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REVIEW 3 major objections 4 minor 25 references

$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For this family, 3-Selmer rank is the 3-class rank of K(√a) up to small prime corrections.

desk verdict Useful extension of the JMS 3-isogeny framework, with real bounds and constructions, but the refined theorem rests on an unproved Tamagawa classification and the Prop 3.10 density calculation is arithmetically wrong. read the letter →

arxiv 2502.01069 v1 pith:T3QBYGXE submitted 2025-02-03 math.NT

Somnath Jha , Dipramit Majumdar , Pratiksha Shingavekar This is my paper · ORCID
classification math.NT MSC 11G0511R2911R3411G4011S25
keywords ellipticcurves3-Selmergroupsidealclass3-isogenylargeranksrootnumberKummermapscyclotomicfieldQ(ζ3)
topics Birch and Swinnerton-Dyer
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

The paper studies the family of elliptic curves $E_{a,b}: y^2=x^3+a(x-b)^2$ over the cyclotomic field $K=\mathbb{Q}(\zeta_3)$, where $a,b$ are integers and the curve carries a rational 3-isogeny $\psi_{a,b}$. Its central result is that the $\mathbb{F}_3$-dimension of the $\psi_{a,b}$-Selmer group of $E_{a,b}$ over $K$ lies in an explicit interval controlled by the 3-rank of the ideal class group of the quadratic field $L=K(\sqrt{a})$, together with the sizes of explicit sets of bad primes. Theorem 3.8 gives matching lower and upper bounds whose only arithmetic input is the 3-ranks of certain $S$-ideal class groups of $L$. This turns a normally expensive Selmer-rank computation into a class-group computation for the whole family. As applications, the paper constructs infinitely many curves $E_{a,1}$ over $K$ with $\dim_{\mathbb{F}_3}\mathrm{Sel}_3(E_{a,1}/K)\ge 2n$ for any $n$, and with no $K$-rational point of order 3, and proves that for a positive proportion of square-free $n$, $E_{n,n}/\mathbb{Q}$ has root number $-1$ and 3-Selmer rank 1.

What carries the argument

The load-bearing objects are the local Kummer maps and the norm-one group $(L_q^*/L_q^{*3})^{N=1}$: the Selmer group is exactly the set of global norm-one classes whose restriction lies in the local Kummer image at every prime. On the algebraic side, the spaces $M(S,a)$ and $N(S,a)$, classes in $L^*/L^{*3}$ whose divisors are cubes away from $S$, have dimensions $h^3_S(L)$ and $h^3_S(L)+|S(L)|+2$, which is why ideal class groups enter the bounds. The paper computes the local Kummer images at all primes using a formula expressing $|\hat E(K_q)/\psi(E(K_q))|$ in terms of local component counts, with the results in Table 1. The refinement uses the ratio identity (13), $|\mathrm{Sel}_{\hat\psi}(\hat E/K)|/|\mathrm{Sel}_{\psi}(E/K)|=3^{|S_2|-|S_3|+1}$ when $3\nmid a$, converting the inclusion bounds into the sharp interval of Theorem 3.8.

What would settle it

Compute the local component counts $c_q(E_{a,b})$ and $c_q(\hat E_{a,b})$ at every bad prime for one explicit curve, say $a=29,b=76$. If any prime outside $S_2$ satisfies $c_q(E_{a,b})=3c_q(\hat E_{a,b})$, or any prime outside $S_3$ satisfies the reverse equality, the identity $S'=S_2$, $S''=S_3$ used in Theorem 3.8 is false and the stated exponent needs correction.

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Extended reading notes

Core claim

At its core the paper claims that for $a\notin K^{*2}$ with $3\nmid a$, the dimension of $\mathrm{Sel}_{\psi_{a,b}}(E_{a,b}/K)$ is bounded below by $\max\{h^3_{S_{1,2}}(L), h^3_{S_{1,3}}(L)+|S_3|-|S_2|-1\}$ and above by $\min\{h^3_{S_{1,2}}(L)+|S_{1,2}(L)|+|S_3|-|S_2|+1, h^3_{S_{1,3}}(L)+|S_{1,3}(L)|+2\}$. Here $L=K(\sqrt{a})$, $h^3_S(L)$ is the 3-rank of the $S$-ideal class group, and $S_1,S_2,S_3$ are explicitly defined finite sets of primes of $K$ determined by the reduction types of $E_{a,b}$ and its dual curve. The argument embeds the isogeny-Selmer group into the norm-one group $L^*/L^{*3}$, compares it with class-group modules of known dimension, and then uses a Selmer-ratio identity involving local component counts to sharpen the crude inclusions. In the special case $S_1=S_2=S_3=\emptyset$, the theorem says the $\psi$-Selmer rank is either $h^3_L$ or $h^3_L+1$.

Load-bearing premise

The refined bounds rest on a local calculation stated without proof in the proof of Theorem 3.8: when $3\nmid a$, the primes at which one curve's local component count is exactly three times the other's are precisely the primes in $S_2$, and the reverse relation holds precisely at the primes in $S_3$. A single misclassified prime would shift the exponent $|S_3|-|S_2|$ and break both the sharp bounds and the large-rank family construction.

Editorial extensions

If this is right

  • Through the exact sequence (12), the class-group bounds in Corollary 3.7 give unconditional two-sided estimates for the full 3-Selmer rank of every curve in the family.
  • For every $n\ge 0$ there are infinitely many $a$ with $\dim_{\mathbb{F}_3}\mathrm{Sel}_3(E_{a,1}/K)\ge 2n$ and $E_{a,1}(K)[3]=0$, so large 3-Selmer rank does not force a rational 3-torsion point.
  • When $S_1=S_2=S_3=\emptyset$, the $\psi$-Selmer rank is $h^3_L$ or $h^3_L+1$, and the root number over $K$ is $-1$.
  • For a positive proportion of square-free $n$, $E_{n,n}/\mathbb{Q}$ has root number $-1$ and 3-Selmer rank 1, and the isomorphic-over-$K$ curve $E_{-3n,-3n}/\mathbb{Q}$ has root number $+1$.

Reading between the lines

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

  • One testable extension is to choose coefficients so that $|S_3|-|S_2|$ is even; Theorem 3.8 would then produce large-rank curves with the opposite root number from the odd case built in Theorem 4.1.
  • A 3-descent computation on a Table 2 curve whose interval has length greater than 1 could show whether the Selmer dimension always sits at an endpoint, indicating whether the class-group and prime-set terms interact additively or cancel.
  • The ratio identity that sharpens the bounds also suggests a way to grow the Tate-Shafarevich group: take large $|S_3|-|S_2|$ while keeping the class-group input small, forcing the extra Selmer dimension into the Tate-Shafarevich term of the isogeny descent sequence.
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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

3 major / 4 minor

Summary. The paper studies the family of elliptic curves E_{a,b}: y^2=x^3+a(x-b)^2 over K=Q(ζ3), which carry a rational 3-isogeny ψ_{a,b}. The main result (Theorem 3.8) gives upper and lower bounds on the F3-rank of the ψ_{a,b}-Selmer group in terms of the 3-ranks of certain S-class groups of L=K(√a), with refined bounds when 3∤a. The authors then use these bounds to construct infinitely many curves E_{a,1} with arbitrarily large 3-Selmer rank over K and no nontrivial K-rational 3-torsion (Theorem 4.1). They also prove, in Proposition 3.10, that for a positive proportion of squarefree n, the curve E_{n,n}/Q has root number -1 and 3-Selmer rank 1. The paper contains extensive local computations in Section 2 and numerical examples in Table 2.

Significance. If the main theorems hold, the paper gives a new explicit family of elliptic curves with unbounded 3-Selmer rank and no rational 3-torsion, complementing earlier constructions of Cassels and others. The bounds relating Selmer ranks to ideal class groups are of independent interest, and the local computations are detailed. The paper also provides a positive-proportion statement for a natural family over Q. However, the refined bounds in Theorem 3.8 depend on an unproved identification of Tamagawa-ratio primes, and the proof of Proposition 3.10 contains a concrete density error and an apparent mismatch in the quadratic algebra used. These issues are load-bearing for the respective claims, though they appear fixable within the scope of the manuscript.

major comments (3)
  1. [Theorem 3.8, proof] The proof asserts without derivation that, when 3∤a, the set S'={q: c_q(E_{a,b})=3c_q(\hat E_{a,b})} equals S2 and S''={q: c_q(\hat E_{a,b})=3c_q(E_{a,b})} equals S3. This identification is the step that converts Cassels' formula (13) into the equality dim_{F3} Sel_ψ = dim_{F3} Sel_{\hatψ} + |S3|-|S2|-1, which is then used for the refined upper and lower bounds, for Corollary 3.9, and for the lower bound in Theorem 4.1. A single misclassified prime would shift the bounds by one. The text says this is 'easy to see from the local theory (Table 1, §2.2.2)', but no verification is supplied. Please provide a complete case-by-case check covering the non-square a∈K_q^{*2} cases, the 2-adic rows T2,2 and T2,3, and the p|3 case 3|b in §2.2.2.
  2. [Proposition 3.10] The density calculation contains a concrete arithmetic error. The displayed equality 30X/(372ζ(2)) · (2^2/(2^2−1)) · (3^2/(3^2−1)) · (31^2/(31^2−1)) = 31X/(28ζ(2)) is false; the left-hand side equals (30·4·9·961)/(372·3·8·960) · X/ζ(2) = (961/7936) X/ζ(2), not 31/28 X/ζ(2). Consequently the subsequent lower bound 'at least 31X/(29ζ(2))' is also invalid and, as written, exceeds the total number of integers up to X. The qualitative conclusion that a positive proportion of n satisfy the stated conditions survives with the corrected constant, but the proof must be corrected.
  3. [Proposition 3.10, proof] The local inclusion stated in the proof uses A_ℓ = Z_ℓ[X]/(X^2+3n), i.e. the field Q_ℓ(√(−3n)). This is the algebra associated with the dual isogeny \hatψ_{n,n} (whose kernel points on \hat E_{n,n}=E_{−27n,31n} are defined over Q(√(−3n))), whereas the Kummer map for ψ_{n,n} takes values in Q_ℓ(√n)^*/Q_ℓ(√n)^{*3} by Proposition 2.2, since the kernel points of ψ_{n,n} are (0, ±n√n). The claimed bound on dim_{F3} Sel_{ψ_{n,n}}(E_{n,n}/Q) by h3_{K_n}, where K_n=Q(√(−3n)), therefore does not follow from the stated local computation. Please correct the isogeny/algebra, or explain a descent argument that justifies the use of Q(√(−3n)) for Sel_{ψ_{n,n}}.
minor comments (4)
  1. [Proposition 3.10, density formula] The expression '∏_{p|M} d 1/(1−p^{-2})' is garbled; it should presumably read ∏_{p|M, p∤d}(1−p^{-2})^{-1} or the correct variant for the specified gcd condition.
  2. [Theorem 4.1] The statement that the curves have no non-trivial K-rational 3-torsion points is not proved in the text; it follows from a∉K^{*2} and −3a∉K^{*2} (the latter from a≡2 mod 3 in the construction), but this argument should be included for clarity.
  3. [Introduction, 'at least 6%'] With the corrected density in Proposition 3.10, the proportion of squarefree n satisfying the stated conditions is approximately 12.1%, so the 'at least 6%' claim remains true; the numerical statement should nevertheless be recomputed after the density correction.
  4. [Table 1] The heading 'a ∈ K_q^{*2} and Tamagawa Numbers q divides Δ_{E_{a,b}}' is unclear; it should be reformatted to indicate that the condition a∈K_q^{*2} applies to all rows.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 3.8 combines Cassels' formula with independent class-group lemmas; the unproved S'=S2 and S''=S3 equality is a missing local check, not a circular reduction.

full rationale

The derivation chain is: Sel_psi(E/K) is identified with norm-one elements of L*/L*3 satisfying local conditions (6); Propositions 2.4-2.8, Table 1 and Remark 2.10 identify the local Kummer images; Proposition 1.13 translates the global conditions into the modules M(S,a), N(S,a); Proposition 1.12 gives their F3-dimensions as h3_S(L) and h3_S(L)+|S(L)|+2; Cassels's formula (13) with Schaefer's formula (8) compares |Sel_psihat|/|Sel_psi| to a product of Tamagawa ratios; Theorem 3.8 combines these ingredients. None of these steps sets its output equal to its input: the Selmer bounds are not used to define S1, S2, S3, and the class-group ranks are not derived from the Selmer dimensions. The same-author citations [JMS] supply auxiliary algebraic and local facts (Propositions 1.12, 1.13, Lemma 2.1), but those facts are parameter-free, do not assume the Selmer bounds, and are not the target result, so they constitute independent support rather than a circular premise. The principal weakness is the unproved assertion in the proof of Theorem 3.8 that, under 3 not dividing a, S' = S2 and S'' = S3; the text says this is 'easy to see from the local theory (Table 1, Section 2.2.2)' but gives no derivation. This is an omitted verification and a correctness risk, not a circular reduction: S2 and S3 are defined by explicit prime-divisibility conditions, and the asserted equality is a local computation feeding into Cassels's formula. Similarly, Proposition 3.10's statement that the required local Kummer inclusions can be shown 'using (8)' is left without detail. These gaps affect the sharpness of the bounds and the large-rank construction if a prime were misclassified, but they do not make the paper's derivation equivalent to its own inputs. Overall, the central claim has independent content and is not forced by definition or by a self-citation chain.

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

No free parameters are fitted; the sets S1,S2,S3 are defined from a and b. The bounds are expressed in terms of class group ranks, which are external arithmetic objects. The key external inputs are Schaefer's and Cassels' formulas, propositions from the authors' prior work [JMS], the Nakagawa-Horie density result, and the known 3-parity conjecture. The paper's explicit density formula in Prop 3.10 contains a numerical typo, noted in red flags.

assumptions (6)
  • standard math Schaefer's formula (8) for the size of E-hat(F)/phi(E(F)) in terms of Tamagawa numbers and formal derivative
    Used throughout Section 2 to compute local quotient sizes at each prime.
  • standard math Cassels' formula (13) for the ratio |Sel_phi-hat| / |Sel_phi|
    Basis for the refined bounds in Theorem 3.8; attributed to Cassels [Ca2].
  • domain assumption Propositions 1.12 and 1.13 on dimensions of M(S,a), N(S,a) and N'(S,a)
    Quoted from the authors' prior paper [JMS, Section 2] without proof; used to convert Selmer group inclusions into class group rank bounds.
  • domain assumption Nakagawa-Horie density result on vanishing of h3 for at least 50% of the specified quadratic fields
    Used in Proposition 3.10 to produce a positive proportion of n with trivial Sel_psi.
  • standard math 3-parity conjecture over K, stated as a known theorem of Nekovar and Dokchitser-Dokchitser
    Used in Corollary 3.9 and Proposition 3.10 to relate Selmer rank parity to root number.
  • standard math Tate's algorithm and p-minimal model computations in Section 2.2
    Used to compute Tamagawa numbers and local Kummer images; standard algorithmic background.

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Pith. "Pith review of $\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks." pith.science (2026). https://pith.science/paper/T3QBYGXE

@misc{pith2026250201069,
  author       = {Pith},
  title        = {Pith review of: $\sqrt-3$-Selmer groups, ideal class groups and large $3$-Selmer ranks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3QBYGXE}},
  note         = {Machine review of arXiv:2502.01069}
}
abstract

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.

Discussion (0). Continue with ORCID to comment.

Reference graph

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