{"id":"c9d26dc5-e818-4459-8768-0cce7e2409d8","arxiv_id":"2509.02291","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A q-expansion based algorithm computes the Hodge filtration needed for quadratic Chabauty on X_0^+(N), with implementations for N=67 and N=193.","lead":"This paper presents an algorithm that computes Hodge filtrations for quadratic Chabauty on Atkin-Lehner quotients X_0^+(N) using weakly holomorphic modular forms, replacing the expensive plane-model input. It implements the method on genus 2 and genus 7 curves and reports new congruences between iterated and single integrals.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 3.8's minimal-jdR step is unproven: basis property for H^1_dR is proved only for j=d-1, while the algorithm uses the smallest j satisfying (14); the §4 examples do not cover the proved case.","rationale":"The reader's weakest assumption identifies exactly the same gap: the algorithm chooses the smallest jdR satisfying (14), while the paper proves existence and basis property only for j=d-1. This is not a manufactured concern; it is visible in the text. Proposition 3.2 proves existence by taking j=d-1, and the subsequent 'always works' argument in §3.1 explicitly takes fdR=s1/sd. In contrast, Algorithm 3.8 Step (3) instructs the user to take the smallest j, and the only evidence offered for that choice is the numerical cup product checks in Section 4. Those checks are genuine and nonzero for N=67 and N=193, but they are not a proof for all primes; moreover, in both examples the minimal jdR is far from d-1, so they do not instantiate the proved case. The gap is load-bearing because the determinant of the cup product matrix is the criterion for having a basis of H^1_dR, and the subsequent computation of Tp, Z, and lambda_Fil depends on that basis. If a degenerate minimal-jdR case existed, Step (3) would output an invalid basis and the algorithm would not compute the Hodge filtration. I am not claiming such a counterexample exists; the worked examples support the method empirically. But the central claim is stated as an algorithm for all prime N of genus at least 2, and its correctness is not fully established. The reader's CONDITIONAL verdict is appropriate: the gap is addressable by either a proof for the minimal jdR or by changing the algorithm to the proved j=d-1 choice, and the current evidence does not contradict the examples. Hence no adjustment to the reader's verdict is needed.","tokens_in":35578,"tokens_out":8428,"duration_ms":97743,"concrete_test":"Run Algorithm 3.8 in Magma for all primes N with genus at least 2 and N <= 1000 (or at least for the genus-7 primes 193, 229, 233, 241, 257, 281): compute the minimal jdR satisfying (14), form the 2g differentials {omega_0,...,omega_{g-1}, (s_{d-jdR}/s_d)omega_0,...}, and compute the determinant of their cup product matrix via Serre's formula. If any determinant vanishes, Algorithm 3.8 fails as stated. If all determinants are nonzero, the concern remains a missing proof rather than a disconfirmed claim; the decisive analytic check is then to prove the dimension statement for the minimal jdR, or amend Step (3) to the fully proved choice j=d-1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 3.8(3) defines jdR as the smallest integer in {1,...,d-1} satisfying (14). The general existence proof, Proposition 3.2, constructs j=d-1, and the 'always works' argument in §3.1 explicitly says 'if we take fdR = s1/sd' (i.e. j=d-1). No proposition shows that the minimal admissible j inherits the basis property. The paper itself concedes after Corollary 3.5: 'by computing the cup product, we have found that taking jdR as in (14) always produces a basis in our examples in §4.' That is numerical evidence, not a theorem. For N=67, jdR=1 while d-1=31; for N=193, jdR=3 while d-1=89, so the worked examples do not exercise the proved case at all. The proof in §3.1 bounds pole orders using s1/sd; the smaller pole order of s_{d-jdR}/s_d can in principle satisfy the valuation condition (14) yet fail to span a complementary g-dimensional subspace of H^1_dR. If some prime N admitted a minimal jdR with degenerate cup product, Step (3) would produce a non-basis and the claimed computation of lambda_Fil would not be defined. This is the load-bearing gap in the central algorithmic claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents an algorithm (Algorithm 3.8) for computing the Hodge filtration on the universal connection AZ for Atkin–Lehner quotients X_0^+(N), N prime, genus at least 2, using only q-expansions of the plus eigenspaces S_2^+(Γ_0(N)) and S_12^+(Γ_0(N)), thereby avoiding an explicit plane model. The first half of the de Rham basis is obtained from holomorphic differentials ω_i attached to S_2^+, and the second half from products s_{d-j_dR}/s_d · ω_i, where s_i are ordered basis elements of S_12^+ and j_dR is the smallest integer satisfying the valuation condition (14). The algorithm then computes T_p, the nice correspondence Z, the gauge matrix ψ_{αβ}, and finally β_Fil and γ_Fil. Worked computations are given for X_0^+(67) (genus 2) and X_0^+(193) (genus 7), together with corank computations for N=67,97,193 suggesting congruences between iterated integrals in the plus eigenspace and single integrals in the minus eigenspace.","tokens_in":35932,"tokens_out":8596,"duration_ms":107792,"significance":"If the correctness gap described below is closed, this is a useful technical advance: it replaces a plane-model input with modular-form data, extends the worked range to genus 7, and gives an explicit, checkable route to a Hodge filtration for quadratic Chabauty. The paper provides many detailed q-expansion matrices and cup-product computations, so the numerical content is concrete and partly verifiable. The empirical congruence observations in §4 are interesting, though they are presented as evidence rather than as a proven theorem. The main theoretical weakness is that the basis-producing step used in the actual algorithm is only verified numerically, not proved.","major_comments":[{"comment":"The central algorithmic claim is not proved for the choice made in Step 3. Proposition 3.2 proves existence of a j satisfying (14) by taking j=d-1, and the argument that the construction 'always works' in §3.1 explicitly uses fdR=s1/sd. Algorithm 3.8 instead chooses jdR to be the smallest j satisfying (14). The paper itself concedes after Corollary 3.5 that the basis property for this minimal choice is checked only numerically in §4. For N=67, jdR=1 while d-1=31, and for N=193, jdR=3 while d-1=89, so the worked examples do not exercise the proved case at all. If a prime N admitted a minimal jdR satisfying (14) but producing a degenerate cup product, Step (7) would not be defined and the claimed λ_Fil would not exist. The algorithm should either be changed to take jdR=d-1, or a proof must be supplied that the minimal admissible j inherits nondegeneracy of the cup product.","section":"§3.1, Corollary 3.5 and Algorithm 3.8(3)"},{"comment":"The normalization γ_Fil(b)=0 requires a rational point b on X in the affine open Y=X\\{∞}, but Algorithm 3.8 does not list b (or the hypothesis X(Q)≠∅) among its inputs, and the text merely says 'we can take b to be the upper half plane representative of a rational CM point of X'. No proof is given that such a point exists for every N in the stated range, and the worked examples do not identify the point used. Since the uniqueness of γ_Fil depends on this normalization, the algorithm as stated is incomplete. Please state the nonemptiness assumption explicitly and, for the examples, specify the point b or explain how γ_Fil was normalized.","section":"§3.2, Algorithm 3.8(7)"},{"comment":"Even for the value j=d-1 covered by Proposition 3.2, the paper does not give a complete proof that {ω_0,...,ω_{g-1}, (s1/sd)ω_0,...,(s1/sd)ω_{g-1}} has nonzero cup-product determinant. The valuation condition (14) is necessary for the displayed differentials not to lie in H^0(X,Ω^1), but it does not by itself imply that they span a complementary g-dimensional subspace of H^1_dR. The text derives a sufficient pole-order supply of functions from Corollary 3.5, then states that the method 'will always work' and appeals to numerical cup-product checks. A theorem establishing nondegeneracy for the chosen j, or an explicit counterexample, is needed for the central claim.","section":"§3.1, after (20)"}],"minor_comments":[{"comment":"Step (2) says 'd = dim S_2^+(Γ_0(N))' but d is used as the dimension of S_12^+(Γ_0(N)); this is surely a typo, but it is confusing in a pseudocode algorithm.","section":"Algorithm 3.8(2)"},{"comment":"The equality w_N(Δ(z)) = Δ(-1/(Nz)) = Δ(Nz) suppresses the automorphy factor and the normalization of the Atkin–Lehner operator; the statement would be cleaner if the slash action and the resulting constant factor were written explicitly. The subsequent quotient construction is unaffected, but the notation should be corrected.","section":"Proposition 3.1"},{"comment":"In Step (7) of Algorithm 3.8 the text refers to '(22 holds)' and the displayed condition is stated twice with slightly different notations; please clean up the formatting and the references to equations.","section":"§3.2, equation (22)"},{"comment":"The displayed symplectic basis elements contain denominators written as '22', '32', '33' (e.g. 22·23·101·617). These are clearly powers of 2 and 3 but should be typeset as 2^2, 3^2, 3^3 to avoid misreading the numerical coefficients.","section":"§4.3"}],"recommendation":"major_revision","confidential_remarks":"The main gap is acknowledged in the paper ('always produces a basis in our examples') and is fixable either by a proof for the minimal j_dR or by modifying the algorithm to use j=d-1. The manuscript would also benefit from an explicit statement of all hypotheses needed for the normalization point b, and from making the Magma code available. I do not see grounds for rejection: the computational contribution is tangible and the correctness issue is local, but it is load-bearing for the central algorithmic claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take. The genuinely new thing is the model-free Hodge filtration algorithm: it replaces the plane model input with q-expansions of S_2^+ and S_12^+, works through a genus 2 and a genus 7 example, and gives a way to compute lambda_Fil entirely in terms of modular forms. That is a real step forward for quadratic Chabauty computations on these curves, and the worked examples are a useful resource.\n\nThe main derivation in Section 3.1 is coherent: the function f_dR = s_{d-j}/s_d has poles only at the cusp, the valuation condition (14) is the right one to avoid holomorphic or residue issues, and the cup product computations verify the basis in the examples. The use of Serre's cup product formula to get the Hecke operator matrix and then the correspondence Z is neat and avoids p-adic Frobenius.\n\nNow the soft spot, and it is significant. Algorithm 3.8 uses the smallest j_dR satisfying (14), but the proof in Section 3.1 only covers j = d-1. The paper itself says after Corollary 3.5 that for the minimal j_dR, 'we have found that taking j_dR as in (14) always produces a basis in our examples in §4.' That is numerical evidence, not a theorem. The examples make this worse: for N=67, j_dR=1 and d-1=31; for N=193, j_dR=3 and d-1=89. So the proved case is never tested. If a prime N admits a minimal j_dR with degenerate cup product, the algorithm would fail silently unless the determinant check catches it (and the paper doesn't describe a loop to handle that). This is a fixable gap: either prove the minimal j_dR works, or modify the algorithm to search increasing j_dR and verify the cup product determinant each time. But as stated, the central claim 'this method always produces a basis' is not established.\n\nMinor issues: the speed claim in the abstract is unsupported by benchmarks; no Magma code is provided despite the implementation being a main selling point; and Algorithm 3.8 has a small typo (d is defined as dim S_2^+ but should be S_12^+). All of these are addressable.\n\nOverall: the paper is a solid computational contribution with a load-bearing but well-contained proof gap. I would send it to peer review. A good referee can get it into good shape. I'd bring it to a reading group if someone in the group works on Chabauty methods; otherwise the gap makes it a 'maybe'.","headline":"Useful model-free Hodge filtration algorithm, but the 'always works' claim is proved only for j=d-1 while the algorithm uses minimal j_dR; the worked examples don't cover the proved case.","tokens_in":36393,"tokens_out":3313,"would_cite":false,"duration_ms":35683,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11F25","11G18","14G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper gives an algorithm that computes the Hodge filtration on X_0^+(N) from q-expansions of modular forms instead of a plane model.","keywords":["quadratic Chabauty","Hodge filtration","Atkin-Lehner quotients","weakly holomorphic modular forms","q-expansions","de Rham cohomology","p-adic heights","X_0^+(N)"],"falsifier":"For a prime N not among 67, 97, 193 with genus(X_0^+(N)) ≥ 2, compute the minimal j_dR from (14), form {ω_i, (s_{d−j_dR}/s_d)ω_i}, and evaluate the exact cup product determinant; a zero determinant would refute the claim that the algorithm always produces a basis of H^1_dR.","tokens_in":35454,"feed_emoji":"🧮","tokens_out":10514,"duration_ms":100813,"temperature":0.7,"pith_summary":"The paper aims to make quadratic Chabauty computations of rational points on the modular curves X_0^+(N) independent of an explicit plane model of the curve. It does this by giving an algorithm, Algorithm 3.8, that computes the Hodge filtration on the universal connection AZ using only q-expansions of weight-2 and weight-12 cusp forms in the plus eigenspace of the Atkin-Lehner involution. A key step is a q-expansion basis of de Rham cohomology built by multiplying holomorphic differentials by a quotient of weight-12 cusp forms; the paper proves such a quotient exists and verifies the basis property numerically in the worked examples. This matters because plane models become impractical as the genus grows, and the Hodge filtration is the main input for the p-adic heights that define the finite set of p-adic points containing the rational points.","feed_headline":"Hodge filtration computed straight from q-expansions","feed_subtitle":"Computes the Hodge filtration from q-expansions for X_0^+(N), removing the plane-model bottleneck in quadratic Chabauty.","key_machinery":"The central object is the Hodge filtration matrix λ_Fil, a (2g+2)-block lower-triangular unipotent matrix recording how the Hodge filtration sits inside the universal connection AZ. The innovation is to build the de Rham basis and λ_Fil from weakly holomorphic modular forms: the function f_dR = s_{d−j_dR}/s_d, a quotient of two weight-12 plus-eigen cusp forms, has poles only at ∞ and supplies the missing g differentials; Serre's cup-product formula verifies the basis; the Hecke operator T_p, computed on q-expansions, produces the correspondence Z and the connection Λ; solving the gauge equation G_∞^{-1} dG_∞ = Λ yields ψ_{αβ}, and the universal property determines β_Fil and γ_Fil by the inte","core_discovery":"The paper claims that for prime N with genus(X_0^+(N)) ≥ 2, the matrix λ_Fil — the data of the Hodge filtration on the rank-(2g+2) unipotent bundle with connection AZ — can be computed from modular form q-expansions alone. Algorithm 3.8 obtains a basis of H^1_dR(X/Q) as {ω_0,...,ω_{g−1}, f_dR ω_0,..., f_dR ω_{g−1}}, where the ω_i come from S_2^+(Γ0(N)) and f_dR = s_{d−j_dR}/s_d is a quotient of weight-12 plus-eigen cusp forms with poles only at the cusp ∞. It then computes the Hecke operator T_p and the nice correspondence Z exactly, solves for the gauge matrix ψ_{αβ}, and determines β_Fil and γ_Fil, with γ_Fil built from the same family of quotients. The worked examples include genus 2 (N=6","pith_inferences":["If the basis construction extends to every j satisfying (14), the same algorithmic pattern would likely apply to other modular curves with a single cusp; for curves with several cusps the single valuation condition would become a system of pole conditions.","The corank difference g−1 in the congruence experiments suggests that the iterated integrals of plus-eigen cusp forms and the single integrals of minus-eigen cusp forms are related by a quotient of rank g−1, possibly a manifestation of the correspondence Z; this could be tested for further primes.","The open gap between the proven choice j = d−1 and the algorithm's minimal j_dR could be closed by proving the cup-product determinant is nonzero for every j satisfying (14), or by a deterministic criterion for choosing j; the algorithm's speed depends on using the minimal choice.","Since the Hecke matrix is computed exactly and rationally, the choice of prime p can be optimized after the cohomology is computed, unlike model-based approaches where p is effectively fixed when computing Frobenius."],"forward_implications":["Algorithm 3.8 computes λ_Fil for any prime N with genus(X_0^+(N)) ≥ 2 using only q-expansions of S_2^+(Γ0(N)) and S_12^+(Γ0(N)), eliminating the plane model as an input.","Because T_p is computed as exact rational numbers rather than p-adic approximations, the same Hodge filtration computation can be reused for multiple primes p.","The method verifiably handles genus 7: for X_0^+(193), the q-expansion cup product matrix has nonzero determinant, yielding a symplectic basis, T_3, Z, β_Fil, and γ_Fil.","The congruence experiments for N=67 and N=97 show a corank difference of g−1 = ρ(J)−1 between matrices built from iterated integrals of plus-eigen cusp forms and single integrals of minus-eigen cusp forms, indicating a systematic relation.","This is the first step toward a model-free quadratic Chabauty algorithm; the remaining step is the matching Frobenius-structure (λ_ϕ) computation."],"supporting_citations":[{"why":"Supplies the model-based quadratic Chabauty algorithm (Algorithm 3.12) that this paper replaces, and Lemma 2.7 showing no height contribution away from p.","marker":"[BDM+23]"},{"why":"Defines the universal connection AZ, the Hodge filtration matrices λ_Fil, and the conditions (Lemma 4.7) determining β_Fil and γ_Fil.","marker":"[BDM+19]"},{"why":"Provides Algorithm 5.20 and the statement Dcris(AZ(b,x)) = x*AZ used in the Hodge-filtration computation.","marker":"[BM23]"},{"why":"Shows p-adic heights give quadratic Chabauty pairs, establishing why the Hodge filtration must be computed.","marker":"[BD18]"},{"why":"Proves finiteness of X(Qp)_2 for X_0^+(N), N prime, and supplies ρ(J)=g, justifying the choice of Z.","marker":"[DLF21]"},{"why":"Hadian's universal property fixes β_Fil and γ_Fil uniquely; the paper solves for them from this condition.","marker":"[Had11]"},{"why":"Valence formula and dimension estimates used to prove existence of a suitable j and the pole-order bound.","marker":"[DS05]"},{"why":"Eta-quotient criterion used to show √Δ·w_N(Δ) is a weight-12 plus-eigen cusp form vanishing only at ∞.","marker":"[RW15]"},{"why":"Dimension bound for weight-12 newforms used to prove the quotients have poles of sufficiently large order.","marker":"[Mar18]"},{"why":"Provides the unipotent Albanese/Selmer theory and the result guaranteeing the trivialisation s0, so the universal connection AZ is well-defined.","marker":"[Kim09]"}],"fun_headline_variants":["Hodge filtration from q-expansions alone","No plane model: compute Hodge filtration from cusp forms","Quicker Chabauty: Hodge filtration from weakly holomorphic forms","Congruence found: iterated integrals reduce to single integrals"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The algorithm chooses the smallest j_dR satisfying the valuation condition (14), but the paper proves the basis property only for j = d−1; for the minimal j_dR the invertibility of the cup product matrix is checked numerically, not proved.","fun_headline_variants_meta":{"raw":{"variants":["Hodge filtration from q-expansions alone","No plane model: compute Hodge filtration from cusp forms","Quicker Chabauty: Hodge filtration from weakly holomorphic forms","Congruence found: iterated integrals reduce to single integrals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000386,"raw_usage":{"total_tokens":1916,"prompt_tokens":825,"completion_tokens":1091,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":1021}},"tokens_in":569,"tokens_out":1091,"duration_ms":11671,"temperature":1.0,"reasoning_tokens":1021,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:42:38.003494+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a prime N not among 67, 97, 193 with genus(X_0^+(N)) ≥ 2, compute the minimal j_dR from (14), form {ω_i, (s_{d−j_dR}/s_d)ω_i}, and evaluate the exact cup product determinant; a zero determinant would refute the claim that the algorithm always produces a basis of H^1_dR.","supporting_citations":[],"review_version":1}