{"id":"cf10a548-0284-410d-a2c9-2bc2fe6b968e","arxiv_id":"2608.10919","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper gives an equationless geometric quadratic Chabauty method for non-split Cartan modular curves and rederives X_ns^+(13)(Q), which consists of the 7 known CM points.","lead":"This paper builds an equationless version of quadratic Chabauty that finds rational points on certain modular curves using only elliptic curves and level structures, not polynomial equations. It recovers the known result that the modular curve X_ns^+(13) has exactly seven rational points, and it may make higher-level cases, relevant to Serre's uniformity question, computationally accessible.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing spanning condition in Remark 4.1.8 is asserted without proof and the paper relies on code termination as evidence, so the p-adic Makdisi model of the Jacobian for the N=13 computation is not rigorously established.","rationale":"The reader's weakest_assumption points to Remark 4.1.8, and this is exactly where I find the greatest load-bearing risk. The entire equationless computation runs inside a p-adic Makdisi model of the Jacobian, and that model is only valid if the traces of Makdisi's moduli-friendly forms span M_2(Γ) in characteristic p. The paper states this as an assumption in Assumption 6.2(2) and offers code termination as implicit evidence. That evidence is not a proof: termination could be due to implementation details, and the span condition is a statement about modular forms over finite fields that can and should be checked by direct computation. The authors' honesty in flagging this is to their credit, but it remains an unproven premise on which the N=13 rederivation depends. The other candidate concerns, such as the dependency on the unpublished work [LSV26] in Proposition 5.3.9, are less severe because the proof is actually written out in the text and the relevant statements from [LSV26] are quoted and used explicitly. The missing spanning proof is therefore the most concrete and checkable weak point. A direct verification for the actual pair (N=13, p=5), and ideally a wider sample, would resolve the concern. Because this is a fixable gap rather than an identified error, the conditional stance of the reader is appropriate; my stress-test does not change that verdict.","tokens_in":70950,"tokens_out":3179,"duration_ms":29161,"concrete_test":"For N=13, p=5 (and for a sample of other admissible pairs, e.g. N=19, p=7), compute dim M_2(Γ_ns^+(N)) over F_p from the genus and level formula, and independently compute the span of traces down to Γ of Makdisi's moduli-friendly forms using his explicit formulas evaluated on representatives (E, P, Q) over F_p. Check that this span equals the full space M_2(Γ) by verifying equality of dimensions and that an evaluation matrix at O(g) sample points has the expected rank. If the span is proper for (13,5), the p-adic Jacobian model in §4.1.4 is not constructible and the N=13 computation is unsupported; if it spans, the key premise is verified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Remark 4.1.8 states that Makdisi's moduli-friendly forms generate M_k(Γ(N)) in characteristic 0 for all k≥2, hence in characteristic p for p large enough, but the paper does not prove that their traces down to level Γ span M_2(Γ) in characteristic p for the chosen p. It says: 'If the traces down to level Γ of Makdisi's forms do not span M_2(Γ) in characteristic p for our chosen p, then our code constructing a p-adic model of the modular Jacobian will loop forever; so our code terminating implicitly proves that this holds for our chosen p.' This condition is the structural input to §4.1.4, where the p-adic Makdisi model of the Jacobian is built, and that model is the substrate for all computations in §6–§8, including the N=13 rederivation with p=5. The condition is also restated as a requirement in Assumption 6.2(2). Termination is not a proof: a nonterminating loop could also arise from an implementation bug, and the span condition is a mathematical input that should be verified independently, not inferred from a successful run. If the span is proper, the subspaces W_D constructed from traces of these forms may not correspond to the intended points of J(F_5), so the ejb and κ computations summarized in Table 2 lose their guaranteed meaning. Unlike Proposition 5.3.9, which is given a written proof (with a stated dependency on [LSV26] whose relevant statements are quoted), the spanning condition is explicitly left open. This is the single most load-bearing unproved premise of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an equationless geometric quadratic Chabauty method for the non-split Cartan modular curves X_ns^+(N) of prime level N. Instead of using projective equations, the method works with the moduli interpretation: points are pairs (E,[φ]) consisting of an elliptic curve and a normalizer-of-non-split-Cartan level structure. The Jacobian and the Mumford torsor are manipulated through Makdisi and Mascot's divisor-arithmetic algorithms, using spaces of modular forms as section spaces. The paper develops the necessary framework over Z and Z_p: formulas for the lifted Abel-Jacobi map e_jb, analytic coordinates on the Mumford torsor, a description of the p-adic closure of the integral points, vertical correction divisors on a regular model, and model-free algorithms for enumerating residue discs, computing Hecke operators, interpolating e_jb, and intersecting with the image of κ. As an illustration, the paper reports a computation for X_ns^+(13) at p=5 that rederives the known result that the Q-rational points are precisely the 7 CM points.","tokens_in":71358,"tokens_out":5420,"duration_ms":55474,"significance":"If the computational claims are fully substantiated, this is a significant methodological advance: it replaces the equation-based cohomological quadratic Chabauty computations for non-split Cartan curves with a directly modular, equation-free geometric method, and it demonstrates the feasibility of combining geometric quadratic Chabauty with Makdisi-Mascot arithmetic on modular Jacobians and Mumford torsors. The paper contains several substantial and carefully stated theoretical contributions, including the general formula for e_jb in Proposition 4.2.31, the integrality statement in Lemma 4.2.36, the analytic-coordinate description of the Mumford torsor in Section 4.3, the description of integral points in Theorem 4.4.6, and the detailed special-fiber calculations of Section 5. It is also a strength that the authors are explicit about several limitations and dependencies, including the unproved spanning condition in Remark 4.1.8 and the reliance on unpublished work of Love, Studnia and Vonk.","major_comments":[{"comment":"The condition that the traces down to level Γ of Makdisi's moduli-friendly modular forms span M_2(Γ) in characteristic p is load-bearing for constructing the p-adic Makdisi model of the Jacobian, and hence for every computation in Sections 6–8. The paper states that this condition is not proved and that termination of the code is the evidence. Termination is not a proof: nontermination could equally arise from an implementation bug, and the characteristic-0 generation statement from [KM12] does not by itself imply the characteristic-p trace-spanning statement for the specific auxiliary prime p=5 used in Section 8. If the span is proper, the subspaces W_D constructed in Section 4.1.1 may fail to represent the intended points of J(F_p), so the θ-polynomials in Table 2 lose their guaranteed meaning. Please provide a proof, or an independent finite verification over F_p for the pairs (Γ,N,p) used, rather than inferring the condition from a successful run.","section":"§4.1.4, Remark 4.1.8, Assumption 6.2(2)"},{"comment":"The central claim that the method 'rederives' the rational points on X_ns^+(13) is a computational assertion, but the PARI/GP and Magma code that produces Table 2 is not included, and the paper provides no execution logs, input files, or reproducible scripts. The textual descriptions in Sections 6.8–6.10 and Example 6.5.3 are detailed, but they are not sufficient for an independent check of the exact θ-polynomials, the higher-residue-disc computations, or the claimed 2-minute runtimes without a substantial reimplementation. For a paper whose novelty is primarily a new computational method, please make the code and data available, or provide a complete certificate (for instance, the matrices γ_i,j, the local equations, and the interpolation data) so that the numerical table can be verified independently.","section":"§8, Table 2"},{"comment":"The proof of Proposition 5.3.9 depends on [LSV26, Theorem A and Section 2], which the preprint cites as unpublished and does not include. This proposition is used in Section 5.4 to justify that ∆^*T_ℓ contains no smooth component of X_{F_N}, which is needed to compute the vertical divisor B in (5.4.2)–(5.4.3); B in turn enters directly into the formula for e_jb in Proposition 4.2.31 and therefore into the N=13 computation. Since the result is load-bearing and the external work is not available to the reader, please either state the needed results from [LSV26] explicitly in the paper, or replace the dependency with a self-contained proof.","section":"§5.3, Proposition 5.3.9"}],"minor_comments":[{"comment":"The bilinear-combination notation ⊤vΠw is introduced very tersely; a short explicit example with small matrices would help the reader verify the compatibility of the two partial group laws.","section":"§2.6"},{"comment":"The component labels in Figure 2 are difficult to read at normal print size; consider a higher-resolution figure or an accompanying table listing the components and their intersection numbers.","section":"§5.1, Figure 2"},{"comment":"The statement that relations among the γ_i were checked 'up to e=10' should specify whether this is a computational verification modulo p^e or an exact identity in J(Z); this distinction matters for what is being asserted about Γ.","section":"§6.5, Example 6.5.3"},{"comment":"The symbols '✓' and 'Inspect' in Table 2 are informal; please define them in the caption or in the surrounding text so that the table is self-contained.","section":"§8, Table 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely to be publishable after the load-bearing gaps are addressed. The most serious issue is Remark 4.1.8, where an explicitly unproved spanning condition is treated as established by code termination; this should be fixed by a proof or by an explicit finite verification for the primes used. The dependence on unpublished work [LSV26] and the absence of the computational code are also significant for a paper whose main result is a numerical rederivation. I would not recommend rejection, because the theoretical framework is coherent, the authors are unusually transparent about limitations, and the N=13 table is internally consistent with the described method. However, the central claim is not yet fully verifiable as submitted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious and original piece of work. The equationless geometric quadratic Chabauty method for X_ns^+(N), computing in the Jacobian and the Mumford torsor via Makdisi's algorithms, is genuinely new and targets exactly where the field is bottlenecked: explicit equations. The authors do not claim a new rational-point theorem; they rederive the known N=13 result as a demonstration, which is a fair and honest framing.\n\nWhat the paper does well: it gives detailed formulas for the map tilde-j_b (Proposition 4.2.31 and Lemma 4.2.36), builds analytic coordinates on the Mumford torsor, and translates the whole method into modular computations with elliptic curves and Cartan structures. The theoretical sections are coherent and, as far as I can tell, correct. The N=13 computation is internally consistent and matches the BDM+19 result; the known CM points are used only to generate the Mordell-Weil subgroup and to compare root counts, so there is no circularity. The citation pattern is reasonable and the related work is properly acknowledged.\n\nThe soft spots are real but addressable. The biggest one is the spanning condition in Remark 4.1.8 and Assumption 6.2(2): the paper needs the traces of Makdisi's moduli-friendly forms to span M_2(Gamma) in characteristic p, and it explicitly does not prove this, instead taking code termination as evidence. Termination is not a proof; the same behavior could come from an implementation bug or an infinite loop of another kind. If the span is proper, the p-adic Makdisi model of the Jacobian in Section 4.1.4 can fail, and the W_D subspaces lose their intended meaning. This is a load-bearing unproved premise, though it is honestly flagged. It is also checkable: for the single prime p=5 used in the N=13 computation, a direct linear algebra verification, or a rigorous argument for why the span holds, should be feasible. Without that, the numerical result is computationally plausible but not mathematically certified.\n\nSecond, no code is shipped. For an algorithmically driven paper, this hurts reproducibility. The authors say they took responsibility, and the AI-disclosure statement is transparent, but reviewers and readers need the Magma/PARI scripts to verify the tables. Third, Proposition 5.3.9 depends on the unpublished [LSV26]; the paper does provide a written proof quoting the needed statements, so this is a lesser concern, but it would still be good to confirm that LSV26 is available or substitute a self-contained argument.\n\nOverall: the central method is solid, the gap is narrow and identified, and the paper deserves serious referee work. I would send it to peer review with instructions to ask for (a) a proof or certificate of the spanning condition for the primes used, (b) the actual code, and (c) a check of the [LSV26] dependency. I would read the revised version with interest, and I would cite the method in my own work.","headline":"A genuinely new equationless quadratic Chabauty framework for non-split Cartan curves, with a clean N=13 rederivation, but one load-bearing computational spanning condition is asserted on faith and the code isn't shipped.","tokens_in":71860,"tokens_out":2002,"would_cite":true,"duration_ms":23826,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","11G50","11F11","14G05","14H40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper describes a fully equationless implementation of geometric quadratic Chabauty for the non-split Cartan modular curves $X_{\\rm{ns}}^+(N)$, and uses it to rederive Theorem 1.1: the rational points of $X_{\\rm{ns}}^+(13)$ are…","keywords":["equationless quadratic Chabauty","non-split Cartan modular curves","rational points","Mumford torsor","moduli interpretation","section-space divisor arithmetic","CM points","uniformity problem"],"falsifier":"A direct test is to compute, for a given $N$ and $p$ satisfying the other assumptions, the $\\mathbb{F}_p$-span of those traces inside $M_2(\\Gamma)$: if its dimension is strictly less than $\\dim M_2(\\Gamma)$, the construction of the $p$-adic section-space Jacobian fails and the method's claimed scope collapses. A second, end-to-end falsifier is to run the level-13 algorithm at $p=5$ to higher precision: if the gcd of the $\\theta$-polynomials for $f_1=T_3-T_2$ and $f_2=2T_7-3T_2$ ever leaves a common root whose higher-residue-disc congruences never contradict and which is not one of the seven CM points, the intersection computation would not separate rational from $p$-adic points.","tokens_in":70767,"feed_emoji":"🧮","tokens_out":9247,"duration_ms":83068,"temperature":0.7,"pith_summary":"Working with the moduli interpretation instead of a projective model, this paper establishes an equationless method for determining the rational points of the non-split Cartan modular curve $X_{\\rm{ns}}^+(N)$: a point is a pair $(E,[\\varphi])$ of an elliptic curve and a normalizer-of-non-split-Cartan level structure, so all computations can be phrased in terms of elliptic curves and their $N$-torsion. The paper shows that the geometric quadratic Chabauty method, which normally requires a plane model of the curve, can be executed inside the Jacobian and the Mumford torsor using divisor arithmetic on spaces of modular-form sections. As an illustration, it rederives the known result that $X_{\\rm{ns}}^+(13)(\\mathbb{Q})$ consists of exactly the seven CM points, independently of the original equation-based computation. The motivation is the uniformity question for elliptic curves, and the article positions the equationless method as a path toward it.","feed_headline":"No equations: level-13 Cartan curve's 7 rational points recovered","feed_subtitle":"A moduli-based quadratic Chabauty method computes X_ns^+(13)(Q) using only elliptic curves and Cartan structures.","key_machinery":"The Mumford torsor $M^\\times$ is the central object: the $\\mathbb{G}_m$-torsor over $J\\times J^0$ associated with the Mumford bundle, whose biextension structure allows bilinear combinations of integral lifts and supports the two maps whose intersection is computed. Carrying the arithmetic are section-space representations of divisor classes, where a point of the Jacobian is a subspace $W_D$ of global sections of a fixed line bundle and the group law and torsor operations become linear algebra; these representations are fed by moduli-friendly modular forms evaluated on triples $(E,P,Q)$, so no $q$-expansions or defining equations of $X$ are needed. The special fiber of a regular model of $X_{\\rm{ns}}^+(N)$ supplies the vertical divisor $B$ and the specialization data that make the integral formulas for $\\widetilde{j}_b$ explicit.","core_discovery":"The central claim is that geometric quadratic Chabauty can be run without equations on $X_{\\rm{ns}}^+(N)$. The implementation represents rational points by their moduli data, computes in the Jacobian and in the Mumford torsor $M^\\times$, the $\\mathbb{G}_m$-torsor attached to the Mumford bundle over $J\\times J^0$, and uses the biextension structure of $M^\\times$ together with section-space divisor arithmetic to make the two relevant maps explicit: $\\widetilde{j}_b$, the lift of $(\\mathrm{id}_J,h)\\circ j_b$ to $M^\\times$, and $\\kappa$, the $p$-adic parameterization of the closure of the integral points of $M^\\times$ in a residue disc. In the chosen analytic coordinates, $M^\\times(\\mathbb{Z})$ closes up to the slice with last coordinate $0$, while the image of $\\widetilde{j}_b$ is given by a power series that is at most quadratic modulo $p$; intersecting the two recovers the rational point set. For $N=13$ and $p=5$ this computation gives Theorem 1.1: $X_{\\rm{ns}}^+(13)(\\mathbb{Q})$ is precisely the set of seven known CM points.","pith_inferences":["The proof of the spanning condition in Remark 4.1.8 is the spot I would check first: an independent verification that traces of moduli-friendly forms span $M_2(\\Gamma)$ modulo $p$ would remove the reliance on code termination and make the method fully conditional.","The same section-space/torsor machinery should transfer to other modular curves whose Jacobian has rank equal to genus and a supply of Hecke endomorphisms, not only the non-split Cartan family.","Because the integral closure of $M^\\times(\\mathbb{Z})$ is linear in these coordinates while $\\widetilde{j}_b$ is quadratic modulo $p$, the method isolates the classical Chabauty dichotomy in a particularly clean form; this coordinate choice may be reusable in other $p$-adic Chabauty settings.","If the approach scales, the natural next stress test is a level where $\\rho>3$ or where some residue disc has no known rational point and the gcd of several $\\theta$-polynomials is needed to close the argument."],"forward_implications":["The level-13 result is recovered without the plane model used in the original computation, confirming the equationless pipeline end to end.","For any prime $N$ with $r=g$ and satisfying the auxiliary-prime assumptions, the rational points of $X_{\\rm{ns}}^+(N)$ can in principle be bounded by enumerating residue discs of elliptic curves and Cartan structures and solving polynomial congruences.","The method extends to levels where a projective model is unavailable or has bad reduction, since the model enters only through the regular model's special fiber and the moduli forms.","Multiple trace-zero endomorphisms can be combined to rule out spurious $p$-adic solutions, so the same framework scales with the Néron–Severi rank.","All computations are performed modulo $p^e$ with explicit precision control, and the approach naturally produces upper bounds on the number of rational points per residue disc."],"supporting_citations":[{"why":"supplies the level-13 result that this paper rederives equationlessly","marker":"[BDM+19]"},{"why":"introduces the geometric quadratic Chabauty method and the Mumford-torsor obstruction used here","marker":"[EL21]"},{"why":"supplies the section-space algorithms for arithmetic in the Jacobian","marker":"[KM04, KM07]"},{"why":"extends the section-space algorithms to the Mumford torsor, making $\\widetilde{j}_b$ and $\\kappa$ computable","marker":"[Mas26]"},{"why":"evaluates modular forms on elliptic curves with level structure, supplying the moduli-friendly sections","marker":"[Mas22]"},{"why":"describes the regular model of $X_{\\rm{ns}}^+(N)$ whose special fiber determines the vertical divisor $B$","marker":"[EP24]"},{"why":"gives the smooth-locus interpretation and intersection multiplicities used to specialize Heegner divisors and compute lifts to $M^\\times$","marker":"[LSV26]"}],"fun_headline_variants":["Quadratic Chabauty without equations: 7 points on Cartan curve","No equations, just elliptic curves: level-13 rational points found","Moduli-only Chabauty computes X_ns^+(13) rational points","Geometric Chabauty skips equations, nets 7 CM points","Equationless Chabauty solves level-13 non-split Cartan"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that, for the chosen auxiliary prime $p$, the traces of the moduli-friendly modular forms to the level of $X_{\\rm{ns}}^+(N)$ span the space $M_2(\\Gamma)$ over $\\mathbb{F}_p$; the authors state this is not proven and cite termination of their code as evidence, so if it fails the $p$-adic model of the Jacobian in Section 4.1.4 cannot be constructed.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic Chabauty without equations: 7 points on Cartan curve","No equations, just elliptic curves: level-13 rational points found","Moduli-only Chabauty computes X_ns^+(13) rational points","Geometric Chabauty skips equations, nets 7 CM points","Equationless Chabauty solves level-13 non-split Cartan"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000585,"raw_usage":{"total_tokens":2751,"prompt_tokens":946,"completion_tokens":1805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":1707}},"tokens_in":562,"tokens_out":1805,"duration_ms":13556,"temperature":1.0,"reasoning_tokens":1707,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:16:01.542584+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test is to compute, for a given $N$ and $p$ satisfying the other assumptions, the $\\mathbb{F}_p$-span of those traces inside $M_2(\\Gamma)$: if its dimension is strictly less than $\\dim M_2(\\Gamma)$, the construction of the $p$-adic section-space Jacobian fails and the method's claimed scope collapses. A second, end-to-end falsifier is to run the level-13 algorithm at $p=5$ to higher precision: if the gcd of the $\\theta$-polynomials for $f_1=T_3-T_2$ and $f_2=2T_7-3T_2$ ever leaves a common root whose higher-residue-disc congruences never contradict and which is not one of the seven CM points, the intersection computation would not separate rational from $p$-adic points.","supporting_citations":[],"review_version":1}