H(m1)-F(m1)=4/3 H(n)-2/3 F(n) should read 4/3 H(n)-2/9 F(n), since F(2)=2/9. The conclusion H(m1)-F(m1)>1 remains true, but these computations need to be corrected.
  3. [§2.3, proof of Lemma 19] The reference 'Proposition 12' in the first sentence should be 'Lemma 12'.
  4. [§3.2, proof of Lemma 20] The last line contains a repeated phrase: 'we obtain the desired results' appears twice before the □. Please clean up the closing sentence.
  5. [§4, Example 47] The notation 'm1=25', 'm1=23', 'm1=24' should be typeset as 2^5, 2^3, and 2^4 respectively; otherwise the sentence is confusing.
  6. [Proof of Theorem 3] In the paragraph after (22), the condition 'v2(m1)≤4' for Δ'_E≡2 mod 4 should be stated as 'v2(m1)∈{3,4}' to match Theorem 3 and Proposition 39; the two are equivalent here because m_E|m forces v2(m1)≥3, but the wording is sloppy.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular construction; central density formulas are computed from first principles, with only minor self-citations that are not load-bearing reductions.

full rationale

The derivation of Theorem 3 starts from the Chebotarev density expression (18), C^{m-div}_E = |G_E(m)∩Ψ(m)|/|G_E(m)|, and then counts local intersections. Lemma 19 factors the count via the CRT, Proposition 27 applies Jones's character ψ_m, and Propositions 39/44 give explicit counts |Y_{ℓ^α,+}|−|Y_{ℓ^α,−}| from elementary determinant/trace computations. The resulting formula is not a fitted input renamed as a prediction: the average density C^{m-div} is computed independently from the Banks–Shparlinski expression (20) in Lemmas 23–24 and Proposition 25, then compared with the Serre-curve formula. The weakest input, the identity H_E(m)=ker ψ_m in (16), is quoted from Jones [13] and is a genuine external theorem; a failure there would be a correctness dependency, not a circular reduction. Citations to [19] and [10] (papers overlapping with the first author) supply standard Galois-image lemmas—Lemma 12, Proposition 16, Lemma 15—whose statements do not contain C^{m-div}_E and which do not force the formula by construction. The Magma checks in Examples 46–48 are illustrative consistency checks rather than fitted inputs. Hence no step reduces to its own input; the only concerns are external validity (Eq. (16)) and minor self-citation, so the score is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; all constants are derived from Galois group structure. The axioms are standard or published results (Serre, Jones, Lombardo–Perucca, Banks–Shparlinski) used as tools, not as the target result.

assumptions (6)
  • standard math Serre's open image theorem: for non-CM E, GE is open in GL2(Ẑ), so an adelic level mE exists (Theorem 10)
    Used to define mE and to reduce GE to finite level; standard in the field.
  • domain assumption Factorization GE(m1m2) ≃ GE(m1) × GL2(Z/m2Z) when gcd(m2,mE)=1 (Lemma 12)
    Cited from [19, Lemma 2.2]; justifies the prime-power split in Lemma 19 and Theorem 3.
  • domain assumption Jones's description HE(m)=ker ψ_m for mE|m|m∞_E (eq. (16))
    The entire correction term in Theorem 3 is computed from this description of the Serre curve Galois image; not proved in this paper.
  • standard math Haar-measure distribution of the 1-eigenspace in GL2(Z_ℓ) (Lombardo–Perucca [22, Lemma 23,25, Theorem 2])
    Used in Lemma 20 to compute |Ψ(ℓ^α)|/|GL2|; the result is cross-checked against Banks–Shparlinski's average formula.
  • domain assumption Banks–Shparlinski averaged density C^{m-div} as defined by (20) and their Theorem 2
    The paper's Theorem 5 compares individual densities to this quantity; the explicit evaluation in Proposition 25 is new.
  • domain assumption Jones's bound on the proportion of non-Serre curves (Theorem 25 of [13]) used in Theorem 5
    Provides the error term for the non-Serre contribution in the average.

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Pith. "Pith review of Divisibility Biases in the Orders of Elliptic Curve Reductions." pith.science (2026). https://pith.science/paper/JYYLUR7Z

@misc{pith2026260625067,
  author       = {Pith},
  title        = {Pith review of: Divisibility Biases in the Orders of Elliptic Curve Reductions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JYYLUR7Z}},
  note         = {Machine review of arXiv:2606.25067}
}
abstract

Let $E$ be an elliptic curve over the rationals. In 2004, Cojocaru proved, using the Chebotarev density theorem, that the set of primes $p \leq x$ for which $m$ divides $\#E_p(\mathbb{F}_p)$ has a natural density. In 2009, Banks and Shparlinski proved an averaged version of this result over families of elliptic curves. In this article, we give a more explicit analysis of these densities. In particular, we show that, for Serre curves, the density of primes $p$ for which $m \mid \#E_p(\mathbb{F}_p)$ is approximately $1/\varphi(m)$, and is always greater than $1/m$ for every $m \geq 2$. Thus, the orders $\#E_p(\mathbb{F}_p)$ exhibit a bias toward divisibility by $m$. Finally, based on Jones' method, we prove that the average of the individual $m$-divisibility densities coincides with the average density proposed by Banks and Shparlinski.

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