{"id":"d63f5f32-7d47-4746-b625-737f4af0ef6b","arxiv_id":"2508.21406","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In many families of elliptic curves with a rational prime-degree isogeny, the logarithmic Tamagawa ratio satisfies a central limit theorem, yielding curves with arbitrarily large ℓ-Selmer groups for ℓ = 2, 3, 5, 7, 13.","lead":"Elliptic curves with a rational isogeny of prime degree are shown to have Selmer ratio fluctuations that follow a bell-curve law as height grows. The paper uses this to prove that for primes 2, 3, 5, 7, and 13, some elliptic curves have arbitrarily large ℓ-Selmer groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"ℓ=13 family's admissibility under (A4) is asserted without exhibiting f,g or the counting asymptotic; Theorem 1.1's ℓ=13 conclusion depends on this unproven check.","rationale":"After a full pass over the argument, the central strategy is coherent: Cassels' ratio, local Tamagawa computations via [15], lattice-point equidistribution (Sections 4-5), and the moment/sieve arguments are structurally sound, with no fitted parameters. The main bridge from these analytic estimates to arithmetic is the admissibility notion of Definition 5.8. For each family in Theorem 1.1, admissibility must be verified explicitly. For the torsion families and the 7-isogeny family this is done (with citations for counts and displayed f,g). The 13-isogeny family is the exception: E is never given in Weierstrass form, and the required hypotheses (A4) are asserted. This is precisely the kind of missing support that should be supplied before relying on the ℓ=13 conclusion. The reader's verdict of CONDITIONAL is appropriate; our check would either upgrade or confirm the gap. We therefore do not alter the verdict.","tokens_in":42651,"tokens_out":15228,"duration_ms":149647,"concrete_test":"Apply Vélu's formula (or [25, Table 7]) to the displayed E′ and the 13-isogeny to obtain explicit f,g for E. Then check: (1) no common real roots and max(deg f/4, deg g/6)=2; (2) factor Δ and Δ′ to confirm D+=a, D−=b and compute 6B(0,1), 6B(1,0) to verify v+=1/2, v−=1; (3) prove the counting asymptotic |{E_t: H(E_t)≤N}|∼cN^{1/12} from the parametrization, showing |S(N)|=(1+O(N^{-ξ}))|F0(N)|. If these hold, the ℓ=13 family is admissible; if not, Theorem 1.1 for ℓ=13 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6.5 defines only E′ via f′,g′, then takes E as the codomain of the dual 13-isogeny and asserts E is admissible under (A4) with m=2. Definition 5.8(A4) requires f,g∈Z[t] for E with no common real roots and m=max(1/4 deg f,1/6 deg g)=2, plus |S(N)|=(1+O(N^{-ξ}))|F0(N)|. None of these are demonstrated: f,g are never displayed, so F0(N) is not explicitly defined and the degree/coprimality conditions cannot be checked; the count |S(N)|∼cN^{1/12} for the dual family is asserted (via injectivity of t) rather than proven; and the values v±=1/2,1 used for constants µ,σ rely on Legendre-symbol evaluations 6B(1,0)=12, 6B(0,1)=-4116 that are stated without giving B. The reader's reference to Lemma 4.10's gcd conditions is not exactly the relevant hypothesis for (A4), but the underlying gap is real and load-bearing: if the check fails, the ℓ=13 family is not admissible, and Theorem 1.1's ℓ=13 case is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for studying the logarithmic Selmer ratio r_phi(E) = dim Sel_phi(E/Q) - dim Sel_{\\hat\\phi}(E'/Q) in families of elliptic curves over Q carrying a rational prime-degree isogeny. The main technical tool is Cassels' comparison of isogeny Selmer groups with Tamagawa ratios, combined with Dokchitser–Dokchitser local formulas and with equidistribution results for lattice points attached to weighted homogeneous parametrizations. In admissible families (Definition 5.8) the authors prove a central limit theorem for r_phi, lower bounds for moments \\sum \\ell^{k r_phi}, and tail lower bounds, and apply these to families with prescribed torsion, to the family with a cyclic 4-isogeny, and to specific 7- and 13-isogeny families. The announced consequence is the existence of elliptic curves with arbitrarily large \\ell-Selmer groups for \\ell \\in \\{2,3,5,7,13\\}.","tokens_in":42997,"tokens_out":15798,"duration_ms":160412,"significance":"If correct, this is a substantial and broad result: it unifies and extends earlier unbounded-Selmer constructions and gives quantitative distribution statements in many parametrized families. A notable strength is that the argument is structural: the constants c_\\pm, \\mu and \\sigma are not fitted but are computed from explicit polynomials and Chebotarev counts. The main technical apparatus in Sections 4--5 is coherent, and the use of Cassels' ratio is a natural and powerful organizing principle. The paper is likely to be influential if the concrete family verifications are completed and made fully checkable.","major_comments":[{"comment":"The \\ell=13 family is declared admissible under type (A4), but the required data for the curve E are never supplied. Definition 5.8(A4) requires f,g \\in Z[t] for E with no common real roots, m = max(1/4 deg f, 1/6 deg g) = 2, and |S(N)| = (1+O(N^{-\\xi}))|F_0(N)|. The paper displays f',g' for the domain curve E' only, and then asserts that the codomain E of the dual 13-isogeny is admissible. The displayed discriminant formula is not enough to determine f,g or to rule out common real roots, and the asymptotic count for the dual family is asserted rather than proved. Since Theorem 1.1's \\ell=13 case depends entirely on this admissibility check, this is a load-bearing gap.","section":"Section 6.5"},{"comment":"With the displayed f(t) = -3(t^2+13t+49)(t^2+5t+1) and g(t) = 2(t^2+13t+49)(t^4+14t^3+63t^2+70t-7), a direct computation gives s = gcd(f^3,g^2) = (t^2+13t+49)^2 up to a rational constant, not (t^2+5t+1)^2 as stated. The verification of condition (A3) therefore contains a false identity. If the intended family uses a different g, the defining equations should be corrected; if the displayed equations are correct, the correct s should be used and the remaining (A3) hypotheses rechecked.","section":"Section 6.4"},{"comment":"The tabulated constants u_\\pm, v_\\pm, \\mu, \\sigma and \\rho are central outputs, yet only the Z/5Z computation is presented in detail. As written, most rows of Tables 1-3 and both rows of Table 5 function as unverified assertions. I am not requesting that every V\\'elu computation be reproduced in the text, but the submission should include either an appendix with the necessary intermediate models and discriminants for each row, or a computer-algebra script that certifies the listed values. This is particularly urgent for Table 5, where the 13-isogeny row depends on the missing verification in Section 6.5.","section":"Section 6, Tables 1-5"}],"minor_comments":[{"comment":"The sentence 'Recall H0 \\leq H' should be 'H \\leq H0'; the argument uses that H0 \\leq N implies H \\leq N.","section":"Section 6.5"},{"comment":"In the proof of Lemma 5.7, 'where M_- and M_- are constants' is a typo; the second occurrence should be M_+.","section":"Section 5.6"},{"comment":"There appear to be OCR-type typos in the displayed discriminant: '49ab' should presumably be '49b^2', and 'a^2+5ab+b' should be 'a^2+5ab+b^2'.","section":"Section 6.4"},{"comment":"The definition of A^\\delta_\\upsilon(N) is ambiguous: the displayed condition reads 'N e12' with no exponent on N, and \\delta is not introduced in Section 2. The formula should be written as max{...} \\leq N^{12\\delta} e^{12} (or the equivalent intended normalization) so that the later use of R_{N e^{12}} in Section 4.2 is clear.","section":"Equation (2.5)"}],"recommendation":"major_revision","confidential_remarks":"I share the reader's conditional assessment. The main obstruction is Section 6.5: the 13-isogeny family is not actually shown to be admissible, and this is load-bearing for one of the headline conclusions. The Section 6.4 gcd inconsistency, even if typographical, also needs a substantive correction because it appears in the verification of an admissibility hypothesis. If the authors can supply the missing coefficients and counting arguments, and provide reproducible verification of the tables, the paper is likely to be acceptable. The reliance on the authors' earlier work [10,11,12] is legitimate and does not by itself raise a circularity concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real result, not a stunt. The main technical theorem and its proof are in good shape. The paper extends the universe of known Selmer unboundedness from ℓ=2,3 to 5,7,13 and gives the first CLT for Tamagawa ratios across all torsion families. No fitted parameters; the constants come from Chebotarev and explicit polynomial data. The use of Cassels' ratio, Dokchitser–Dokchitser local formulas, lattice point counting, and the method of moments is coherent. I checked the central chain (Lemmas 3.6, 4.10, 5.7, Theorem 5.9) and found no structural contradiction. The authors are honest about the dependence on their earlier work, and the self-citation is benign.\n\nThe soft spot is Section 6.5. For the ℓ=13 family, the authors define E′ explicitly but only give the dual family E via its discriminant and isogenous curve. They assert E is admissible under (A4) with m=2, but they never display f and g for E. That means the reader cannot check the degree condition, the no-common-real-roots condition, or the count |S(N)| ~ c N^{1/12}. The values v±=1/2,1 also depend on Legendre-symbol computations for B(1,0) and B(0,1), where B is not written down. This is load-bearing: if E does not satisfy (A4), Theorem 5.9 does not apply and the ℓ=13 unboundedness claim is unsupported. I think the gap is fixable—the parametrization from Maier's table should give f,g in principle—but it has to be filled, not asserted. The other tables in Section 6 are also thin: several constants are stated without derivation, but those are routine and can be checked; less concerning.\n\nVerdict: worth refereeing seriously. The general theorem and the ℓ=5,7 parts (and the torsion families) are solid and important. I would send it to a good number theory journal, but only after the authors supply the missing ℓ=13 data and the counting argument. If they do that, this is a substantial contribution.","headline":"Strong general theorem for Selmer ratios; the ℓ=13 application is missing its admissibility check and needs a revision before the unboundedness claim for 13 is fully supported.","tokens_in":43488,"tokens_out":4580,"would_cite":true,"duration_ms":45578,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11N36"],"pacs":[],"model":"deepseek-v4-flash","headline":"For families of elliptic curves with a rational prime-degree isogeny, the logarithmic Selmer ratio is asymptotically Gaussian, and for ℓ ∈ {2,3,5,7,13} there exist curves with arbitrarily large ℓ-Selmer groups.","keywords":["elliptic curves","Selmer groups","isogeny","Tamagawa ratio","central limit theorem","lattice point counting","rational isogeny","unbounded Selmer ranks"],"falsifier":"For the $\\ell=13$ family in Section 6.5, derive the polynomials $f$ and $g$ for the curve $E$ (the domain of the dual isogeny, not displayed in the paper) and verify explicitly that $\\max(\\deg f/4, \\deg g/6) = 2$ and that $f$ and $g$ have no common real root; if either condition fails, the family is not admissible under (A4) and the equidistribution propositions do not apply.","tokens_in":42517,"feed_emoji":"🔢","tokens_out":13180,"duration_ms":112749,"temperature":0.7,"texified_at":"2026-08-05T20:18:49.730864+00:00","pith_summary":"This paper proves that in several parametrized families of elliptic curves over the rationals that come with a rational isogeny of prime degree $\\ell$, the logarithmic Selmer ratio — the difference between the dimensions of the Selmer group attached to the isogeny and the Selmer group attached to its dual — is not bounded but fluctuates like a Gaussian random variable as the curves are ordered by height. In particular, for the primes $\\ell = 2, 3, 5, 7, \\text{and } 13$, the $\\ell$-Selmer group itself attains arbitrarily large size inside these families. The reason is that the Selmer ratio is, up to bounded factors, the product of local Tamagawa numbers of the two isogenous curves, and this product can be understood as a multiplicative function of the two parameters that define the curve; the paper proves the needed equidistribution of these parameters in residue classes and then applies the method of moments. The result matters because it shows that unbounded Selmer ranks occur persistently inside thin isogeny families, even though the average size of the Selmer group over all elliptic curves is finite.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":11018,"prompt_tokens":975,"completion_tokens":10043,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":975,"completion_tokens_details":{"reasoning_tokens":9058}},"feed_headline":"Five primes give elliptic curves with unbounded Selmer groups","feed_subtitle":"In torsion and isogeny families, the Selmer ratio is Gaussian and ℓ-Selmer ranks grow without bound.","key_machinery":"A classical identity (Lemma 3.1) bounds $|\\operatorname{Sel}_\\phi(E/Q)|/|\\operatorname{Sel}_{\\hat{\\phi}}(E'/Q)|$ between two constant multiples of $\\prod_p c_p(E')/c_p(E)$. A local classification (Lemma 3.6) determines each factor $c_p(E')/c_p(E) \\in \\{\\ell^{-1}, 1, \\ell\\}$ from the reduction type, so the product becomes $\\ell^{\\sum Y_p(a,b)}$, where $Y_p$ are local functions of the two parametrizing integers $(a,b)$. The paper then shows the pairs $(a,b)$ are equidistributed in residue classes (via lattice-point counting and, in the non-coprime case, a refined sieve-type argument), which fits the product into a central limit theorem for independent random variables and into known averages of multiplicative functions. The constants $\\mu, \\sigma, \\rho(k)$ are read off from","core_discovery":"Theorem 5.9 is the engine: for any admissible family $S$ (Definition 5.8) of curves with a rational degree-$\\ell$ isogeny, the logarithmic Selmer ratio $r_\\phi(E) = \\dim \\operatorname{Sel}_\\phi(E/Q) - \\dim \\operatorname{Sel}_{\\hat{\\phi}}(E'/Q)$, ordered by naive height, converges after centering and scaling to a standard Gaussian, with mean $\\mu \\log \\log N$ and variance $\\sigma^2 \\log \\log N$. The same theorem gives $\\ell^{k r_\\phi}$-moment lower bounds and a Paley–Zygmund tail bound: for every $A > 0$, at least $|S(N)|(\\log N)^{-\\delta(A)}$ curves have $r_\\phi(E) \\geq A \\log \\log N$. Since $|\\operatorname{Sel}_\\ell(E/Q)| \\geq \\ell^{r_\\phi(E)}/O_\\ell(1)$, the $\\ell$-Selmer group is unbounded in each admissible family. Theorem 1.1 applies this to torsion-subgroup families, the cyclic-4-isogeny family, and infinite $\\ell=7$ a","pith_inferences":["Editorial extension: the same Tamagawa-ratio sandwich should apply to any family of curves over ℚ with a rational ℓ-isogeny satisfying the admissibility hypotheses; the principal bottleneck is the lattice-point equidistribution, not the Selmer-group argument.","Editorial extension: the Gaussian shape suggests that in isogeny families the Selmer rank fluctuates on the log-log scale, in contrast with the bounded average over all curves; this gives a quantitative heuristic for Selmer-statistics conjectures in thin families.","Editorial extension: a direct numerical check for the ℓ=7 and ℓ=13 families — computing r_ϕ for curves up to height N and measuring the decay of the tail P(r_ϕ ≥ A log log N) — would confirm the predicted power-of-logarithm decay and the constants δ(A), ρ(k)."],"forward_implications":["For ℓ ∈ {2,3,5,7,13}, there exist elliptic curves over ℚ with arbitrarily large ℓ-Selmer groups.","In every admissible family, the Selmer ratio r_ϕ has Gaussian fluctuations of order sqrt(log log N) with explicit mean and variance.","The ℓ^{k r_ϕ}-moment grows like |S(N)| (log N)^{ρ(k)}; in all listed families ρ(2) > 0, and ρ(1) > 0 except for the torsion types Z/2Z × Z/2mZ with m = 1,3,4 and ℓ = 2.","All families of elliptic curves with a prescribed non-trivial torsion subgroup T, with ℓ dividing |T|, are covered, so unbounded Selmer ranks hold across each of these torsion strata.","The lower-bound method does not require computing the full Selmer group: a large Tamagawa ratio alone forces a large Selmer group."],"supporting_citations":[{"why":"Supplies the identity relating |Sel_ϕ|/|Sel_ϕ̂| to the product of local Tamagawa numbers c_p(E')/c_p(E), the backbone of the whole argument.","marker":"[9]"},{"why":"Gives the local classification of c_p(E')/c_p(E) ∈ {ℓ^{-1}, 1, ℓ} in terms of reduction type, which turns the Tamagawa product into a multiplicative function.","marker":"[15]"},{"why":"Provides the explicit Weierstrass polynomials f,g for elliptic curves with prescribed torsion subgroups, used in the torsion-family computations.","marker":"[3]"},{"why":"Supplies the asymptotic count of elliptic curves with prescribed torsion, needed to verify the admissibility of the torsion families.","marker":"[17]"},{"why":"Gives the rational parametrizations of curves with cyclic n-isogenies (Table 7), used for the ℓ=7 and ℓ=13 families.","marker":"[25]"},{"why":"Provides the count of elliptic curves with a 7-isogeny and the lattice-counting approach adapted for the non-coprime gcd(f^3,g^2) case.","marker":"[27]"},{"why":"Parametrizes and counts elliptic curves with a cyclic 4-isogeny, making that family admissible.","marker":"[30]"},{"why":"Davenport's lattice point theorem supplies the main term in the equidistribution estimates for the pairs (a,b).","marker":"[14]"},{"why":"The fundamental lemma of sieve from this reference is used in the lower-bound proof of Theorem 5.2.","marker":"[19]"},{"why":"Supplies the theorem on averages of multiplicative functions along equidistributed sequences that yields the moment and tail bounds.","marker":"[12]"}],"fun_headline_variants":["ℓ-Selmer groups unbounded for five primes","Gaussian Selmer ratio drives unbounded ℓ-Selmer groups","Central limit theorem yields arbitrarily large Selmer groups","Unbounded Selmer groups from five prime isogenies"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The argument depends on the admissibility hypotheses of Definition 5.8 — most crucially, for families where $f$ and $g$ share a factor, that $\\gcd(f^3, g^2)$ has degree below $24m/(2+m)$ and is an $r$-th power of a squarefree polynomial with $r \\in \\{2,3,4,6\\}$; if a concrete family fails this, the equidistribution estimates and hence the whole theorem collapse for that family.","fun_headline_variants_meta":{"raw":{"variants":["ℓ-Selmer groups unbounded for five primes","Gaussian Selmer ratio drives unbounded ℓ-Selmer groups","Central limit theorem yields arbitrarily large Selmer groups","Unbounded Selmer groups from five prime isogenies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001298,"raw_usage":{"total_tokens":5104,"prompt_tokens":684,"completion_tokens":4420,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":428,"completion_tokens_details":{"reasoning_tokens":4361}},"tokens_in":428,"tokens_out":4420,"duration_ms":30014,"temperature":1.0,"reasoning_tokens":4361,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T14:19:50.700160+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the $\\ell=13$ family in Section 6.5, derive the polynomials $f$ and $g$ for the curve $E$ (the domain of the dual isogeny, not displayed in the paper) and verify explicitly that $\\max(\\deg f/4, \\deg g/6) = 2$ and that $f$ and $g$ have no common real root; if either condition fails, the family is not admissible under (A4) and the equidistribution propositions do not apply.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the identity relating |Sel_ϕ|/|Sel_ϕ̂| to the product of local Tamagawa numbers c_p(E')/c_p(E), the backbone of the whole argument."},{"cited_title":"Dokchitser and V","cited_arxiv_id":null,"evidence_quote":"Gives the local classification of c_p(E')/c_p(E) ∈ {ℓ^{-1}, 1, ℓ} in terms of reduction type, which turns the Tamagawa product into a multiplicative function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit Weierstrass polynomials f,g for elliptic curves with prescribed torsion subgroups, used in the torsion-family computations."},{"cited_title":"Harron and A","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic count of elliptic curves with prescribed torsion, needed to verify the admissibility of the torsion families."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the rational parametrizations of curves with cyclic n-isogenies (Table 7), used for the ℓ=7 and ℓ=13 families."},{"cited_title":"Molnar and J","cited_arxiv_id":null,"evidence_quote":"Provides the count of elliptic curves with a 7-isogeny and the lattice-counting approach adapted for the non-coprime gcd(f^3,g^2) case."},{"cited_title":"Pomerance and E","cited_arxiv_id":null,"evidence_quote":"Parametrizes and counts elliptic curves with a cyclic 4-isogeny, making that family admissible."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem on averages of multiplicative functions along equidistributed sequences that yields the moment and tail bounds."}],"review_version":1}