{"id":"043dbb99-2f4b-471c-a2e4-26ea8829856e","arxiv_id":"2502.01069","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The 3-Selmer ranks of the elliptic curves y^2=x^3+a(x-b)^2 over Q(zeta_3) are controlled by ideal class group 3-ranks, yielding large-rank families and a positive proportion of rank-one curves.","lead":"Mathematicians studied a family of elliptic curves with a special 3-fold symmetry and found that their 3-Selmer groups, a measure of rational point complexity, are controlled by ideal class groups of related quadratic fields. They used this to construct infinitely many examples with arbitrarily large 3-Selmer rank and to prove that many simple curves have 3-Selmer rank exactly one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.8's refined bounds hinge on an unproved equality of Tamagawa-ratio prime sets; a misclassified prime would shift the Selmer exponent and break the large-rank construction.","rationale":"The reader's weakest assumption is correctly identified: the equality S' = S2 and S'' = S3 is load-bearing for the refined bounds and the large-rank theorem, and the proof merely asserts it. I agree that this needs to be written out or checked computationally. The rest of the central argument appears coherent: the Selmer description in (6), the local Kummer computations in Section 2, and the class-group bounds in Theorem 3.5 fit together, and the local Tamagawa table does make the asserted classification plausible. There is also a separate, concrete numerical error in Proposition 3.10: the displayed count of squarefree integers in the specified congruence classes, 31X/(28 zeta(2)), exceeds the total number of squarefree integers up to X, which is about 0.608X; hence the claimed density is certainly overcounted. That error affects the positive-proportion statement but does not invalidate Theorem 3.8 or Theorem 4.1. Since the main structural claim has not been shown false, only insufficiently justified at one key step, the reader's conditional verdict should stand.","tokens_in":23171,"tokens_out":33843,"duration_ms":317008,"concrete_test":"For a set of examples satisfying gcd(a,b) square-free in K and 3 not dividing a, with representatives covering every row of Table 1 plus the special cases 2 divides b, 4 divides b, 8 divides b, and 3 divides b, compute the Tamagawa numbers c_q(E_{a,b}) and c_q(E_{-27a,4a+27b}) using Tate's algorithm in SageMath or Magma. Verify that the ratio c_q(E_{a,b})/c_q(hat{E}_{a,b}) is 3 exactly for q in S2, is 1/3 exactly for q in S3, and is 1 for every other prime q, including q = p and q = 2. If any example violates this equality, Theorem 3.8 and Corollary 3.9 fail as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 3.8, the paper asserts without derivation that, under 3 not dividing a, the set S' of primes where c_q(E_{a,b}) = 3 c_q(hat{E}_{a,b}) is exactly S2, and the set S'' where c_q(hat{E}_{a,b}) = 3 c_q(E_{a,b}) is exactly S3. The text says this is 'easy to see from the local theory (Table 1, Section 2.2.2)', but no derivation is supplied. This identification is the step that converts Cassels' formula (13) into the equality dim_{F3} Sel_phi(E_{a,b}/K) = dim_{F3} Sel_hatphi(hat{E}_{a,b}/K) + |S3| - |S2| - 1. That equality is then used for the refined upper and lower bounds, for the root-number parity formula in Corollary 3.9, and for the lower bound dim_{F3} Sel_phi >= 2n in Theorem 4.1. A single prime misclassified between S2 and S3 would change the exponent by one, shifting the bounds and potentially destroying the large-rank conclusion. The classification is plausible from the rows of Table 1, but the non-square cases, the 2-adic cases in T2,2 and T2,3, and the p-adic case with 3 dividing b all need to be checked explicitly; the paper does not provide that check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23451,"tokens_out":40824,"duration_ms":367909,"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":[{"comment":"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.","section":"Theorem 3.8, proof"},{"comment":"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.","section":"Proposition 3.10"},{"comment":"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}}.","section":"Proposition 3.10, proof"}],"minor_comments":[{"comment":"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.","section":"Proposition 3.10, density formula"},{"comment":"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.","section":"Theorem 4.1"},{"comment":"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.","section":"Introduction, 'at least 6%'"},{"comment":"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.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are plausible and the paper contains useful new results, but the proof of Theorem 3.8 is incomplete at a load-bearing point, and Proposition 3.10 has both a concrete arithmetic error and an apparent mismatch in the quadratic algebra. These issues are fixable but require substantive rewriting. The algebra mismatch in Proposition 3.10 may be more than a typo; if it cannot be repaired, the positive-proportion theorem would be unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine advance in the 3-isogeny descent program, and the main theorems are probably right, but two things stand between the paper and being reliable as written. The refined bounds in Theorem 3.8 rest on an unproved identification of Tamagawa-ratio prime sets, and Proposition 3.10 contains a density calculation that is numerically wrong.\n\nWhat's new: the two-parameter family E_{a,b}: y^2 = x^3 + a(x-b)^2, with a rational 3-isogeny, is a natural next step after the authors' earlier work on E_a. The paper gives upper and lower bounds for Sel_phi over K=Q(ζ3) in terms of 3-ranks of S-class groups of K(√a), uses Cassels' formula to sharpen them, derives the root number parity, constructs infinite families with arbitrarily large 3-Selmer rank and no K-rational 3-torsion (Theorem 4.1, an elegant choice of primes), and proves a positive proportion of E_{n,n}/Q have root number -1 and 3-Selmer rank 1. The local computations in Section 2 are substantial and the table of Tamagawa data is useful. The linkage to class groups via M(S,a) and N(S,a) is clean and the examples in Table 2 help.\n\nSoft spots, in order of severity. First, the proof of Theorem 3.8 defines S' and S'' as the primes where the Tamagawa ratio is 3 or 1/3, and then says 'it is easy to see' that S'=S2 and S''=S3. That identification is the step that converts Cassels' formula into the equality dim Sel_phi = dim Sel_hatphi + |S3|-|S2|-1. Every refined bound, the root number formula, and the large-rank construction depend on it. The stress-test is right: a single misclassified prime shifts the exponent by one. The classification is plausible from Table 1 and the p-adic cases, but the non-square, 2-adic, and 3|b cases need to be checked explicitly. This is a load-bearing gap, not a cosmetic one; the referee should demand the derivation.\n\nSecond, Proposition 3.10 has a concrete arithmetic error. The displayed simplification '30X/372ζ(2) · 4/3 · 9/8 · 961/960 = 31X/28ζ(2)' is wrong: the left side equals 31X/256ζ(2). The subsequent 'at least 31X/29ζ(2)' is off by a factor of about nine. The qualitative claim of a positive proportion survives, but the stated quantitative estimate is not correct. The proof also asserts, without demonstration, the local Kummer image inclusion for every ℓ that gives dim Sel_psi ≤ h3(K_n); that too should be written out.\n\nOn citation pattern: the paper leans on the authors' prior [JMS] for several propositions. That is acceptable since those results are cited, but it does make the paper less self-contained. No misattribution issues.\n\nWho should read it: people working on Selmer ranks, isogeny descent, and class group links. It deserves a serious referee: the main ideas are novel and the constructions are real. I'd send it to review, with a request for major revision to fill the Tamagawa check and fix the density.","headline":"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.","tokens_in":24040,"tokens_out":6468,"would_cite":true,"duration_ms":56325,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11R29","11R34","11G40","11S25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For this family, 3-Selmer rank is the 3-class rank of K(√a) up to small prime corrections.","keywords":["elliptic curves","3-Selmer groups","ideal class groups","3-isogeny","large 3-Selmer ranks","root number","Kummer maps","cyclotomic field Q(ζ3)"],"falsifier":"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.","tokens_in":22953,"feed_emoji":"🧮","tokens_out":23673,"duration_ms":207946,"temperature":0.7,"pith_summary":"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.","feed_headline":"3-Selmer rank equals 3-class rank, up to prime corrections","feed_subtitle":"The φ-Selmer rank of Ea,b over Q(ζ3) sits inside an explicit interval given by class-group ranks of K(√a).","key_machinery":"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.","core_discovery":"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 $h^3_L+1$.","pith_inferences":["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."],"forward_implications":["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 $+1$."],"supporting_citations":[{"why":"It supplies the algebraic backbone, including the identification of the isogeny-Selmer group with norm-one classes and the dimension formulas for the class-group modules.","marker":"[JMS]"},{"why":"It provides the local formula expressing $|\\hat E(K_q)/\\psi(E(K_q))|$ in terms of local component counts, used for every local Kummer image.","marker":"[Sc]"},{"why":"It supplies the Selmer-ratio identity (13) that compares the two isogeny Selmer groups and yields the refined bounds.","marker":"[Ca2]"},{"why":"It gives the explicit local Kummer-map formulas used to describe the Selmer condition at each prime.","marker":"[Ca3]"},{"why":"It supplies the exact sequence connecting the $\\psi$-Selmer group and the dual-isogeny Selmer group to the full 3-Selmer group.","marker":"[SS]"},{"why":"It provides the class-number vanishing proportion for the chosen residue classes used in the positive-proportion rank-one result.","marker":"[By]"},{"why":"It establishes the group-scheme isomorphism between the isogeny kernel and the kernel of a restriction-of-scalars norm map, on which the norm-one description rests.","marker":"[BES]"}],"fun_headline_variants":["Selmer rank tied to class group rank","3-Selmer rank bounded by class group ranks","Arbitrarily large 3-Selmer ranks with no 3-torsion","Class groups control 3-Selmer dimensions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Selmer rank tied to class group rank","3-Selmer rank bounded by class group ranks","Arbitrarily large 3-Selmer ranks with no 3-torsion","Class groups control 3-Selmer dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00076,"raw_usage":{"total_tokens":3429,"prompt_tokens":1052,"completion_tokens":2377,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":668,"completion_tokens_details":{"reasoning_tokens":2309}},"tokens_in":668,"tokens_out":2377,"duration_ms":19241,"temperature":1.0,"reasoning_tokens":2309,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:42:03.765373+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}