The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

Lazar Radi\v{c}evi\'c; Matteo Verzobio; Stevan Gajovi\'c

The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

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Pith's one-line read For primes $p \ge 3$, the paper proves exact rational-function formulas for the Haar-measure density of Weierstrass equations over $\mathbb{Z}_p$ that define elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point, and…

desk verdict Exact p-adic densities for 3-torsion and 3-isogeny, with two independent proofs in the p>3 case and one externally supported p=3 entry that a referee should check. read the letter →

arxiv 2502.08583 v2 pith:7Y3R37MP submitted 2025-02-12 math.NT

Stevan Gajović , Lazar Radičević , Matteo Verzobio This is my paper · ORCID

classification math.NT MSC 11G0714H5211S80

keywords ellipticcurves3-torsionpoints3-isogeniesp-adicfieldsHaarmeasuremodularX(3)andX_1(3)integrationKodairatypes

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

For primes $p \ge 3$, this paper computes exact probabilities for two arithmetic properties of a random elliptic curve over the $p$-adic numbers. Sampling long Weierstrass equations with coefficients in $\mathbb{Z}_p$ by Haar measure, the authors prove that the chance the curve has a nonzero $\mathbb{Q}_p$-rational 3-torsion point is the explicit rational function of $p$ in Theorem 1.1, and the chance it carries a $\mathbb{Q}_p$-rational cyclic 3-isogeny is the explicit rational function in Theorem 1.2 for $p > 3$. Which of the two formulas applies depends only on whether $p$ is $1$ or $2$ modulo $3$, with a separate value at $p = 3$. These are exact densities, not asymptotics, and they record the full system of congruences that detect or exclude rational 3-torsion. The paper also derives a large-sieve upper bound for counting rational 3-torsion curves over $\mathbb{Q}$ in boxes and an asymptotic for $\ell$-torsion points as $p$ tends to infinity.

What carries the argument

The central machinery is the upgrade of the classical level-3 modular parametrizations to spaces of Weierstrass equations equipped with a regular differential. Tate normal form $y^2 + uxy + vy = x^3$ parametrizes curves with a marked nonzero 3-torsion point, the Hesse pencil $u(x^3+y^3+z^3) - 3vxyz = 0$ parametrizes curves with full level-3 structure, and a twisted version with the point defined over a quadratic extension parametrizes the nontrivial characters of a rational 3-isogeny. Forgetting the level structure yields covering maps $\pi_1 : A^2_{X_1(3)} \to A^2_X$ and $\pi_2 : A^2_{X(3)} \to A^2_X$ whose Jacobian determinants are $256v^2$ and $-559872\,u^2(u-v)^2(u^2+uv+v^2)^2$, respectively. The $p$-adic change-of-variables formula turns the counting of integral preimages into integrals of the $p$-adic absolute values of these Jacobians, which are then evaluated by a residue-class calculation. Bad-reduction contributions are supplied by pre-existing Kodaira-type densities, and the case $p = 3$ uses congruences for the 3-division polynomial $\psi_3$ modulo higher powers together with the canonical lift.

What would settle it

Fix $p=5$, for which Theorem 1.1 predicts density $25/62$ and Theorem 1.2 predicts $401/781$, and compute by exhaustive enumeration the fraction of tuples in $(\mathbb{Z}/5^m\mathbb{Z})^5$ for increasing $m$ whose Weierstrass equation has a rational 3-torsion point, respectively a root of $\psi_3$ in $\mathbb{Q}_5$; convergence to the predicted rational numbers would support the theorems, and a definite mismatch at any single $m$ would refute them. Independently, recompute Proposition 4.3's $\mathbb{F}_p$-point counts on $X_1(3)$ and $X(3)$ for $p=5$ or $p=7$ by direct computer algebra to check the good-reduction step.

Watch

Extended reading notes

Core claim

The paper's central claim is that two local probabilities are exact rational functions of the prime $p$. For $p \ge 3$, the Haar measure of the set of tuples $[a_1,a_2,a_3,a_4,a_6]$ in $\mathbb{Z}_p^5$ defining an elliptic curve with a nonzero $\mathbb{Q}_p$-rational 3-torsion point is $\frac{p^2(3p^6+4p^2-4p+4)}{8(p^8+p^6+p^4+p^2+1)}$ when $p \equiv 1 \pmod 3$, $\frac{p^2}{2(p^2+p+1)}$ when $p \equiv 2 \pmod 3$, and $\frac{3}{26}$ when $p = 3$. For $p > 3$, the measure of the set defining a $\mathbb{Q}_p$-rational cyclic 3-isogeny is $\frac{3p^4+3p^3+4p^2+4}{4(p^4+p^3+p^2+p+1)}$ when $p \equiv 1 \pmod 3$ and $\frac{p^4+p^3+2p^2+2}{2(p^4+p^3+p^2+p+1)}$ when $p \equiv 2 \pmod 3$. The proof treats good and bad reduction separately: for good reduction, the density is obtained from $\mathbb{F}_p$-point counts on the modular curves $X_1(3)$ and $X(3)$; for bad reduction, it is obtained from the Kodaira type, using pre-existing densities for each type. A second proof via $p$-adic integration computes the same torsion density from the Jacobian determinants of the forgetful maps, and the case $p = 3$ requires a separate analysis of the 3-division polynomial and the canonical lift.

Load-bearing premise

The load-bearing premise is that the Kodaira-type densities quoted from the external local-density computation (for split multiplicative reduction of type $I_m$ and for additive types $IV$ and $IV^*$) are correct; the final formulas inherit these numbers, and the paper does not re-derive them.

Editorial extensions

If this is right

For $p \ge 3$, the complete system of congruences characterizing elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point is known, and its total Haar volume is the explicit rational function in Theorem 1.1.
If the density theorems are correct, the large-sieve argument gives an upper bound $\#S \ll N_1N_2/\sqrt{\min\{N_1,N_2\}}$ for elliptic curves with rational 3-torsion in an arbitrary box $[M_1,M_1+N_1] \times [M_2,M_2+N_2]$, and a nontrivial saving for short intervals.
For a fixed ordinary elliptic curve $\bar{E}$ over $\mathbb{F}_p$ with an $\mathbb{F}_p$-rational $p$-torsion point, exactly one lift in $p$ of the coefficient pairs modulo $p^2$ gives a $\mathbb{Q}_p$-rational $p$-torsion point, so the lifting probability is $1/p$; these lifts are cut out by congruences modulo $p^2$ and coincide with the canonical lift.
For a prime $\ell > 3$ with $p \neq \ell$, the density of curves with a $\mathbb{Q}_p$-rational $\ell$-torsion point tends to $\frac{1}{\ell-1} - \frac{\delta}{\ell^2-1}$ as $p \to \infty$ through a fixed residue class modulo $\ell$, where $\delta = 1$ for $p \equiv 1 \pmod \ell$ and $\delta = 0$ otherwise; the paper also expresses the finite-$p$ density in terms of $\mathbb{F}_p$-point counts on

Reading between the lines

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

Editorial inference: the same $p$-adic integration scheme should yield exact densities for $\ell = 2, 3, 4, 5$, where $X(\ell) \cong \mathbb{P}^1$, with formulas depending on $p$ through the residue class modulo $\ell$; the paper expects this but does not compute it.
Editorial inference: since the torsion density is assembled from local point counts on modular curves, one should expect the density of curves with a prescribed local Galois structure on $E[\ell]$ to be a rational number whenever the relevant moduli space is rational over $\mathbb{Q}_p$, and not otherwise; testing $\ell = 7$ would discriminate between these regimes.
Editorial inference: combining the exact $p$-adic densities with the large sieve suggests that the short-interval count of elliptic curves with a rational 3-torsion point satisfies the improved bound in Corollary 7.2(2), a regime where the trivial geometry-of-numbers estimate is worse and the local $p$-adic information carries the argument.

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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

Steve,

Quick take on arXiv:2502.08583. The paper finds exact Haar-measure densities for the set of Z_p^5 Weierstrass equations defining an elliptic curve over Q_p with a rational 3-torsion point (Theorem 1.1) and with a rational 3-isogeny (Theorem 1.2). These are genuinely new; prior work covered 2-torsion via root counting and height-counting over Q. The formulas are clean rational functions of p, split by p mod 3, and the p≡2 mod 3 torsion density is simply p^2/(2(p^2+p+1)). That is a natural explicit question and the answer is what people will quote.

What is good: the paper does the job twice. Section 4 splits by reduction type and uses Kodaira-type probabilities plus F_p-point counts on X_1(3) and X(3). Section 5 redoes the p>3 case by p-adic integration, with Igusa change-of-variables lemmas, and the two routes agree. That agreement is real evidence; there are no fitted constants anywhere. Theorem 1.2 is derived purely by p-adic integration, including a careful treatment of the twisted level-3 moduli spaces. I checked the structure around Lemmas 5.7–5.10 and the Jacobian computations; they are detailed and the integrals are evaluated with the stated Lemma 5.4. The paper also gives a nice asymptotic for ℓ-torsion (Theorem 7.4) and a sieving corollary for short intervals (Corollary 7.2), which is a sensible application.

Soft spots, in proportion. The p=3 entry of Theorem 1.1 depends on Cremona–Sadek [11] for the densities of Kodaira types IV, IV*, and split multiplicative I_m. That is a citation, not a re-derivation, so if any of those numbers is wrong, the 3/26 changes. This is not circularity; it is standard external input and the paper flags it. But it does mean the p=3 case is not independently verified inside the paper, whereas the p>3 torsion case and the isogeny case are. A referee should check the [11] inputs against the formulas in Section 6 of the cited paper, and ideally test the 3/26 value by a direct finite computation modulo 3^k for small k. That would settle it cheaply. The stated exclusions (p=2 for torsion, p=2,3 for isogenies) are declared and look like tedium rather than a hidden gap.

Verdict: this deserves a serious referee. The central claims are well-supported, the exposition is competent, and the two-proof agreement is the kind of thing that catches systematic errors. A specialist in arithmetic statistics will get real value. Send it to peer review; the p=3 caveat is a verification issue, not a rejection issue.

Referee Report

0 major / 4 minor

Summary. This paper determines, for each prime p≥3, the Haar measure of the subset of Z_p^5 consisting of coefficients [a1,...,a6] for which the corresponding Weierstrass equation defines an elliptic curve over Q_p with a nonzero Q_p-rational 3-torsion point (Theorem 1.1), and for p>3 the analogous density for rational cyclic 3-isogenies (Theorem 1.2). The densities are explicit rational functions of p, depending only on p mod 3, with the exceptional value 3/26 at p=3. The proof strategy has three parts: (i) reduction to short Weierstrass form via measure-preserving changes of variables, (ii) counting F_p-points on modular curves X(3), X_1(3), and twisted versions, and (iii) a p-adic integration method based on Igusa's change-of-variables formula applied to the Jacobians of the covering maps. The paper also contains a self-contained treatment of p-torsion over Q_p (Proposition 6.13), an asymptotic for ℓ-torsion as p→∞, a large-sieve counting application for elliptic curves over Q with rational 3-torsion, and a family-of-twists analysis.

Significance. If correct, the two main theorems give exact unconditional local densities rather than asymptotics, and they are the first such results for torsion of order greater than 2 in this family. The paper is particularly strong in that Theorem 1.1 for p>3 is proved in two independent ways that agree: a reduction-type count (§4) and a p-adic integration computation (§5.2). The modular-curve parametrizations are explicit and the Jacobian determinants are stated in closed form, making the calculations reproducible. All numerical constants arise from a finite set of explicit inputs; no fitted parameters appear. The main external input is the set of Kodaira-type probabilities from Cremona–Sadek [11], which is used in the p=3 part of Theorem 1.1 and in the bad-reduction cases for p>3; the other parts of the argument are self-contained. The paper is clearly written and will be of interest to arithmetic statisticians and to researchers studying local-global questions for elliptic curves.

minor comments (4)

[Section 6, first paragraph] The citation '[31, Theorem .6.4]' has a missing theorem number; it should refer to a valid statement in Silverman's book, for example the formal logarithm or the structure of E_1(Q_p).
[Theorem 6.9] The typo 'Propostion' should read 'Proposition'. In the same theorem, the additive and split-multiplicative contributions are imported from [11, Proposition 2.2 and Proposition 2.5]; since the p=3 value 3/26 is the only part of Theorem 1.1 not corroborated by the independent p-adic integration proof, please add a sentence identifying the exact numerical densities taken from [11] (for Kodaira types IV, IV^*, and split I_m) so that this dependence can be checked without consulting the external paper.
[Lemma 6.11] The phrase 'the smoothness of a map End(E_can) → End(E) is an isomorphism' is mangled; presumably the intended statement is that the reduction homomorphism on endomorphism rings is an isomorphism. Please rephrase.
[Proposition 7.1 and Corollary 7.2] Proposition 7.1 assumes M_1 and M_2 are positive integers, but Corollary 7.2(1) applies it with M_1 = M_2 = 0; the hypotheses should be amended to allow nonnegative M_i.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main densities are derived by independent modular-curve counting and p-adic integration, with the only self-citation being contextual and not load-bearing.

full rationale

The paper's central claims, Theorems 1.1 and 1.2, are derived through two independent routes: reduction-type counting using modular curves X1(3) and X(3), and p-adic integration via explicit bijections with Tate and Hesse normal forms. The p-adic integration proof does not import the conclusion of Theorem 1.1; instead it computes the same quantity from Jacobian determinants and Igusa's change-of-variables theorem, and Remark 5.11 notes agreement as a sanity check. No fitted parameters appear, and no quantity is defined in terms of the target density. The reliance on Cremona and Sadek [11] for Kodaira-type densities is an external, cited benchmark with no author overlap, and the paper explicitly states when it is used; this is ordinary mathematical citation, not circularity. The only self-citation is to [8] by Bhargava, Cremona, Fisher, and the first author, but it appears solely as context for related p-adic density results and is not load-bearing for the proofs here. The p=3 case uses [11] for IV and IV* densities, but those are independent external inputs, so any sensitivity there is a correctness/fragility concern, not circularity.

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

No free parameters are fitted: all formulas are exact functions of p. The paper introduces no new entities. It does rely on several external pillars: Igusa's change-of-variables theorem, the Cremona-Sadek Kodaira statistics, the Tate/Hesse parametrizations of level-3 moduli spaces, and Chebotarev equidistribution for the asymptotic section.

assumptions (6)

standard math Igusa's p-adic change of variables formula (Theorem 5.1), including the Jacobian determinant rule for Haar measure.
Used throughout Section 5 for the reduction from long Weierstrass to short and for the integrals defining the densities.
domain assumption Cremona-Sadek local density formulas for Kodaira types, especially split multiplicative type I_m density (p-1)^2/(2 p^{m+2}) and the densities of types IV and IV^* with Tamagawa number 3.
Black-boxed in Propositions 4.4, 4.6, and Theorem 6.9; the final densities are linearly affected by these numbers.
domain assumption The bijections between A^2_{X1(3)}, A^2_{X(3)}, A^2_{Xpsi(3)}, A^2_{XF(3)} and moduli of (E,omega,P) from Tate and Hesse normal forms, valid over perfect fields with char not 2 or 3.
Used in Section 3 and then Lemmas 5.5, 5.6, and 5.13 to transfer counting to Weierstrass coefficients.
standard math Hensel's lemma and Nagell-Lutz integrality for torsion points over Z_p.
Used in Lemma 5.5 to show preimages under pi_1 are integral and in Section 6 for p-torsion lifting.
domain assumption Serre-Tate canonical lift and Serre's ramification criterion for ordinary elliptic curves (Theorem 6.12).
Basis for Proposition 6.13 that random lifts have a Q_p-rational p-torsion point with probability 1/p; not needed for the main 3-torsion or 3-isogeny formulas.
standard math Geometric Chebotarev equidistribution of Frobenius conjugacy classes in GL2(F_l) as p varies.
Used in Section 7.2 to derive the asymptotic probability for l-torsion; not used in Theorems 1.1 and 1.2.

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Pith. "Pith review of The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny." pith.science (2026). https://pith.science/paper/7Y3R37MP

@misc{pith2026250208583,
  author       = {Pith},
  title        = {Pith review of: The density of elliptic curves over $\mathbbQ_p$ with a rational 3-torsion point or a rational 3-isogeny},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Y3R37MP}},
  note         = {Machine review of arXiv:2502.08583}
}

abstract

We determine the probability that a random Weierstrass equation with coefficients in the $p$-adic integers defines an elliptic curve with a non-trivial $3$-torsion point, or with a degree $3$ isogeny, defined over the field of $p$-adic numbers. We determine these densities by calculating the corresponding $p$-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of $\ell$-torsion for $\ell>3$ prime.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Counting odd genus $2$ curves with a marked rational $3$-torsion point
math.NT 2026-07 accept novelty 7.0 of 10

The number of minimal monic odd-degree genus-2 Weierstrass models of height ≤ X with a marked rational Jacobian 3-torsion point is c X^10 + o(X^10) for an effectively computable constant c.

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [11]

Local and global densities for Weierstrass models of elliptic curves

John Cremona and Mohammad Sadek. Local and global densities for Weierstrass models of elliptic curves. Math. Res. Lett. , 30(2):413–461, 2023

work page 2023

[2] [1]

Jacobians of genus one curves

Sang Yook An, Seog Young Kim, David C Marshall, Susan H Marshall, W illiam G McCallum, and Alexander R Perlis. Jacobians of genus one curves. Journal of Number Theory , 90(2):304–315, 2001

work page 2001

[3] [2]

The Hesse pencil of plane cubic curves

Michela Artebani and Igor V Dolgachev. The Hesse pencil of plane cubic curves. L’Enseignement Math´ ematique, 55(3):235–273, 2009

work page 2009

[4] [3]

Shadow line distributions

Jennifer S Balakrishnan, Mirela C ¸ iperiani, Barry Mazur, and Karl Rubin. Shadow line distributions. arXiv preprint arXiv:2409.00891 , 2024

work page Pith review arXiv 2024

[5] [4]

How often does a cubic hypersurface have a rational point?

Lea Beneish and Christopher Keyes. How often does a cubic hype rsurface have a rational point? arXiv preprint arXiv:2405.06584, 2024

work page Pith review arXiv 2024

[6] [5]

Higher composition laws IV: The parametrizat ion of quintic rings

Manjul Bhargava. Higher composition laws IV: The parametrizat ion of quintic rings. Annals of Math- ematics, 167:53–94, 2008

work page 2008

[7] [6]

The proport ion of plane cubic curves over Q that everywhere locally have a point

Manjul Bhargava, John Cremona, and Tom Fisher. The proport ion of plane cubic curves over Q that everywhere locally have a point. International Journal of Number Theory , 12(04):1077–1092, 2016. 31

work page 2016

[8] [7]

The proport ion of genus one curves over Q deﬁned by a binary quartic that everywhere locally have a point

Manjul Bhargava, John Cremona, and Tom Fisher. The proport ion of genus one curves over Q deﬁned by a binary quartic that everywhere locally have a point. International Journal of Number Theory , 17(04):903–923, 2021

work page 2021