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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.

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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.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection 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. the 3 major comments →

arxiv 2508.21406 v1 pith:EHENWRXP submitted 2025-08-29 math.NT

Selmer groups of families of elliptic curves with an $\ell$-isogeny

Stephanie Chan , Matteo Verzobio This is my paper
classification math.NT MSC 11G0511N36
keywords elliptic curvesSelmer groupsisogenyTamagawa ratiocentral limit theoremlattice point countingrational isogenyunbounded Selmer ranks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 proves that in several parametrized families of elliptic curves over the rationals that come with a rational isogeny of prime degree ℓ, 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 ℓ = 2, 3, 5, 7, and 13, the ℓ-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.

Core claim

Theorem 5.9 is the engine: for any admissible family S (Definition 5.8) of curves with a rational degree-ℓ isogeny, the logarithmic Selmer ratio r_ϕ(E) = dim Sel_ϕ(E/Q) − dim Sel_{\hat ϕ}(E'/Q), ordered by naive height, converges after centering and scaling to a standard Gaussian, with mean μ log log N and variance σ^2 log log N. The same theorem gives ℓ^{k r_ϕ}-moment lower bounds and a Paley–Zygmund tail bound: for every A > 0, at least |S(N)|(log N)^{−δ(A)} curves have r_ϕ(E) ≥ A log log N. Since |Sel_ℓ(E/Q)| ≥ ℓ^{r_ϕ(E)}/O_ℓ(1), the ℓ-Selmer group is unbounded in each admissible family. Theorem 1.1 applies this to torsion-subgroup families, the cyclic-4-isogeny family, and infinite ℓ=7 a

What carries the argument

A classical identity (Lemma 3.1) bounds |Sel_ϕ(E/Q)|/|Sel_{\hat ϕ}(E'/Q)| between two constant multiples of ∏_p c_p(E')/c_p(E). A local classification (Lemma 3.6) determines each factor c_p(E')/c_p(E) ∈ {ℓ^{−1}, 1, ℓ} from the reduction type, so the product becomes ℓ^{∑ 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 μ, σ, ρ(k) are read off from

Load-bearing premise

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 ∈ {2,3,4,6}; if a concrete family fails this, the equidistribution estimates and hence the whole theorem collapse for that family.

What would settle it

For the ℓ=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.

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If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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).
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

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\}.

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 (3)
  1. [Section 6.5] 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.
  2. [Section 6.4] 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.
  3. [Section 6, Tables 1-5] 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.
minor comments (4)
  1. [Section 6.5] The sentence 'Recall H0 \leq H' should be 'H \leq H0'; the argument uses that H0 \leq N implies H \leq N.
  2. [Section 5.6] In the proof of Lemma 5.7, 'where M_- and M_- are constants' is a typo; the second occurrence should be M_+.
  3. [Section 6.4] 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'.
  4. [Equation (2.5)] 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.

Circularity Check

0 steps flagged

No circular derivation; constants computed, not fitted; self-citations auxiliary; ℓ=13 admissibility asserted without proof is a gap, not a circle.

full rationale

The paper's derivation chain from admissibility conditions (Definition 5.8) to the CLT and Selmer-ratio bounds (Theorem 5.9) is not circular. The constants c±, μ, σ, and ρ(k) are computed from explicit Chebotarev counts attached to polynomials D± obtained by factoring the discriminant polynomials, with values such as 6B(1,0)=12 and 6B(0,1)=-4116 evaluated directly; no parameter is fitted to Selmer-group data. The equidistribution statements (Propositions 4.5, 4.11, and Lemma 5.7) are proven in the paper under explicit hypotheses on f,g, and the averaging theorem (Theorem 5.2) is proved internally, with the upper bound importing the published, independent [12, Thm 1.9] and sieve ingredients from [19]. The self-citations [10,11,12] are used as tools and do not smuggle in the target result: [10] is a related prior result, [11] is a comparable CLT in another family, and [12] is an external averaging theorem. None of these citations is the only justification for the main theorem. The most substantive concern is in Section 6.5: the ℓ=13 family is asserted to be admissible under (A4) with m=2 without displaying the f,g of the dual family or verifying the hypotheses (no common real roots, degree condition, and |S(N)|=(1+O(N^{-ξ}))|F0(N)|). This is an omitted verification, not a circular step: if the checks fail, the conclusion for ℓ=13 is unsupported, but the theorem is not reduced to its own input. The score of 2 reflects minor non-load-bearing self-citations rather than an actual circular dependency.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The proof introduces no fitted numerical parameters: μ, σ, and c± are computed from Chebotarev counts of explicit polynomials. It does rest on substantial imported machinery and on admissibility hypotheses tailored to the counting method.

axioms (6)
  • domain assumption Cassels' Selmer-ratio identity and Dokchitser-Dokchitser classification of local Tamagawa ratios for isogenous curves
    Used in Lemmas 3.1 and 3.2 to replace the Selmer ratio by the product of Tamagawa numbers; if these local formulas fail, the random variable model does not track r_phi(E).
  • domain assumption The parametrizations of torsion families and n-isogeny families from [3], [25], [30], [17] are exhaustive and have the stated fiber sizes
    Theorem 1.1 covers all curves in a family S, so S must be exactly, up to O(N^-ξ), the image of the parametrization; Lemma 6.2 and Sections 6.3-6.5 import this from the literature.
  • domain assumption Counting asymptotics for curves with prescribed torsion or isogeny from [17], [13], [29], [30], [2], [27] have the error quality required by Definition 5.8
    Admissibility asks for |S(N)| = (1+O(N^-ξ))|F(N)|; the equidistribution propositions inherit this error term.
  • standard math Chebotarev density theorem and Davenport's lattice point theorem
    Used in Section 5.4 and Section 4 to compute c± and the equidistribution of (a,b) in congruence classes.
  • standard math The fundamental lemma of sieve theory and the averaging theorem of [12], [39] for multiplicative functions
    Theorem 5.2 and Theorem 5.3 rest on these external averaging and sieve results.
  • ad hoc to paper Structural conditions (2.2) and Definition 5.8 (A3)/(A4): no common real roots, weighted-degree normalization, and gcd(f^3,g^2) an r-th power of a squarefree polynomial with degree bounds
    These hypotheses are imposed to make the lattice-point counting uniform; they are not derived from the Selmer problem and exclude many natural families.

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Pith. "Pith review of Selmer groups of families of elliptic curves with an $\ell$-isogeny." pith.science (2026). https://pith.science/paper/EHENWRXP

@misc{pith2026250821406,
  author       = {Pith},
  title        = {Pith review of: Selmer groups of families of elliptic curves with an $\ell$-isogeny},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHENWRXP}},
  note         = {Machine review of arXiv:2508.21406}
}
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abstract

For certain families of elliptic curves admitting a rational isogeny of prime degree $\ell$, we establish a central limit theorem for the Tamagawa ratio and derive bounds on its average value. By using the Tamagawa ratio to bound the size of the $\ell$-isogeny Selmer group from below, we show that for $\ell \in\{ 2, 3, 5, 7, 13\}$, there exist elliptic curves with arbitrarily large $\ell$-Selmer groups.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tamagawa ratios and unbounded Selmer moments

    math.NT 2026-06 unverdicted novelty 6.0

    Framework gives conjectural characterization of geometric families of abelian varieties with unbounded average l-Selmer sizes, proven correct when l-torsion module is constant across the family.

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.