{"id":"91518edd-9573-4171-b3cc-69918ea2f12f","arxiv_id":"2502.05705","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For elliptic curves with large 3-torsion Galois image, the Selmer-rank distribution over S3-cubic extensions with a fixed quadratic resolvent is a parity mixture of a universal Markov-chain law, giving a 31.95% lower bound for small rank growth.","lead":"This paper studies how often a fixed elliptic curve gains rank when moved to a cubic extension whose Galois closure has symmetric group S3 and a fixed quadratic subfield. It derives a limiting distribution for the relevant Selmer ranks under a special non-standard ordering, yielding a 31.95% lower bound for rank growth at most one.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.3 identifies fan characters with S3-cubic extensions via a 'fixed field' that cannot exist for non-Galois K, so the density counts characters, not extensions.","rationale":"The single most load-bearing step in the paper is the map from S3-cubic extensions K/k to global characters chi_K with fixed field K. This is the only bridge between the Markov-chain/fan machinery of Sections 5 and 6 and the object-level theorem about S3-cubic extensions. The reader and I independently identify the same obstruction: a homomorphism G_k -> mu_3 has a normal kernel, so its fixed field is Galois over k with abelian Galois group of order dividing 3, while an S3-cubic extension is non-Galois. Thus the asserted character cannot exist. This is an internal inconsistency in the proof, not a disagreement with a heuristic or with current consensus. The limit interchange over w and m is also not justified in the text, but even a fully justified fan limit would not repair the object-level mismatch. The Markov-chain computation itself and the numerical constant 31.95% appear plausible and are not what fails; the failure is that (86) is stated as a density over S3-cubic extensions, whereas the fan in Definition 6.4 is a set of characters. I therefore support rejection of the central claim as written, while noting that a corrected parametrization might allow a revised version of the theorem.","tokens_in":28369,"tokens_out":9785,"duration_ms":106041,"concrete_test":"Take k = Q, F = Q(zeta_3) (admissible case B) and p = 2, and let K be the cubic subfield of F(2^(1/3)); K = Q(2^(1/3)) is a non-Galois S3 extension with quadratic resolvent F. Check whether there exists chi in Hom(G_Q, mu_3) whose fixed field is K: because ker(chi) is normal, any fixed field is Galois over Q, whereas K/Q is not Galois. No such chi exists, contradicting the first sentence of the proof of Theorem 7.3. More generally, test the asserted identification by replacing the sets in Definition 7.2 with the cyclic cubic extensions cut out by the characters in Definition 6.4 and comparing the count in (86); the two counts describe different families unless an explicit bijection is written down.","verdict_should_be":"REJECT","load_bearing_attack":"The proof of Theorem 7.3 starts: 'Given an S3-cubic extension K/k, denote by chi_K in Hom(G_k, mu_3) the global character whose fixed field is K.' Definition 6.1 fixes C(k) = Hom(G_k, mu_3). For any such chi, ker(chi) is normal in G_k, so its fixed field is Galois over k with cyclic Galois group of order dividing 3. But K/k is non-Galois: its Galois closure has group S_3. Hence no such chi_K exists. Consequently the fan B_{m,*}(X), defined in Definition 6.4 as a union of character spaces C(d), is not a collection of S3-cubic extensions; Definition 7.2 simply declares it one. The isomorphism Sel_{1-sigma_K}(B_{K/k}/k) ≅ Sel(E[3], eta_d(chi_K))_F and the density in (86) are then asserted for objects that are not the ones counted. Characters in C(k) can at best cut out cyclic cubic Galois extensions of k; the S3 setting would require characters of G_F cutting out the cubic subextension of the Galois closure over F, or another explicit parametrization, which the paper does not supply. Because Theorems 1.1, 1.2, and Corollary 7.4 inherit this identification, the central claim is unsupported as stated. The Markov-chain computation may be salvageable if such a parametrization is supplied, but that is not done in this manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28754,"tokens_out":8626,"duration_ms":87012,"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":[{"comment":"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.","section":"§7, Definition 7.2 and proof of Theorem 7.3"},{"comment":"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.","section":"§7, Eq. (87)"},{"comment":"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.","section":"§6–§7, Theorem 6.7 to (86)"}],"minor_comments":[{"comment":"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.","section":"§3, Definition 3.4"},{"comment":"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.","section":"§5.2, Lemma 5.17"},{"comment":"The reference [GZ86] contains a typo in the author list ('Benedict Gross, , and Don Zagier').","section":"References"},{"comment":"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.","section":"§5.2, Proposition 5.20"},{"comment":"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.","section":"§7, Definition 7.1"}],"recommendation":"reject","confidential_remarks":"To the editor: the core difficulty is not a gap in a routine argument but an obstruction in the statement of Theorem 7.3 as written: the fan is built from abelian characters of G_k, whose fixed fields are Galois, while the claimed family consists of non-Galois S3 extensions. This is not a presentation issue. In my view the paper would need a substantially new parametrization of the fan over S3 extensions before the Markov-chain results can be applied, and hence I recommend rejection rather than major revision. The auxiliary-variety construction and the Markov-chain computations may be reusable in a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one before you cite it: the main theorem is not supported as written. In the proof of Theorem 7.3 the authors write: 'Given an S3-cubic extension K/k, denote by chi_K in Hom(G_k, mu_3) the global character whose fixed field is K.' That character cannot exist. Any chi in Hom(G_k, mu_3) has a normal kernel, so its fixed field is Galois over k with cyclic Galois group of order dividing 3; it can never be a non-Galois S3-cubic field. The fan structure B_{m,*}(X) from Definition 6.4 is a union of character spaces C(d), so Definition 7.2 simply declares it to be a set of S3 extensions. The density in (86), together with Theorems 1.1, 1.2 and Corollary 7.4, therefore counts cyclic cubic characters, not S3-cubic extensions. The stress-test note is accurate; I checked the manuscript and found no repair of this step later.\n\nCredit where it is due: the paper is the first distributional attempt for rank growth in S3-cubic families with fixed quadratic resolvent, and the auxiliary four-dimensional abelian variety B_{K/k} and the reduction to 1-sigma Selmer groups are a real adaptation of MR07 and Park's thesis. Sections 3-5 contain a careful extension of the KMR14 Markov-chain and local-Selmer machinery, including the corestriction control in Lemma 5.16 and the parity-preserving operator of Proposition 5.21. The 31.95% lower bound is a clean consequence of the stationary distribution, and the authors are honest that the ordering is non-standard.\n\nTwo further soft spots, in decreasing order of importance. The limit interchange over w in Definition 7.2 and Theorem 7.3 is not justified. And the final distribution depends on rho_E, a constant the paper defines but never computes, so the numerical distribution is left conditional on an unknown parameter.\n\nThis is a promising framework with a load-bearing gap. A serious referee could tell the authors exactly what to fix, so I would send it out rather than desk-reject, but I would not accept it in current form. I would not cite it yet.","headline":"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.","tokens_in":29226,"tokens_out":3976,"would_cite":false,"duration_ms":36754,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11R16","11R32","11R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["elliptic curves","rank growth","S3-cubic extensions","quadratic resolvent","Selmer groups","Markov chain","fan structure","3-torsion"],"falsifier":"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.","tokens_in":28160,"feed_emoji":"📈","tokens_out":14603,"duration_ms":125408,"temperature":0.7,"pith_summary":"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.","feed_headline":"At least 31.95% of S3-cubic extensions cap rank growth at one","feed_subtitle":"The formula pins down how often cubic base change boosts a curve's rank, with large jumps rare.","key_machinery":"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.","core_discovery":"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 $\\mathrm{rk}(E(K))-\\mathrm{rk}(E(k))\\le\\dim_{\\mathbb{F}_3}\\mathrm{Sel}_{1-\\sigma_K}(B_{K/k}/k)$.","pith_inferences":["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."],"forward_implications":["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 $\\rho_E=0$."],"supporting_citations":[{"why":"Supplies the parametrization of $S_3$-cubic extensions with a fixed quadratic resolvent field, the backbone of the family being ordered.","marker":"[CM11]"},{"why":"Provides the Markov-chain/fan-structure framework, the effective Chebotarev theorem, and the stationary-distribution computation the paper adapts.","marker":"[KMR14]"},{"why":"Gives the local-constant/Selmer machinery and the torsion identification behind the auxiliary abelian variety $B_{K/k}$.","marker":"[MR07]"},{"why":"Supplies the random maximal-isotropic-subspace viewpoint used to characterize the local conditions that define $\\mathrm{Sel}_{1-\\sigma_K}(B_{K/k}/k)$.","marker":"[PR12]"},{"why":"Yields the character-theoretic rank identity $\\mathrm{rk}(E(K))-\\mathrm{rk}(E(k))=\\frac{1}{2}(\\mathrm{rk}(E(\\tilde K))-\\mathrm{rk}(E(F)))$ that connects rank growth to $B_{K/k}$.","marker":"[DD10]"},{"why":"Provides the alternative proof of the torsion isomorphism $B_{K/k}[1-\\sigma_K]\\cong(\\mathrm{Res}^F_k E)[3]$ used in Proposition 4.4.","marker":"[Par24]"}],"fun_headline_variants":["S3-cubic extensions cap rank gain at one for ≥31.95% of cases","Rank growth over S3-cubic fields: at most one, with probability ≥31.95%","≥31.95% of S3-cubic extensions keep elliptic rank growth at one"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["S3-cubic extensions cap rank gain at one for ≥31.95% of cases","Rank growth over S3-cubic fields: at most one, with probability ≥31.95%","≥31.95% of S3-cubic extensions keep elliptic rank growth at one"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000777,"raw_usage":{"total_tokens":3462,"prompt_tokens":1000,"completion_tokens":2462,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":2387}},"tokens_in":616,"tokens_out":2462,"duration_ms":18962,"temperature":1.0,"reasoning_tokens":2387,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:18:26.223081+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}