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Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents

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Pith's one-line read The paper gives an exact distribution for rank growth of elliptic curves over cubic S3-extensions with a fixed quadratic resolvent, and a 31.95% lower bound for growth by at most one.

desk verdict A promising Markov-chain framework undermined by a central identification that confuses characters of G_k with S3-cubic extensions; the main density statement as written counts the wrong objects. read the letter →

arxiv 2502.05705 v1 pith:YOOHS4TQ submitted 2025-02-08 math.NT

Daniel Keliher , Sun Woo Park This is my paper · ORCID
classification math.NT MSC 11G0511R1611R3211R45
keywords ellipticcurvesrankgrowthS3-cubicextensionsquadraticresolventSelmergroupsMarkovchainfanstructure3-torsion
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

This paper studies how often a fixed elliptic curve $E$ over a number field $k$ gains rank when lifted to an $S_3$-cubic extension $K/k$—a cubic extension whose Galois closure has Galois group $S_3$—with a fixed quadratic resolvent field $F/k$. The main result is an exact formula for the density, in a non-standard 'fan' ordering, of the $\mathbb{F}_3$-vector-space dimension of the Selmer group $\mathrm{Sel}_{1-\sigma_K}(B_{K/k}/k)$ attached to an auxiliary four-dimensional abelian variety $B_{K/k}$. Because this dimension is an upper bound for the rank growth $\mathrm{rk}(E(K))-\mathrm{rk}(E(k))$, the formula implies that rank growth is at most one for at least $31.95\%$ of the fan-ordered extensions, and that the density of large rank growth decays quadratic-exponentially. The limiting distribution is a mixture of an even and an odd component, with the mixture weight $\rho_E$ determined by $E$ alone.

What carries the argument

The load-bearing machinery is a Markov chain on local Selmer structures, organized by a fan ordering that makes the chain converge. Starting from the local Selmer structure of $E[3]$ over the quadratic resolvent $F$, adding a ramified local character at a prime where $\dim_{\mathbb{F}_3} E[3](k_v)$ is $1$ or $2$ changes the Selmer dimension by $\pm 2$ or $0$, with transition probabilities controlled by an alternating mod-3 Lagrangian Markov operator (Definition 5.21); the operator preserves parity of the Selmer dimension. The fan structure $B_{m,*}(X)$ (Definition 7.2) orders $S_3$-cubic extensions by a sequence of bounding functions rather than by discriminant, so that sufficiently many primes enter the support to force convergence. The link between the Markov chain and the original curve is the isomorphism $\mathrm{Sel}_{1-\sigma_K}(B_{K/k}/k) \cong \mathrm{Sel}(E[3],\eta_d(\chi_K))_F$, obtained from Proposition 4.10 and Shapiro's lemma, which carries the Selmer dimension of the auxiliary four-dimensional variety to a Selmer group of $E$ over $F$ with local character conditions.

What would settle it

The theorem would be settled by enumerating the fan sets $B_{m,*}(X)$ for one explicit admissible pair $(E,F)$ with increasing $m$ and $X$, computing the proportions of extensions with $\dim_{\mathbb{F}_3} \mathrm{Sel}_{1-\sigma_K}(B_{K/k}/k)=s$, and comparing them to $\rho_E E_{\mathrm{even}}(s)+(1-\rho_E)E_{\mathrm{odd}}(s)$; any mismatch at a single $s$ would show the formula is wrong. A direct check of the proof's parametrization is also decisive: a homomorphism $G_k \to \mu_3$ has an abelian fixed field, so if the argument treats such characters as cutting out non-Galois $S_3$-cubic extensions, that bijection cannot hold as stated.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 7.3: for every admissible quadratic extension $F/k$ and every elliptic curve $E/k$ with $\mathrm{Gal}(k(E[3])/k) \supset \mathrm{SL}_2(\mathbb{F}_3)$ and $F$ linearly disjoint from $k(E[3])$, the fan-structure density of $S_3$-cubic extensions with quadratic resolvent $F$ for which $\dim_{\mathbb{F}_3} \mathrm{Sel}_{1-\sigma_K}(B_{K/k}/k)=s$ equals $\rho_E E_{\mathrm{even}}(s)+(1-\rho_E)E_{\mathrm{odd}}(s)$. Here $E_{\mathrm{even}}(s)=\prod_{k=0}^{\infty}(1+3^{-k})^{-1}\prod_{k=1}^{s/2}3/(3^k-1)$ for even $s$ and $0$ otherwise, and $E_{\mathrm{odd}}(s)$ is the corresponding shifted distribution for odd $s$; $\rho_E\in[0,1]$ is a computable constant depending only on $E$. The rank-growth statement follows from the identity $\mathrm{rk}(B_{K/k}(k))=2(\mathrm{rk}(E(K))-\mathrm{rk}(E(k)))$ together with the inequality $\mathrm{rk}(E(K))-\mathrm{rk}(E(k))\le\dim_{\mathbb{F}_3}\mathrm{Sel}_{1-\sigma_K}(B_{K/k}/k)$.

Load-bearing premise

The proof assumes that the fan-ordered family of global characters is in bijection with the $S_3$-cubic extensions with the given quadratic resolvent, and that the limit over the fan size can be interchanged with the limit over the field-bound $X$; if either assumption fails, the density statement does not follow.

Editorial extensions

If this is right

  • For every admissible pair $(E,F)$, the fan-structure density of $S_3$-cubic extensions with a given Selmer dimension $s$ is exactly $\rho_E E_{\mathrm{even}}(s)+(1-\rho_E)E_{\mathrm{odd}}(s)$, with the constants and distributions written out explicitly.
  • At least $31.95\%$ of the fan-ordered $S_3$-cubic extensions satisfy $\mathrm{rk}(E(K))-\mathrm{rk}(E(k))\le 1$, and at least $\rho_E\cdot 31.95\%$ satisfy $\mathrm{rk}(E(K))=\mathrm{rk}(E(k))$.
  • The density of extensions with rank growth at least $s$ decays quadratic-exponentially in $s$: Corollary 7.4 gives the explicit bound $C\cdot 3^{-s(s-2)/8}$ for even $s\ge 4$ and the analogous odd bound.
  • The parity of $\dim_{\mathbb{F}_3} \mathrm{Sel}_{1-\sigma_K}(B_{K/k}/k)$ is forced by the Markov process: the total limiting mass on even dimensions is $\rho_E$, and the total mass on odd dimensions is $1-\rho_E$.
  • The numerical distribution is concentrated on dimensions $0$ through $7$, with the largest single mass at dimension $2$ when $\rho_E=1$ and at dimension $3$ when $\rho_E=0$.

Reading between the lines

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

  • Because the fan ordering is non-standard, the $31.95\%$ figure is not a statement about the same family ordered by discriminant; translating it to discriminant order would require an additional equidistribution assumption not proved here.
  • The proof's character-to-extension bijection may be repairable: the paper's Section 3 parametrization of $S_3$-cubic extensions by ideal pairs and $3$-Selmer units is independent of the global-character description, so the theorem could in principle be re-derived on that parametrization if the character map fails.
  • One testable prediction is that in function-field analogues, where the effective Chebotarev bound can be made explicit, the parity mixture $\rho_E E_{\mathrm{even}}+(1-\rho_E)E_{\mathrm{odd}}$ should already be visible for small fan parameters before the double limit is taken.
  • The same Markov-operator scheme should give analogous rank-growth distributions for other extension types whose Galois closure has a fixed quadratic subfield, such as $D_4$ or $A_4$ quartic extensions, with the $3$-torsion Selmer group replaced by the appropriate isogeny Selmer group.
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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 / 5 minor

Summary. This paper claims to determine, under a non-standard fan ordering, the density of S3-cubic extensions K/k with fixed quadratic resolvent F/k for which dim_{F3} Sel_{1-σ_K}(B_{K/k}/k) equals s, where B_{K/k} is a 4-dimensional abelian variety constructed from E. The authors prove an exact relation between rank growth of E and the rank of B_{K/k}, and they bound rank growth by the Selmer dimension. They adapt the Klagsbrun-Mazur-Rubin Markov-chain method to an 'alternating' operator, compute its stationary distribution, and express the limiting distribution as ρ_E E_even + (1-ρ_E) E_odd, with an unspecified parameter ρ_E. As a corollary they claim a 31.95% lower bound for the density of rank growth at most one. The main theorem depends on identifying the fan B_{m,*}(X), defined as a union of spaces Hom(G_k, μ_3), with S3-cubic extensions.

Significance. If the theorems were correct, they would be a natural extension of the KMR14 Markov-chain method to non-abelian cubic extensions and would give a quantitative answer to a natural rank-growth question. The paper contains several useful components: the construction of B_{K/k} and the identity rk(E(K))-rk(E(k)) = (1/2) rk(B_{K/k}(k)); the reduction of its Selmer group to a 3-Selmer group over F via Proposition 4.4; the local Lagrangian subspace counts; and the explicit stationary distribution of the alternating Markov operator. The 31.95% lower bound is a clean consequence of the mixture formula (86) and is independent of ρ_E. However, the central parametrization tying these components to S3-cubic extensions is invalid as stated, so the advertised theorems are not established for the claimed family.

major comments (3)
  1. [§7, Definition 7.2 and proof of Theorem 7.3] The proof begins by taking, for an S3-cubic extension K/k, a character χ_K ∈ Hom(G_k, μ_3) whose fixed field is K. For every such character, ker(χ_K) is normal in G_k, so its fixed field is a Galois extension of k with cyclic Galois group of order 1 or 3. An S3-cubic extension is not Galois over k and has Galois closure with Galois group S3; it contains no degree-3 Galois subextension. Hence no such χ_K exists. Since Definition 6.1 defines C(d) as a subset of Hom(G_k, μ_3), the fans in Definition 6.4 are unions of character spaces whose fixed fields are cyclic cubic Galois extensions, not S3-cubic extensions. Definition 7.2's assertion that B_{m,*}(X) is a collection of S3-cubic extensions is therefore not established, and the density in (86) counts characters rather than the S3 extensions appearing in Theorems 1.1, 1.2, and Corollary 7.4. A correct treatment would need a different parametrization of S3 cubics, such as the one in Section 3, and a fan structure adapted to it; this is not supplied.
  2. [§7, Eq. (87)] The isomorphism Sel_{1-σ_K}(B_{K/k}/k) ≅ Sel(E[3], η_d(χ_K))_F is asserted only after the invalid identification of K with the fixed field of χ_K. Consequently the right-hand side is not defined for a non-Galois S3 extension. If one tried to repair this by taking a character of G_F cutting out the C3-extension K̃/F of Section 3, the local conditions at places of k and the fan definition would have to be reworked, because the current C(d) consists of characters of G_k, not of G_F.
  3. [§6–§7, Theorem 6.7 to (86)] The passage from Theorem 6.7 to (86) omits a nontrivial limiting argument. Theorem 6.7 fixes both m and w and gives a limit as X→∞ for B_{m,w,D*,X}; Definition 7.2 then takes a union over w, and (86) takes m→∞. The proof gives no justification for interchanging the X and m limits and no control of the distribution of w among the fan. Even after the parametrization issue is resolved, the limiting statement in (86) would require such estimates.
minor comments (5)
  1. [§3, Definition 3.4] The table in Definition 3.4 has a typographical issue in the last column label ('Further Conditions on F/k ?') and the table formatting makes the rows harder to parse than necessary.
  2. [§5.2, Lemma 5.17] Lemma 5.17 assumes Gal(k(E[3])/k) = GL_2(F_3), whereas Theorem 1.1 assumes only that Gal(k(E[3])/k) contains SL_2(F_3); in the cases where the lemma is used this follows from F = k(ζ_3), but the paper should state this implication explicitly.
  3. [References] The reference [GZ86] contains a typo in the author list ('Benedict Gross, , and Don Zagier').
  4. [§5.2, Proposition 5.20] Proposition 5.20 labels the table entries c_{i,j}(r_ω), but the displayed entries depend on r_ω through r; the notation should be made uniform with Definition 5.21.
  5. [§7, Definition 7.1] The parameter ρ_E is introduced and called computable, but it is not evaluated or bounded anywhere in the paper; Theorem 1.2 is therefore a conditional distribution in terms of an unspecified input, which should be stated more prominently.

Circularity Check

1 steps flagged · score 6.0 of 10

The main density theorem is self-definitional: Definition 7.2 declares the fan union of characters to be the set of S3-cubic extensions, so the extension-counting statement is by construction rather than derivation.

  1. self definitional [Section 7, Definition 7.2 and the proof of Theorem 7.3 (Eqs. (85)-(87))]
    "Definition 7.2: "Let B_{m, ∗}(X) := ⋃_w B_{m,w, D∗,X (85) be a collection of S3-cubic extensions K of k with a fixed quadratic resolvent field F/k. The right hand side is a union of fan structures as in Definition 6.4." Proof of Theorem 7.3: "Given an S3-cubic extension K/k, denote by χ_K ∈ Hom(G_k, µ_3) the global character whose fixed field is K.""

    The object whose density Theorem 7.3 claims to compute—the "collection of S3-cubic extensions"—is defined in (85) as the union of the character fan spaces B_{m,w,D∗,X} from Definition 6.4, which are subsets of C(k) = Hom(G_k, μ_3). The theorem then evaluates the density over this character fan and calls it the density over S3-cubic extensions. This identification is not derived; it is put in by definition. The proof's asserted correspondence χ_K → K is also circular in its role: every χ ∈ Hom(G_k, μ_3) has kernel a normal subgroup, so its fixed field is Galois and cyclic over k, not a non-Galois S3-cubic extension.

full rationale

The Markov-chain portion of the paper is genuinely self-contained and not circular: Proposition 5.23 derives the stationary law E_even/E_odd from the operator M_L, the 31.95% constant is an honest numerical evaluation of the infinite product, and the per-step rank-change rules in Section 5 are imported from KMR14's independent framework rather than from a fit. The constant ρ_E is an explicitly defined input (the even-mass of E_1), not a fitted parameter disguised as output, so the formula is a one-parameter family rather than a full prediction. No load-bearing self-citation of the present authors occurs: [Par24] is used for one isomorphism, but Proposition 4.4 also gives an independent proof sketch, and [Kel22] is unrelated background. However, the central translation from fan characters to S3-cubic extensions is effected by Definition 7.2 and the impossible 'fixed field is K' character, so the main theorem's counted object is the character fan by definition. That is a genuine self-definitional collapse of the paper's central claim, although the underlying Markov computation is independent.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the KMR14 Markov model, the fan ordering, and an unproved identification between fan characters and S3-cubic extensions. The scalar rho_E is an uncomputed input, and even if the character identification is repaired, the distribution statement remains conditional on rho_E and on the non-standard fan ordering.

free parameters (1)
  • rho_E = not computed
    Defined in Definition 7.1 as the probability that dim Sel(E[3], E1)_F is even; it enters the main distribution formula in Theorem 7.3. The paper calls it computable but gives no value or algorithm, so the distribution is only determined up to this scalar.
assumptions (4)
  • domain assumption Gal(k(E[3])/k) contains SL2(F3) and F is linearly disjoint from k(E[3]) over k.
    Used throughout, for example in Theorems 1.1, 1.2, and 7.3, and in Lemma 5.14.
  • ad hoc to paper Density is taken with respect to the fan structure, with limits in m and then X, not with respect to a natural ordering by discriminant.
    Definition 6.4 and Remark 6.5 state the ordering is non-standard and not by discriminant; all density claims are relative to this ordering.
  • domain assumption The KMR14 effective Chebotarev theorem applies to the fields k_{d,omega} and hat-k_{d,omega} constructed in Section 5.2.
    Invoked in Proposition 5.20 and Theorem 6.7; the paper cites KMR14 Theorem 8.1 for this.
  • ad hoc to paper Every character in the fan corresponds to exactly one S3-cubic extension with resolvent F, and the multiplicity from choices of u in S3(F_z)[T] is uniform.
    Asserted implicitly in Definition 7.2 and the proof of Theorem 7.3; not proved. This is the flagged weak point.
invented entities (1)
  • Auxiliary abelian variety B_{K/k} independent evidence
    purpose: A 4-dimensional abelian variety whose k-rational rank equals twice the rank growth of E from k to K, defined in Definition 4.3 and used throughout.
    It is constructed via the kernel of a norm map on Weil restriction, a standard operation, and its 1 - sigma_K torsion is identified with (Res_F^k E)[3] in Proposition 4.4, so it is not an unexplained ad hoc entity.

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Pith. "Pith review of Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents." pith.science (2026). https://pith.science/paper/YOOHS4TQ

@misc{pith2026250205705,
  author       = {Pith},
  title        = {Pith review of: Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOOHS4TQ}},
  note         = {Machine review of arXiv:2502.05705}
}
abstract

We study the probability with which an elliptic curve $E/k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-cubic extension $K/k$ with quadratic resolvent field $F/k$, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to $E$ and $S_3$-cubic extensions $K/k$ following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that $E$ gains rank by at most one upon base extension to $K$ with probability at least $31.95\%$.

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