The size of $2$-Selmer groups for the $\frac{\pi}{3}$-congruent number problem

Aprameyo Pal; Kushal Bhowmick

REVIEW 3 major objections 5 minor 12 references

The size of $2$-Selmer groups for the $\frac{\pi}{3}$-congruent number problem

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The average size of the 2-Selmer group in two infinite families of π/3-congruent twists is exactly 9.

desk verdict First average 2-Selmer result for the pi/3-congruent family, worth a serious referee, but the proof as written has two unproved load-bearing steps that need to be filled before the constant 9 is established. read the letter →

arxiv 2602.08912 v2 pith:3YHCDINY submitted 2026-02-09 math.NT

Kushal Bhowmick , Aprameyo Pal This is my paper

classification math.NT MSC 11G0511G4011L40

keywords 2-Selmergroupπ/3-congruentnumberaveragerank2-descentquadratictwistsellipticcurvesSelmerdensitytheta-congruentnumbers

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 computes the average size of the 2-Selmer group for the elliptic curves attached to the π/3-congruent number problem. It proves that for square-free n ≤ X in the two families n ≡ 5 or 13 (mod 24) with all prime divisors ≡ 1 (mod 4), the sum of 2^{s(n)} over the family is 9 times the size of the family, up to a power-saving error. That fixes the average 2-Selmer size at 9, hence the average 2-Selmer rank at 3 as a real number. Using parity results, the paper then derives unconditional positive densities for 2-Selmer rank 0 or 2 (n ≡ 5 family) and 1 or 3 (n ≡ 13 family), and the same average bounds for the Mordell–Weil rank. The constant 9 is notable because it disagrees with the usual random-matrix heuristic for Selmer groups.

What carries the argument

The central identity is Lemma 2.7: 2^{s(n)} equals a sum over the 16 factorizations n = ∏ n_{ij} of a product of Jacobi symbols and powers of 4^{-ω}. This identity comes from a 2-descent on the curve E_{n,π/3}: y^2 = x(x+3n)(x-n), which has full rational 2-torsion. A naturally defined 2-isogeny lets the local solvability of the descent system be encoded by Legendre symbols, and the global Selmer condition becomes a sum of local counting factors. The paper then averages this identity over n using a character-sum method; the main terms contribute exactly 9.

What would settle it

Enumerate all square-free n ≤ 10^5 with n ≡ 5, 13 (mod 24) and all prime factors ≡ 1 (mod 4); for each, compute 2^{s(n)} by explicit 2-descent and compare the average with 9. If the average does not tend to 9 as X grows, Theorem 1.2 is false. Alternatively, for a single n, list all factorizations n = n1 n2 n3 n4 and test whether the Q2-solvability of the descent system matches the four cases of Lemma 2.4; any mismatch would break the identity.

Watch

Extended reading notes

Core claim

The central result is Theorem 1.2: for h = 5, 13, the sum of 2^{s(n)} over square-free n ≤ X with n ≡ h (mod 24) and all prime factors ≡ 1 (mod 4) equals 9 times the number of such n, with an error of O(X (log X)^{-5/8} (log log X)^8). Here s(n) is the 2-Selmer rank, the exponent such that the 2-Selmer group has size 2^{2+s(n)}. The constant 9 is the core arithmetic content: it is not a power of 2 and it reflects that, on average, the 2-Selmer condition in this family behaves like the intersection of three independent affine F2-hyperplanes. The paper also derives from this the corollaries that the density of s(n) = 0 or 2 in the h = 5 family is at least 7/16, and the density of s(n) = 1 or 3

Load-bearing premise

The paper's proof that the descent system is solvable over Q2 reduces to a 'careful case-by-case study' in Lemma 2.4, and the further assertion that only cases (ii) or (iv) occur for n in the family is not demonstrated; the factorization identity (Lemma 2.7) and hence the constant 9 depend on this unproved 2-adic classification.

Editorial extensions

If this is right

For the h = 5 family, the average of s(n) is at most 9/2, and the density of s(n) = 0 or 2 is at least 7/16.
For the h = 13 family, the average of s(n) is at most 10/3, and the density of s(n) = 1 or 3 is at least 1/3.
Since the Mordell–Weil rank r(n) satisfies r(n) ≤ s(n), the same average bounds apply to r(n); with a parity assumption on r(n) and s(n), positive densities of ranks 0/2 and 1/3 follow for r(n).
The error term is a power-saving O(X (log X)^{-5/8} (log log X)^8), so the average 9 is stable under local variations of the family.

Reading between the lines

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

If the same 2-descent identity holds for the 2π/3-congruent family, as the paper suggests, a similar constant should emerge and could be computed by the same method.
The constant 9 suggests that the distribution of 2-Selmer ranks in this family is not Poisson-like; a natural next step is to determine the full limiting distribution of s(n), not just its average.
A direct computational check of Lemma 2.7 for individual small n in the family would verify the factorization identity; if it holds, the remaining difficulty is purely analytic.
The paper's restriction to prime factors ≡ 1 (mod 4) appears load-bearing; dropping it may change the 2-adic classification and thus the constant 9.

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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

The main result is new: for squarefree n with n ≡ 5 or 13 mod 24 and all prime factors ≡ 1 mod 4, the average of 2^{s(n)} is 9 with a respectable error term. The density corollaries for s(n) = 0 or 2 and s(n) = 1 or 3 are the first of their kind for this family, and they follow cleanly once Theorem 1.2 is accepted. The paper is also honest about its limits: Remark 1.3 says the method only reaches these residue classes with the p ≡ 1 mod 4 restriction, and that further progress needs new ideas.

Credit where due: the Selmer-to-systems count in Lemma 2.7 is a genuine adaptation of Heath-Brown's strategy, and the paper correctly sees that the exceptional index structure changes from 4 types in H93 to 9 here. The reliance on H93 and the parity theorem of Wei-Guo is transparent, and there are no fitted parameters or assumed conclusions.

The soft spots are real. The stress-test note is right about Lemma 2.4: the Q_2-solvability classification of system (4) is asserted after 'a careful case-by-case study' and the proof is not supplied. This is load-bearing because Lemma 2.7 implicitly builds the p = 2 condition into the formula, so a missing or extra case would change the constant 9. The specific consequence 'either ii) or iv) holds for our n' is probably elementary from n ≡ 5 mod 8 and p ≡ 1 mod 4, but the full four-case list needs to be demonstrated. The 'one can check' claims extending H93's lemmas from modulus 8 to 24 are less worrying, but still need enough detail for a referee to verify.

The second gap is in Section 5. The paper lists nine exceptional index types, computes the 10,20,30,40 contribution, and then says 'Therefore Theorem 1.2 follows.' The other eight contributions are not shown. They may be routine analogues, but as written this is an unproved structural step, not a cosmetic omission. The SageMath checks are also invoked without code or tables; if they are load-bearing for restricting the family, they should be reproducible.

The central strategy is sound and the paper is a serious extension, not a paradigm shift. The missing pieces look repairable, but the proof as submitted is incomplete in exactly the places where the final constant could change. I would send this to a competent arithmetic-geometry referee and ask for a revision with complete proofs of Lemma 2.4 and Section 5. It should not be desk-rejected; it should be refereed, but with a tight request for details.

Referee Report

3 major / 5 minor

Summary. The paper studies the average size of the 2-Selmer group of the elliptic curves E_{n,\pi/3}: y^2=x(x+3n)(x-n) associated with the \pi/3-congruent number problem. Following Heath-Brown's strategy for the congruent number problem, the authors derive an expression for 2^{s(n)} as a sum over factorizations of n into sixteen variables (Lemma 2.7), average this expression over the family S(X,h) of square-free n\equiv h \pmod{24} (h=5,13) all of whose prime divisors are 1 mod 4, and obtain the asymptotic formula \sum_{n\in S(X,h)} 2^{s(n)} = 9\#S(X,h)+O(X(\log X)^{-5/8}(\log\log X)^8) (Theorem 1.2). From this they derive average bounds and unconditional positive lower-density results for 2-Selmer rank being 0 or 2 (for h=5) and 1 or 3 (for h=13), conditional on the parity theorem of Wei-Guo.

Significance. If the proof is completed, the main theorem is a significant result: it gives a parameter-free average value 9 for 2^{s(n)} in a natural family, thereby providing a concrete instance where the distribution of 2-Selmer ranks diverges from the Bhargava-Kane-Lenstra-Poonen-Rains heuristics, and it yields the first unconditional positive-density statements for the 2-Selmer ranks in this \pi/3-congruent setting. The paper has no fitted parameters; the constant 9 is structurally derived from enumerating exceptional configurations. The reliance on Heath-Brown's lemmas and Wei-Guo's parity theorem is explicit and external. However, two load-bearing steps are currently asserted rather than proved, and one of them is a delicate 2-adic classification whose failure would change the main constant.

major comments (3)

[§2, Lemma 2.4] Lemma 2.4 gives a complete classification of Q_2-solvability of the system (4), but no proof is supplied; the text says only that it follows from 'a careful case-by-case study'. This is load-bearing: in Lemma 2.7, the expression g(n) contains no factor enforcing the Q_2 condition, and the identity 2^{s(n)}=\sum_n g(n) relies on Lemma 2.4 to make the Q_2 condition automatic. If the classification is incomplete, or if one of the four congruence cases is misstated, the local-solubility indicator changes for a positive proportion of factorizations and the constant 9 in Theorem 1.2 would change. The subsequent assertion that 'either ii) or iv) is satisfied as n\in S(X,h)' is also not shown; it is true and can be proved from n\equiv5\pmod8 together with the fact that each n_j is a product of primes 1 mod 4, but the proof should be included. A complete proof of Lemma 2.4, or a reference contain
[§5, after Eq. (11)] Lemma 4.4 lists nine exceptional index types whose sums are not negligible. Section 5 computes the main term only for the first type, i.e. indices 10,20,30,40, deriving \sum 4^{-\omega(n)} = \#S(X,h)+O(\cdots). It then says 'Therefore Theorem 1.2 follows.' But the theorem's constant 9 is the sum of the main terms of all nine exceptional types. The other eight types (30,40,34,43; 30,31,32,34; 40,41,42,43; 10,20,12,21; 31,32,41,42; 12,21,34,43; 10,21,31,41; 20,12,32,42) are not evaluated. If any of them has a main term different from \#S(X,h), the final constant changes. A direct calculation for each of the nine types, or a uniform argument showing each contributes \#S(X,h)+o(X), is needed.
[§3 and §4, Lemmas 3.1, 4.2, 4.3] Several central estimates are imported from Heath-Brown [H93] with the comment that the proof 'one can check' also works for modulus 24 instead of modulus 8. These lemmas are used to control the vast majority of the sums S(A) and to obtain the final error O(X(\log X)^{-5/8}(\log\log X)^8). The adaptation is not completely routine: the character group modulo 24 has different structure from that modulo 8, and the proof must handle the congruence conditions and the additional factors 4^{-\omega(n)}. Since these lemmas are load-bearing for the error term, the authors should either provide the adaptation in detail or give a precise reference to a version that covers modulus 24.

minor comments (5)

[§6, Remark 6.4] Remark 6.4 states that the positive density of s(n)=0 or 2 holds among n\equiv13\pmod{24}, but Corollary 6.3 and the surrounding text are for h=5. This appears to be a typographical error and should be corrected.
[§2] The paper repeatedly invokes SageMath calculations to justify the restriction to n\equiv5,13\pmod{24} and to the condition that all prime divisors are 1 mod 4. No code, output, or precise statement of the computation is provided. Since these calculations motivate the main family, it would be helpful to include the relevant code or a documented verification.
[§4, Lemma 4.1] The classification of the 24 exceptional index types is argued in words rather than in a formal proof. Some steps, such as 'we can show that a similar conclusion holds if A_{i0}, A_{j0}\ge C', are not fully written out. Please expand this into a complete enumeration.
[§3, Lemma 3.5] The proof uses the bound \sum_{n\le N}\gamma^{\omega(n)}\ll N(\log N)^{\gamma/2-1}; the range of \gamma and the exact assumptions on n (square-free, primes 1 mod 4) should be stated precisely. Also, C=\exp(\kappa(\log\log X)^2) is not necessarily a power of 2, so the opening sentence 'Let's take that C is some power of 2' needs justification or a rounding argument.
[Global] There are several minor typos and notation issues, e.g. 'n_{ij} n_{kl} \equiv h' (mod 24)' without spacing, and the definition of \omega(n) is used before it is explicitly defined. These should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: main Selmer identity comes from 2-descent and local conditions; constant 9 comes from counting exceptional index cases, not from fitting or self-citation.

full rationale

The paper's central identity, Lemma 2.7, expresses 2^{s(n)} as a sum over 16 factorizations of a product of Jacobi symbols and 4^{-\omega} factors. This identity is derived from the definition of the 2-Selmer group via the 2-descent map and local solvability conditions, not from the desired average. The main theorem then follows by dyadic decomposition, Heath-Brown's character-sum lemmas, and an enumeration of the nine exceptional index configurations in Lemma 4.1 and Remark 4.5; the constant 9 is the count of these configurations, not a fitted parameter. No parameter is calibrated to the target average, and no 'prediction' is a renamed input. The parity theorem of Wei-Guo, used in the corollaries, is an external result by different authors. The potentially load-bearing Lemma 2.4, classifying Q2-solvability, is asserted via 'a careful case-by-case study' without a full proof, and the application to S(X,h) is also abbreviated; this is a verification gap and a correctness risk, not a circular reduction. The SageMath-based observations about which residue classes admit a suitable representative are heuristic restrictions, but they do not assume Theorem 1.2 and the final theorem is confined to the classes h=5,13. Overall, the derivation chain does not reduce to its own inputs.

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

The proof rests on standard 2-descent machinery plus several unproved local computations. No fitted parameters or new entities are introduced. The most fragile axioms are the Q2-solvability classification and the mod-24 extension of Heath-Brown's lemmas.

assumptions (6)

standard math E_{n,pi/3} has full rational 2-torsion and no other Q-torsion for n not dividing 6
Stated in Section 1 and used to set #Sel2 = 2^{2+s(n)}.
domain assumption The Kummer map theta is injective and Im(theta) restricted to the chosen coset representatives has size 2^{r(n)}; local Kummer images give Selmer size via the systems (4)
Section 2 uses the standard 2-descent setup; the identification of 2^{s(n)} with the number of locally solvable systems is asserted rather than proved in detail.
ad hoc to paper Lemma 2.4's Q2-solvability classification is correct
Proof deferred to 'careful case-by-case study'; no details are given.
ad hoc to paper For n∈S(X,h), either condition ii) or iv) of Lemma 2.4 holds
Stated as 'We can easily observe' with no proof; the conditions involve mod 8 coincidences among the n_i that need not hold for arbitrary factorizations.
domain assumption Heath-Brown's Lemmas 4, 6, and 10 extend from modulus 8 to modulus 24 with the same proof
Used in Lemmas 3.1, 3.3, 4.2, and 4.3; the paper says 'one can check' but gives no verification.
domain assumption Wei-Guo parity theorem (Theorem 6.1) is correct
Parity of s(n) by n modulo 24 is taken from [WG2022] and drives the density corollaries.

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Cite this review

Pith. "Pith review of The size of $2$-Selmer groups for the $\frac{\pi}{3}$-congruent number problem." pith.science (2026). https://pith.science/paper/3YHCDINY

@misc{pith2026260208912,
  author       = {Pith},
  title        = {Pith review of: The size of $2$-Selmer groups for the $\frac\pi3$-congruent number problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3YHCDINY}},
  note         = {Machine review of arXiv:2602.08912}
}

abstract

Our main objective in this paper is to study the average rank of the $2$-Selmer group of the elliptic curve associated with the $\frac{\pi}{3}$-congruent number problem. Following Heath-Brown's strategy, we could find an asymptotic formula for the size of the relaxed $2$-Selmer groups, which has several consequences towards the average of $2$-Selmer ranks and $\frac{\pi}{3}$-congruent number problem. Indeed, we could find an unconditional positive density of $2$-Selmer rank being $1$ or $3$, among the positive square-free integers $n\equiv 13\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$ and an unconditional positive density of $2$-Selmer rank being $0$ or $2$, among the positive square-free integers $n\equiv 5\pmod{24}$ having all the prime divisors congruent to $1$ modulo $4$.

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Works this paper leans on

12 extracted references · 3 linked inside Pith

[1]

Bhargava, D

M. Bhargava, D. Kane, H. Lenstra, B. Poonen, E. Rains., Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves , Camb. J. Math. 3 (2015), no. 3, 275—321

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[2]

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[9]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Bhargava, D

M. Bhargava, D. Kane, H. Lenstra, B. Poonen, E. Rains., Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves , Camb. J. Math. 3 (2015), no. 3, 275—321

2015

[2] [2]

K. Feng, Q. Liu, J. Pan, Y. Tian, Quadratic twists of tiling number elliptic curves , https://arxiv.org/pdf/2405.11132v1

arXiv

[3] [3]

Fujiwara, -congruent numbers , Number theory (Eger, 1996), 235–241

M. Fujiwara, -congruent numbers , Number theory (Eger, 1996), 235–241. Walter de Gruyter & Co., Berlin, 1998

1996

[4] [4]

HeathBrown, The size of Selmer groups for the congruent number problem , Invent

D.R. HeathBrown, The size of Selmer groups for the congruent number problem , Invent. Math.111(1993), no.1, 171–195

1993

[5] [5]

HeathBrown, The size of Selmer groups for the congruent number problem II , Invent

D.R. HeathBrown, The size of Selmer groups for the congruent number problem II , Invent. Math.118(1994), no.2, 331–370

1994

[6] [6]

Hibino T., Kan M., -congruent numbers and Heegner points , Arch. Math. (Basel). 77.4 (2001), 303-308

2001

[7] [7]

Science in China(Series A)

He We, Hu Yirong and Tian Ye., Tiling number problem and -congruent number problem (in Chinese). Science in China(Series A). 51.10 (2021), 1579-1594

2021

[8] [8]

Daniel Kane, On the ranks of the 2 -Selmer groups of twists of a given elliptic curve , Algebra Number Theory7(2013), no.5, 1253–1279

2013