The density and distribution of CM elliptic curves over $\mathbb{Q}$

Adrian Barquero-Sanchez; Jimmy Calvo-Monge

The density and distribution of CM elliptic curves over $\mathbb{Q}$

T0 review · 0 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that CM elliptic curves over $\mathbb{Q}$ have natural density 0 in the family of all elliptic curves ordered by naive height, and among CM curves, those with $j$-invariant 0 have natural density 1.

desk verdict A clean, correct counting argument that turns the folklore heuristic into a theorem; send to a serious referee. read the letter →

arxiv 2411.13526 v1 pith:P6KSHC2E submitted 2024-11-20 math.NT math.AG

Adrian Barquero-Sanchez , Jimmy Calvo-Monge This is my paper · ORCID

classification math.NTmath.AG MSC 11G0511G1511N45

keywords ellipticcurvescomplexmultiplicationnaturaldensitynaiveheightj-invariantCMordersofclassnumberoneintegralpointsoncuspidalcubicspower-freeintegers

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

Complex multiplication is rare among elliptic curves over $\mathbb{Q}$: the paper proves that when all curves $y^2=x^3+Ax+B$ in minimal short Weierstrass form are ordered by the naive height $h_{\mathrm{naive}}=\max\{4|A|^3,27|B|^2\}$, the CM curves have natural density $0$. It then proves that among the thirteen CM orders of class number one, the curves with $j$-invariant $0$ have natural density $1$ inside the CM family, so asymptotically all CM curves over $\mathbb{Q}$ are the $j=0$ curves with CM by $\mathbb{Z}[( -1+\sqrt{-3})/2]$. The proof gives asymptotic counts for the number of curves with a fixed $j$-invariant: the $j=0$ family contributes a main term of order $X^{1/2}$, the $j=1728$ family of order $X^{1/3}$, and each of the other eleven CM $j$-invariants at most $O(X^{1/6})$, against a total of order $X^{5/6}$. These exponents force both density statements, and the same 100% conclusion is obtained in an alternative family of representatives built from the theory of twists.

What carries the argument

The central mechanism is counting integral points on cuspidal cubics. For a fixed $j\neq 0,1728$, the condition $j(E_{A,B})=j$ is equivalent to the point $(A,B)$ lying on the curve $y^2=\frac{1728-4j}{27j}x^3$. Lemma 2.3 shows that any rational cuspidal cubic $y^2=a x^3$ with $a=p/q$ in lowest terms has at most $2\sigma_0(q)\sqrt{|pq|}\,T^{1/2}$ integral points with $|x|\le T$; the proof parametrizes the curve by $t\mapsto (t^2/a,t^3/a)$ and observes that integrality forces the denominator of $t$ to divide $q$. Applied with $a=(1728-4j)/(27j)$ and $T=X^{1/3}/2^{2/3}$, this bounds each of the eleven non-exceptional CM $j$-invariants by $O(X^{1/6})$. For $j=0$ and $j=1728$, the curves are forced to have $A=0$ or $B=0$, respectively, so the count reduces to counting $6$-th and $4$-th power-free integers, giving the $X^{1/2}$ and $X^{1/3}$ main terms. The total $\#\mathcal{E}(X)$ is Brumer's lattice-point count of order $X^{5/6}$, and the thirteen CM $j$-invariants come from the classification of orders of class number one.

What would settle it

Compute $\#\mathcal{E}_j(X)$ for one fixed CM $j$-invariant $j\neq 0,1728$, such as $j=-3375$, up to $X=10^{12}$ using the counting code described in the paper; if the count grows faster than a constant multiple of $X^{1/6}$ (for instance like $X^{1/3}$ or $X^{1/2}$), the pivotal Lemma 2.3 bound fails and the 100% $j=0$ density conclusion would not follow. A more direct check is to search for integral points on $y^2=\frac{1728-4j}{27j}x^3$ with $|x|\le T$ and confirm whether their number exceeds the bound $2\sigma_0(q)\sqrt{|pq|}\,T^{1/2}$ from the lemma.

Watch

Extended reading notes

Core claim

The paper establishes that the natural density of the set $\mathcal{E}^{\mathrm{cm}}$ of CM elliptic curves inside the family $\mathcal{E}$ of minimal short Weierstrass models over $\mathbb{Q}$ is $0$, and that the natural density of the subfamily $\mathcal{E}_0$ with $j$-invariant $0$ inside $\mathcal{E}^{\mathrm{cm}}$ is $1$. In concrete terms, asymptotically none of the elliptic curves over $\mathbb{Q}$, ordered by naive height, have complex multiplication, and asymptotically all of the CM curves that do occur have $j=0$. This follows from the asymptotic formulas $\#\mathcal{E}_0(X)=\frac{2}{3^{3/2}\zeta(6)}X^{1/2}+O(X^{1/12})$, $\#\mathcal{E}_{1728}(X)=\frac{2^{1/3}}{\zeta(4)}X^{1/3}+O(X^{1/12})$, $\#\mathcal{E}_j(X)=O(X^{1/6})$ for the other eleven CM $j$-invariants, and Brumer's total $\#\mathcal{E}(X)=\frac{2^{4/3}}{3^{3/2}\zeta(10)}X^{5/6}+O(X^{7/12})$. Combining these gives $\#\mathcal{E}^{\mathrm{cm}}(X)=\frac{2}{3^{3/2}\zeta(6)}X^{1/2}+\frac{2^{1/3}}{\zeta(4)}X^{1/3}+O(X^{1/6})$, so the CM subfamily grows like $X^{1/2}$ inside an $X^{5/6}$ family and the $j=0$ subfamily dominates the CM subfamily at the density level.

Load-bearing premise

The load-bearing estimate is Lemma 2.3's bound that a cuspidal cubic $y^2=a x^3$ with fixed rational $a$ has only $O(T^{1/2})$ integral points with $|x|\le T$; if this bound failed, the eleven CM $j$-invariants other than $0$ and $1728$ could contribute a larger share and the claim that the $j=0$ curves form 100% of the CM family asymptotically could fail.

Editorial extensions

If this is right

The natural density of CM elliptic curves in the full family $\mathcal{E}$ is $0$, so complex multiplication is asymptotically negligible among minimal short Weierstrass models ordered by naive height.
Inside the CM family, the curves with $j=0$ have natural density $1$, so asymptotically every CM elliptic curve over $\mathbb{Q}$ has endomorphism ring $\mathbb{Z}[( -1+\sqrt{-3})/2]$.
The number of CM curves up to height $X$ grows like $\frac{2}{3^{3/2}\zeta(6)}X^{1/2}$, much slower than the $X^{5/6}$ growth of the total family.
In the twist-based family $\mathcal{E}^{\mathrm{cm}}_T$, the $j=0$ curves again have density $1$, and the counts for every $j\in J^{\mathrm{cm}}$ have explicit main terms $C(j)X^{1/m(j)}$ with $C(j)=\frac{2}{\zeta(n(j))}h_{\mathrm{naive}}(E_j)^{-1/m(j)}$.

Reading between the lines

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

Since the proof gives only an $O(X^{1/6})$ bound in the minimal-Weierstrass family for the eleven non-exceptional CM $j$-invariants, while the twist family yields an explicit main term, a natural next step is to determine the true asymptotic for $\#\mathcal{E}_j(X)$ for these $j$; the constants $C(j)$ from the twist family provide a candidate for the leading coefficient.
The density conclusions are tied to the box-shaped height balls defined by $h_{\mathrm{naive}}$; under other standard orderings, such as by conductor or Faltings height, the relative growth of CM curves could differ, since the lattice-point mechanism depends on the precise box geometry.
The twist-based family $\mathcal{E}_T$ separates the $j$-invariant from a power-free twist parameter and may be the more convenient setting for statistical questions about CM curves, such as average ranks or Selmer groups, because the counting constants $C(j)$ are explicit in that family.
The dominance of the order $\mathbb{Z}[( -1+\sqrt{-3})/2]$ among the thirteen class-number-one orders suggests a general phenomenon: among CM structures over $\mathbb{Q}$, the order with the largest unit group dominates at the level of natural density; one could test whether analogous dominance holds for CM abelian varieties of higher dimension.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Short version: this is a serious paper and the main claims hold. The authors prove that CM elliptic curves over Q have natural density zero in the standard minimal Weierstrass family ordered by naive height, and that among CM curves, 100% asymptotically have j=0. The explicit asymptotic for #E_cm(X) — with constants 2/(3^{3/2}ζ(6)) X^{1/2} + 2^{1/3}/ζ(4) X^{1/3} — and the fixed-j counts are not in Brumer or anywhere else I know.

What it does well: the counting is clean and self-contained. The j=0 and j=1728 cases reduce to k-free integer counts with zeta constants; the other eleven j-invariants are handled by a lattice-point bound on y^2 = a x^3. The error terms are tracked, no fitted constants, no circularity. The density-zero result and the 100% j=0 statement follow directly from those asymptotics. Section 4 redoes the count in a twist-based family and gets a genuine X^{1/6} leading term for j≠0,1728, which is a nice complement.

Soft spots, all minor. There is a slip in equation (2.25): the coefficient should be 4(1728-j)/(27j), not (1728-4j)/(27j). Since j is fixed and nonzero for the eleven CM values, the exponent in the O(X^{1/6}) bound is unaffected, so no repair is needed beyond fixing the typo. Section 4's constants C(j) depend on the chosen representative E_j; the authors acknowledge this only in a remark, and a careful statement about which quantities are representation-independent would help. The code is provided but not versioned; a commit hash would make the numerics reproducible. The numerics themselves match the main terms well.

One thing I pushed on: the stress-test concern about Lemma 2.3 being load-bearing. It is sound, and the parametrization argument is correct. The 100% j=0 conclusion really does rest on that O(X^{1/6}) bound for the other eleven j-invariants, but the bound is proved properly, so I don't consider it a hidden assumption.

Who should read this: arithmetic geometers and number theorists interested in CM curves or average ranks. It is a modest but real contribution, and the 100% j=0 fact is likely to be cited. Verdict: accept, with the typo fixed and representative-dependence made explicit. I would send it to a serious referee; it deserves one.

Referee Report

0 major / 6 minor

Summary. The paper studies the density and distribution of CM elliptic curves over Q in the family E of minimal short Weierstrass models, ordered by the naive height h = max{4|A|^3, 27|B|^2}. The main results are: (1) the natural density of CM curves in E is zero (Theorem 1.1); (2) among CM curves, those with j-invariant 0 have asymptotic density 1 (Theorem 1.5); (3) for j=0 and j=1728, precise asymptotic counts are obtained, and for the other eleven CM j-invariants an upper bound O(X^{1/6}) is proved (Theorem 1.6/2.4); and (4) an alternative family of representatives built from twists is analyzed, yielding leading terms for every CM j-invariant (Theorem 4.4). The proofs are elementary, using k-free counts and a lattice-point estimate on cuspidal cubics.

Significance. If correct, the paper establishes a striking and previously unquantified phenomenon: although infinitely many Q-isomorphism classes of CM elliptic curves exist for each of the thirteen class-number-one orders, the j=0 family absorbs all the mass when curves are ordered by naive height, and CM curves are negligibly rare among all elliptic curves. The arguments are internally consistent and free of fitted parameters; the leading constants are exact expressions in zeta values. A particular strength is that the main load-bearing estimate (Lemma 2.3 on integral points on cuspidal cubics) is proved in detail and is sound. The paper also provides reproducible code, which is a plus. The claimed results follow from the stated lemmas, and I found no circular reasoning.

minor comments (6)

[§2, Eq. (2.25)] In the proof of Theorem 2.4, solving j = 1728 * 4A^3 / (4A^3 + 27B^2) for B^2 gives B^2 = 4(1728-j)/(27j) A^3, so the coefficient displayed in Eq. (2.25) should be 4(1728-j)/(27j), not (1728-4j)/(27j). The error does not affect the subsequent O(X^{1/6}) bound because the corrected constant is still a fixed nonzero rational for the eleven j-invariants in question, but the displayed equation should be corrected.
[§2, proof of Theorem 2.4, Eq. (2.24)] In the line following Eq. (2.24), the simplification 2/(2^{2/3}ζ(4))X^{1/3} = 2^{1/2}/ζ(4) X^{1/3} is incorrect; the correct exponent is 2^{1/3}, consistent with the theorem statement. This is a typographical slip only.
[§2, Lemma 2.3] The explicit constant in Eq. (2.10), 2σ0(q)√|pq| T^{1/2}, is not valid uniformly for every T>0 because the count of integers in a bounded interval gives 2N+1 rather than 2N; the O(T^{1/2}) statement in Eq. (2.11) is correct, but the lemma should restrict to T ≥ 1 or use a slightly larger constant.
[§4, Table 6] The row for dK=-7, f=1 (j=-3375) in Table 6 reports ET_j(10^10)=0, which is inconsistent with Table 4 (E_j(10^10)=8) and with Theorem 4.4 using the base curve y^2=x^3-35x+98 of height 259308, which would predict 8 twists. This entry should be checked and corrected.
[§2, proof of Theorem 2.1, Eq. (2.2)] The decomposition D'(X) = ⨆_{d≤X^{1/12}} d * M(d^{-12}X) is correct, but the uniqueness of the representation is not justified in the text; a one-sentence explanation would help the reader.
[Title] The running title contains the typo 'CUR VES'; it should be 'CURVES'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the CM-count estimates are derived from Möbius inversion, k-free counting, and a self-contained lattice-point estimate, with no fitted parameter renamed as a prediction.

full rationale

The derivation is self-contained rather than circular. Theorem 1.3 and Theorem 1.5 are obtained by counting, for each of the thirteen class-number-one CM j-invariants, the lattice points (A,B) satisfying the j-invariant equation and the minimality condition. For j=0 and j=1728 these counts are reduced to 6-free and 4-free integers via the standard estimate Q_k(X)=X/zeta(k)+O(X^(1/k)); for the other eleven j-invariants the count is bounded by Lemma 2.3, whose proof is given in the paper and does not appeal to the theorem it supports. The total number #E(X) in the denominator is taken from Brumer's 1992 result, and the argument is also reproduced in Theorem 2.1. The leading constants are exact zeta values, and no constant is fitted to the data whose density is being predicted. The external classification of the thirteen CM orders (Heegner, Baker, Stark, Cox) is independent background, and no load-bearing step reduces to a self-citation. The paper even discloses the dependence of the twist-family counts on the choice of representative E_j (Remark 4.3), and that dependence is harmless because the X^(1/2) term from j=0 dominates the X^(1/3) and X^(1/6) terms. One minor algebraic slip occurs at Eq. (2.25), where the coefficient should be 4(1728-j)/(27j) rather than (1728-4j)/(27j); for each fixed nonzero, non-1728 CM j-invariant this remains a fixed nonzero rational, so Lemma 2.3 still yields the same O(X^(1/6)) bound and the conclusion is unaffected. This is a correctness typo, not a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest entirely on standard analytic number theory and standard CM and twist theory; no new constants are fitted and no new entities are introduced. The only non-canonical choice is the representative family E and the twist representatives in Section 4, which are explicitly acknowledged in the paper.

assumptions (5)

standard math If E/Q has CM by an order O, then [Q(j(E)):Q] = h(O), so the endomorphism order must have class number one; there are exactly thirteen such orders.
Used in Section 1.2 to reduce E_cm to the thirteen j-invariants in J_cm. This is standard CM theory (Silverman, Cox) and the class-number-one classification of Heegner, Baker, Stark.
standard math Every Q-isomorphism class of elliptic curves over Q has a unique representative of the form y^2=x^3+Ax+B with integers A,B, Delta != 0, and no prime p with p^4|A and p^6|B.
Defines the family E and underlies the natural density in Theorems 1.1 and 1.5; cited to Washington, Silverman, Poonen.
standard math The number of positive k-free integers up to Y is Y/zeta(k) + O(Y^(1/k)) (Gegenbauer).
Used in Proposition 2.2 and in the proofs of Theorems 2.4 and 4.4 to pin down the leading constants.
standard math For a curve with j-invariant j, the set of twists over Q is classified by Q^*/ (Q^*)^{n(j)}, with n(j)=2,4,6, and the explicit twist equations alter coefficients as in Silverman X.5.4.
Used in Sections 3 and 4 to build the alternative family E^T and to compute h_naive(E_j^D) = |D|^{m(j)} h_naive(E_j).
standard math Mobius inversion and 1/zeta(s) = sum mu(n)/n^s.
Used in the proof of Theorem 2.1 to pass from all lattice points to minimal representatives, the family E.

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Pith. "Pith review of The density and distribution of CM elliptic curves over $\mathbb{Q}$." pith.science (2026). https://pith.science/paper/P6KSHC2E

@misc{pith2026241113526,
  author       = {Pith},
  title        = {Pith review of: The density and distribution of CM elliptic curves over $\mathbbQ$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P6KSHC2E}},
  note         = {Machine review of arXiv:2411.13526}
}

abstract

In this paper we study the density and distribution of CM elliptic curves over $\mathbb{Q}$. In particular, we prove that the natural density of CM elliptic curves over $\mathbb{Q}$, when ordered by naive height, is zero. Furthermore, we analyze the distribution of these curves among the thirteen possible CM orders of class number one. Our results show that asymptotically, $100\%$ of them have complex multiplication by the order $\mathbb{Z}\left[\frac{-1 + \sqrt{-3}}{2} \right]$, that is, have $j$-invariant 0. We conduct this analysis within two different families of representatives for the $\mathbb{Q}$-isomorphism classes of CM elliptic curves: one commonly used in the literature and another constructed using the theory of twists. As part of our proofs, we give asymptotic formulas for the number of elliptic curves with a given $j$-invariant and bounded naive height.

Figures

Figures reproduced from arXiv: 2411.13526 by the authors.

**Figure 1.** A rational parametrization of the cuspidal cubic Ca : y 2 = ax3 . The graph shows a typical curve with a < 0. Indeed, note that when we substitute y = tx into the equation for Ca we get t 2x 2 = ax3 ⇐⇒ x 2 (ax − t 2 ) = 0 ⇐⇒ x = 0 or x = t 2 a . This parametrization gives us a bijection ϕ: R −→ Ca t 7−→ Pt := t 2 a , t 3 a . (2.12) Moreover, note that restricting ϕ to Q also gives us a bijection ϕe: Q −→ Ca(Q), … view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Introduction to Analytic Number Theory

Tom Apostol. Introduction to Analytic Number Theory . Springer-Verlag, 1976

work page 1976

[2] [2]

A. Baker. Linear forms in the logarithms of algebraic numbers. I , II , III . Mathematika , 13:204--216; ibid. 14 (1967), 102--107; ibid. 14 (1967), 220--228, 1966. https://doi.org/10.1112/s0025579300003843 doi:10.1112/s0025579300003843

work page doi:10.1112/s0025579300003843 1967

[3] [3]

The average rank of elliptic curves

Armand Brumer. The average rank of elliptic curves. I . Invent. Math. , 109(3):445--472, 1992. https://doi.org/10.1007/BF01232033 doi:10.1007/BF01232033

doi:10.1007/bf01232033 1992

[4] [4]

Python code for the paper The density and distribution of CM elliptic curves over , 2024

Adrian Barquero-Sanchez and Jimmy Calvo-Monge. Python code for the paper The density and distribution of CM elliptic curves over , 2024. GitHub repository. URL: https://github.com/JimmyCalvoMonge/ellipticCM

work page 2024

[5] [5]

David A. Cox. Primes of the form x^2+ny^2 --- F ermat, class field theory, and complex multiplication . AMS Chelsea Publishing, Providence, RI, [2022] 2022. Third edition [of 1028322] with solutions, With contributions by Roger Lipsett

work page 2022

[6] [6]

Coates and A

J. Coates and A. Wiles. On the conjecture of B irch and S winnerton- D yer. Invent. Math. , 39(3):223--251, 1977. https://doi.org/10.1007/BF01402975 doi:10.1007/BF01402975

doi:10.1007/bf01402975 1977

[7] [7]

Asymptotische Gesetze der Zahlentheorie

Leopold Gegenbauer. Asymptotische Gesetze der Zahlentheorie . Denkschriften Akad. Wiss. Wien , 49:37--80, 1885

work page

[8] [8]

Diophantische A nalysis und M odulfunktionen

Kurt Heegner. Diophantische A nalysis und M odulfunktionen. Math. Z. , 56:227--253, 1952. https://doi.org/10.1007/BF01174749 doi:10.1007/BF01174749

doi:10.1007/bf01174749 1952