REVIEW 4 major objections 4 minor 25 references
Relative $p$-class groups and $p$-Selmer groups
T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper claims that for CM elliptic curves with j-invariant 0 or 1728, the p-Selmer dimension of every twist is forced to be $r_p$ or $r_p+1$ by a relative p-class group, and parity then selects the exact value.
desk verdict Solid Rubin-refinement paper whose 'determines' claim rests on an unproved parity input; fix that and it's publishable. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the relative p-class group $\mathrm{Cl}(L_{C,p}/F_{C,p})[p]$ and its $\chi_C$-isotypic component, the p-torsion classes of $L_{C,p}$ that transform by the same character as the p-torsion representation of $C$ and are not inherited from $F_{C,p}$. The paper splits the p-part of the class group as $\mathrm{Cl}(L_{C,p})[p] \cong \mathrm{Cl}(L_{C,p}/F_{C,p})[p] \times \mathrm{Cl}(F_{C,p})[p]$, so a descent-theoretic dimension $\dim_{\mathfrak{k}_p}\mathrm{Hom}(\mathrm{Cl}(L_{C,p}), C[p])^G$ becomes a dimension of that relative component. Division polynomials for $y^2=x^3-Dx$ and $y^2=x^3+A$ supply explicit polynomials $f_{C,p}$ such that $F_{C,p}$ is presented as $K[X]/f_{C,p}(X)$ and $L_{C,p}$ is generated by a root of $f_{C,p}(X^k)$, with the same $F_{C,p}$ for all twists; these identities carry the argument that the class-group input is a single invariant attached to the fixed field.
What would settle it
Compute $s_p(E_t)$ and $r_p(E_t)$ for a twist with $r_p(E_t)=1$ and root number $\omega(E_t)=+1$: the paper's central claim predicts $s_p(E_t)=1$, so a computation returning $s_p(E_t)=2$ would refute the claimed uniqueness. Equivalently, any single good prime and twist with $s_p(E_t) \not\equiv \omega(E_t) \pmod 2$ would falsify the parity step on which the exact determination rests.
Extended reading notes
Core claim
For each twist $C$ of the base curve, let $K$ be $\mathbb{Q}(i)$ (when $j=1728$) or $\mathbb{Q}(\zeta_3)$ (when $j=0$), and let $L_{C,p}=K(C[p])$ be the field generated by the $p$-torsion points of $C$. The Galois group of $L_{C,p}/K$ is cyclic, and its natural character $\chi_C$ describes the action on $C[p]$. The paper shows that for a cyclic subextension $F_{C,p}$ of degree $2$, $3$, $4$, or $6$ (according to the twist type), the dimension $r_p(C)=\dim_{\mathfrak{k}_p}\mathrm{Cl}(L_{C,p}/F_{C,p})[p](\chi_C)$ satisfies $r_p(C) \leq s_p(C) \leq 1+r_p(C)$, and that the fields $F_{C,p}$ themselves are independent of the twist up to isomorphism. Combined with the parity relation $s_p(C) \equiv \omega(C) \pmod 2$, which the paper invokes, this pins $s_p(C)$ exactly. The proof passes from the full class group of the division field to the relative class group of a fixed extension, using explicit division-polynomial generators for the intermediate fields.
Load-bearing premise
The load-bearing premise is the unproved p-parity conjecture for these CM twists, used to decide between the two consecutive integers allowed by the bound; if that parity fails, the same theorems only give $r_p \le s_p \le 1+r_p$, not an exact value.
Editorial extensions
If this is right
- For a twist $E_t$ with $r_p(E_t)=0$, the bound forces $s_p(E_t)$ to be $0$ or $1$, and the paper's parity step then decides whether the curve has rank zero or rank one.
- Rank-zero and rank-one p-converse theorems for these CM curves become equivalent to statements about the $\chi$-component of a relative p-class group: trivial for rank zero, cyclic for rank one, checked at a single good prime.
- For squarefree $n \equiv 1,2,3 \pmod 8$, the congruent-number status of $n$ is controlled by the relative quadratic class group: if $n$ is congruent, every prime $p\nmid 2n$ divides the relative class number $|\mathrm{Cl}(L_{n^2,p}/K_{1,p})|$, while one prime with trivial component proves $n$ is not congruent.
- For the cube-sum problem, a cube-free $n$ with root number $+1$ that is a rational cube sum forces every prime $p\nmid 6n$ to divide the relative cubic class number $|\mathrm{Cl}(L_{-432n^2,p}/K_{-432,p})|$.
- Large-rank elliptic curve families yield infinitely many number fields of degree 4 (for $j=1728$) or degree 6 (for $j=0$) over the fixed field $F_{1,p}$ with p-class rank at least $12+\mathrm{rk}\,\mathrm{Cl}(F_{1,p})[p]$ or the corresponding $6+\mathrm{rk}$ bound, giving concrete progress on the unboundedness conjecture for class group p-ranks.
Reading between the lines
- Implicit in the paper, the bottleneck is the p-parity conjecture: if that parity is proved for these CM twists, the main theorems become unconditional, while one parity failure would collapse the exact determination back to the two-point bound.
- Because $F_{C,p}$ is twist-independent, the method suggests a practical test: computing the $\chi$-component dimension in one fixed field should predict Selmer ranks across an entire twist family, so numerical discrepancies between class-group predictions and Selmer computations would pinpoint where the parity assumption fails.
- The same relative-decomposition strategy may extend to other CM elliptic curves or to higher-degree twist types, provided division-polynomial identities of the same shape can be produced; the paper does not claim such extensions.
- If the Selmer rank is unbounded in either CM twist family, Proposition 4.8 transfers that unboundedness to p-ranks of class groups in degree 4 and 6 extensions, linking two longstanding open problems; this conditional consequence is the paper's own observation, not a proof.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies elliptic curves with j-invariant 0 or 1728 and their quadratic, cubic, quartic, and sextic twists. For each twist E_t and each good odd prime p, the authors define a relative p-class group attached to a subfield of the p-division field and denote by r_p(E_t) the dimension of the character component corresponding to the Galois action on E_t[p]. Using Rubin's descent theorem they prove inequalities r_p(E_t) ≤ s_p(E_t) ≤ 1+r_p(E_t), where s_p(E_t) is the F_p-dimension of the p-Selmer group. They then assert that the root number ω(E_t) and r_p(E_t) uniquely determine s_p(E_t), relying on the congruence s_p(E_t) ≡ ω(E_t) mod 2, which is invoked as the 'p-parity conjecture' without proof or citation. This determination is used to reformulate rank 0 and rank 1 p-converse theorems in terms of relative class groups, to derive criteria for congruent-number and cube-sum problems, to construct number fields with large p-rank class groups, and to produce numerical tables of ranks.
Significance. If the parity step were justified, the paper would provide an attractive and concrete bridge between Selmer ranks and relative class groups for CM elliptic curves, with explicit division-polynomial descriptions of the relevant fields and a clean numerical method. The construction of infinite families of degree 4 and 6 extensions with lower bounds on p-class group rank is also a potentially interesting contribution. The paper uses standard machinery (Rubin's descent, division polynomials, root-number formulas) and no free parameters are fitted; the claimed Selmer-to-class-group inequality is a genuine theorem of Rubin. However, the central 'complete determination' conclusion is conditional: the parity congruence is the only step that turns the two-element interval [r_p, r_p+1] into a unique value, and for the inert primes considered (p ≡ 3 mod 4 for j=1728 and p ≡ 5 mod 6 for j=0) the reduction is supersingular, where p-parity is not a standard unconditional consequence of Iwasawa main conjectures. The paper therefore currently overstates what is proved.
major comments (4)
- [Theorems 2.3, 2.4, 3.2, 3.3] The sentence 'From p-parity conjecture, we know that s_p(E) ≡ ω(E) (mod 2)' appears in the proofs of all four theorems with no proof and no citation. This congruence is the only step that allows the authors to conclude that ω(E_t) and r_p(E_t) 'uniquely determine' s_p(E_t). Without it, the Selmer dimension is only known to lie in {r_p, r_p+1}, so the advertised conclusion is not established. The problem is particularly acute for inert primes, where the reduction is supersingular and p-parity is not an automatic consequence of the usual main conjectures. The authors should either provide a proof or a precise reference establishing the parity congruence in this supersingular CM setting, or replace the unconditional claims by explicitly conditional statements.
- [Corollaries 2.5, 2.6, 3.4, 3.5 and Corollary 2.8] The equivalences in these corollaries use the same parity step to pass from r_p=0 to s_p=0 and from r_p≤1 to s_p=1. For example, in the proof of Corollary 2.5, the statement 'if r_p(E_D)=0, then by Theorem 2.3 or Theorem 2.4, we have s_p(E_D)=0' is only valid if the parity congruence is available. Likewise, in Corollary 2.8 the assertion that a congruent number forces s_p(E_n^1) ≥ 2 needs the parity input in addition to positivity of rank. These applications should be labeled as conditional on the same parity assumption, or the missing parity result should be supplied.
- [Theorems 4.2 and 4.6] The proof of Theorem 4.2 uses the step 'since for n∈S_p, we have ω(E_n)=-1 and hence s_p(E_n) is odd, and moreover as rk_Q E_n ≥ 6, we see that s_p(E_n) ≥ 7'. This uses the unproved parity congruence to exclude s_p(E_n)=6. If parity is not known for these supersingular primes, one only obtains s_p(E_n)≥6, hence r_p(E_n)≥5 instead of the stated r_p≥6, and the displayed lower bounds 12+rkCl(F) and 6+rkCl(F) in Theorem 4.2 must be reduced accordingly. The same issue affects Theorem 4.6. Thus the class-group construction is quantitatively dependent on the same load-bearing assumption.
- [Proofs of Theorems 2.4 and 3.3] The field-identification arguments in the split/inert cases contain several asserted irreducibility and factorization statements that are not fully justified as written. For example, in the proof of Theorem 2.4 the sentence 'Consequently, f_{D,p}(X^4) has at-least one irreducible factor of degree (p-1) over K' and the subsequent case analysis require the reader to fill in nontrivial steps; similarly, in Theorem 3.3 the phrase 'proceeding as in the proof of Theorem 2.4, we conclude...' skips the analogous argument. Since the field identifications F_{D,p} ≅ F_{1,p} and L_{D,p} ≅ Q[X]/(f_{D,p}(X^4)) are used in the numerical applications and in the independence-of-D statements, these arguments should be expanded to a verifiable level of detail.
minor comments (4)
- [Corollary 2.8, proof] The phrase 'which in-turn implies that s_p ≥ 2' is not immediate and should mention that the root number is +1 and that the parity congruence is being used.
- [Section 2, notation] The notation for primes p and p over p in the split case (Theorems 2.4 and 3.3) is sometimes ambiguous; for instance, statements such as 'p is unramified in L_{D,p}/K' and 'p is totally ramified in L_{D,p}/K' should be indexed by the chosen conjugate prime to avoid confusion.
- [Section 5, tables] The tables do not indicate which entries depend on the p-parity congruence in addition to GRH; given the central role of parity, the table headers or footnotes should make this dependence explicit.
- [Throughout] There are several minor typographical and grammatical issues, e.g. 'p division field' should be 'p-division field' and 'deg ree six' should be 'degree six'; these do not affect the mathematics.
Circularity Check
No circularity: the Selmer-to-class-group bounds are imported from Rubin's theorem and the field identifications are independently derived; the main caveat is an unproved p-parity assumption, not a circular step.
full rationale
No circular step is exhibited. The central inequality r_p(E) <= s_p(E) <= 1 + r_p(E) in Theorems 2.3, 2.4, 3.2 and 3.3 is imported from [Rub87a, Theorem 1], and the relative-class-group reformulation is obtained by the character decomposition Cl(L_D,p)[p] = Cl(L_D,p/F_D,p)[p] x Cl(F_D,p)[p] under p not dividing [L_D,p : K], together with the definition of the chi-component. Neither step defines the class-group rank in terms of the Selmer rank, nor fits any parameter to data. The field isomorphisms such as F_D,p = F_1,p are derived from explicit division polynomials, homogeneity, and degree/ramification arguments rather than assumed. The statement 'From p-parity conjecture, we know that s_p(E_D) = omega(E_D) (mod 2)' is load-bearing for the advertised conclusion that omega(E_D) and r_p(E_D) uniquely determine s_p(E_D), and the paper gives no proof or citation for this congruence; however, this is an unproved external conjecture, not a self-referential reduction, so it is a correctness/conditionality concern rather than circularity. The self-citations [JMS22, JMS23] appear only in peripheral cube-sum corollaries and do not support the main derivation. The paper is self-contained against external benchmarks for its main inequalities, so the circularity score is 0.
Assumptions & free parameters
assumptions (7)
- domain assumption Rubin's descent theorem (Rub87a): for E with j=0 or 1728 and prime p of good reduction, dim Hom(Cl(L), E[p])^G ≤ dim Sel_p(E/Q) ≤ dim Hom(Cl(L), E[p])^G + 1.
- domain assumption p-parity conjecture: dim Sel_p(E/Q) ≡ ω(E) (mod 2) for the twisted CM curves.
- domain assumption Coates-Wiles and Rubin results on L-values and finiteness of Tate-Shafarevich for rank 0 and 1 CM curves.
- domain assumption Burungale-Tian p-converse theorem (BT20).
- domain assumption Kihara's rank-6 family and Mestre's rank-7 family over Q(r,s,t) and Q(v).
- standard math Silverman's specialization theorem and Faltings' finiteness theorem for families of elliptic curves.
- standard math Greaves' power-free values theorem for binary forms.
Cite this review
Pith. "Pith review of Relative $p$-class groups and $p$-Selmer groups." pith.science (2026). https://pith.science/paper/Q6HIVT75
@misc{pith2026241213022,
author = {Pith},
title = {Pith review of: Relative $p$-class groups and $p$-Selmer groups},
year = {2026},
howpublished = {\url{https://pith.science/paper/Q6HIVT75}},
note = {Machine review of arXiv:2412.13022}
}
abstract
Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group.
Reference graph
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