{"id":"ac4d1327-be7b-45a7-923d-74c7d4dc8762","arxiv_id":"2506.14026","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An algorithm with proven bound B = 19g+48 reconstructs equations of curves over Q from truncated 1-form expansions at a nonrational point, covering hyperelliptic and nonhyperelliptic cases.","lead":"This paper gives a provably correct algorithm that recovers the equation of an algebraic curve from power series expansions of its regular 1-forms at a nonrational point, extending a 2005 method that worked only at rational points. The algorithm has already produced equations for over 4700 modular curves in the LMFDB, including curves that previously lacked equations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 7 asserts deg U = 2g+2 without proof; this degree claim is load-bearing for the odd-genus hyperelliptic output.","rationale":"I read the paper's central claim as the existence of a correct algorithm recovering curve equations from finite truncations. The precision analysis in Section 6 is intricate but, on close inspection, internally consistent: the Table 1 relative and absolute error bounds combine correctly, and Lemma 6.2's inequality B = 19g+48 provides enough clearance for Steps 3, 5, and 6. I therefore do not find a concrete flaw in the reader's identified weakest assumption. Instead, the most concrete unproved step I found is in Step 7: the assertion deg U = 2g+2 is both necessary for the construction of H and J and not derived from the preceding statements. The claim is likely true, since h = y^2 forces the odd part of the divisor of h to have degree 2g+2, but the paper does not supply this proof. Because the odd-genus hyperelliptic case is a central part of Theorem 3.1 and the algorithm's output depends on the exact degree of U, I would not accept the proof as written without this gap being closed. The computation over 4700 modular curves provides empirical support, but it does not replace the missing argument. The verdict should be CONDITIONAL: accept once Step 7's degree claim is proved or the algorithm is adjusted to guarantee it.","tokens_in":9664,"tokens_out":48952,"duration_ms":506318,"concrete_test":"Independently re-derive Step 7 analytically: starting from F/G = h and h = y^2 on the hyperelliptic cover X -> C, compute the degree of the reduced part U of D = Z_C(F) + Z_C(G) using the divisor of h and the Riemann-Hurwitz branch divisor. If the derivation yields deg U = 2g+2 without extra hypotheses on the chosen F,G, the concern is resolved; if it requires additional conditions not stated in Step 5, then Step 7 needs a revised argument or a modified choice of F,G.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Section 5, Step 7, the proof needs to decompose D := Z_C(FG) as U + 2V with U reduced, and then asserts without justification: \"We have deg U = deg U' = 2g+2, so deg V = deg V' = g+5.\" This is not immediate from the preceding lines: D has degree 4g+12, and a decomposition into U + 2V with U reduced can in general produce reduced parts of many possible even degrees. The subsequent Riemann-Roch construction of H in Q[a,b,c]_{g+1} and J in Q[a,b,c]_{(g+5)/2} requires exactly deg U = 2g+2. If Step 5's linear algebra produced F,G for which the odd part of Z(F)+Z(G) had a different degree, Step 7 would fail to produce the promised model. The missing argument presumably uses that F/G = h = y^2, so the odd-multiplicity part of div(h) consists of the 2g+2 branch points of the hyperelliptic cover; but this is not stated or proved. Thus Theorem 3.1's odd-genus hyperelliptic case contains an unverified structural claim that is independent of the precision analysis in Section 6. The reader's focus on the precision table is reasonable, but this omitted degree computation is at least as load-bearing.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents an algorithm that, given the truncated q-expansions to order B = 19g + 48 of a Q-basis of regular 1-forms on a nice genus-g curve X over Q at a nonrational point P, returns defining equations for X over Q. For nonhyperelliptic X, the output is a set of homogeneous equations cutting out the canonical model; for hyperelliptic X of even genus, a separable f in Q[x] with X birational to y^2 = f(x); for odd genus, a quadratic form Q and a form H presenting X as a double cover of the conic Q = 0. The proof is constructive: Section 4 derives vanishing-order and dimension lemmas, Section 5 gives the infinite-precision algorithm, and Section 6 provides a q-adic error analysis demonstrating that B = 19g + 48 suffices.","tokens_in":9957,"tokens_out":37988,"duration_ms":365819,"significance":"If correct, this is a valuable extension of the BGGP05 algorithm: the nonrational case requires trace constructions to descend from K to Q, and the hyperelliptic quotient can be a non-split conic, which the paper handles for odd genus. The error analysis in Table 1 and Lemma 6.2 is detailed and the claimed bound is explicit. The paper also reports successful use on over 4700 hyperelliptic modular curves without rational cusps, which gives practical evidence. No circularity is apparent: the unknown equations are recovered from the expansions by solving overdetermined linear systems, not fitted by using the target equations. The main concerns are two missing justifications in Section 5, detailed below.","major_comments":[{"comment":"The proof asserts, without justification, that in the decomposition D = U + 2V with U reduced one has deg U = 2g + 2; this equality is load-bearing because it implies deg V = g + 5 and because the later Riemann-Roch step constructs H in degree g + 1, whose zero divisor on the degree-2 conic must have degree 2g + 2. I could not find this fact elsewhere in the paper. The missing argument should be: since F/G = h = y^2, the coefficient of Q in D is |ord_Q(h)|, and the points Q for which ord_Q(h) is odd are exactly the branch points of the double cover X -> C; Riemann-Hurwitz gives 2g + 2 such points counted once. Please write out this argument, as the current text leaves a gap in the proof of Theorem 3.1's odd-genus case.","section":"5, Step 7"},{"comment":"The step opens with 'Suppose that g is even. In this case, C is isomorphic to P^1' but no proof is given. The assertion is necessary because the output f in Q[x] presupposes a rational parametrization of C. It follows from the parity of the degree: the section omega_1 defines a divisor D of degree g - 1 on C, and a non-split conic over Q admits line bundles only of even degree, so g even forces C to be split. A one-line justification should be added here.","section":"5, Step 6"}],"minor_comments":[{"comment":"The sentence 'find three of them that are K-linearly independent and hence Q-linearly dependent' should read 'Q-linearly independent'; as written it contradicts the fact that three independent elements of the 3-dimensional space V form a basis.","section":"5, Step 2"},{"comment":"The notation nu^*D should be nu_*D, or the intended meaning should be explained, since D is a divisor on C and nu maps from C to P^1; as written, pulling back a divisor on C along nu is not defined.","section":"5, Step 7"},{"comment":"The indexing in the proof of Lemma 4.4 is terse; it would help to state explicitly that omega'_g = J_0(x)dx/y with J_0 constant, so that t = J_1/J_0 is a degree-one polynomial rather than a polynomial of degree g - 1.","section":"4, Lemma 4.4"},{"comment":"Several references to 'Theorem 4.2', 'Theorem 4.3', and 'Theorem 4.6' should be to Corollary 4.2, Lemma 4.3, and Lemma 4.6 respectively; the numbering as printed makes the proof harder to follow.","section":"5, Steps 3 and 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the main construction is convincing. The two missing arguments in Section 5 are localized and can be repaired without changing the algorithm or the bounds; I recommend a major revision primarily to require those proofs. I do not see grounds for rejection. The reported LMFDB computations support the practical claim, but the theoretical proof should be complete on its own."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ray — this one is worth a referee's time. It extends BGGP05 from rational to nonrational points, and the real weight is in the hyperelliptic case, especially the odd-genus case where the conic can be pointless and you output the double cover of a conic. The proof is constructive and unusually careful about q-adic precision; Lemma 6.2's B=19g+48 bound is a genuine contribution. The LMFDB usage—4700+ curves, 1500+ as double covers of pointless conics—is real evidence the algorithm runs at scale, even though no code ships.\n\nThe math is sound in the parts I checked. Lemma 4.7 is the right tool for ramification-dependent valuations, Table 1's error propagation holds together, and the linear algebra recovery has no circularity: the equations are solved for, not fit to. Credit is fairly given to Baran and Mercuri–Schoof for the nonhyperelliptic examples.\n\nNow the soft spot. The stress-test flagged Step 7: the assertion deg U = deg U' = 2g+2 is not justified. They decompose D = Z_C(FG) as U+2V with U reduced and immediately claim deg U = 2g+2. That degree is exactly what makes the Riemann–Roch construction of H in degree g+1 work, so it is load-bearing. The claim is true—it follows from h = y^2, so the odd-multiplicity part of div(h) is the 2g+2 branch points of X→C, and the extra common zeros of F and G are doubled—but the paper never says that. It's a one-sentence fix, not a structural flaw. Worth mentioning to the authors, not worth rejecting over.\n\nMinor: no code or data in the submission. The description is detailed enough to re-implement, but a Magma or Sage script would have removed doubt.\n\nWho is this for? People computing equations of modular curves without rational cusps, and anyone extending power-series reconstruction methods. I'd accept it. It deserves serious peer review; with the degree justification added and maybe code, it should be a clean accept.\n\nRecommendation: send to a good editor, accept.","headline":"Extends BGGP05 to nonrational points with a solid, scale-tested algorithm; the only real gap is a missing justification for the degree of U in Step 7, a one-sentence fix.","tokens_in":10481,"tokens_out":7236,"would_cite":true,"duration_ms":67192,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G30","14H45","14Q05"],"pacs":[],"model":"deepseek-v4-flash","headline":"From power-series expansions of regular 1-forms at a single nonrational point, the algorithm returns defining equations over ℚ for the curve.","keywords":["algebraic curves","regular 1-forms","power series expansions","nonrational point","hyperelliptic curves","canonical model","characteristic 0","modular curves"],"falsifier":"A direct check is to take a known hyperelliptic curve over $\\mathbb{Q}$, choose a nonrational point $P$ defined over a quadratic field, compute the first $19g+47$ terms instead of the advertised $19g+48$, and see whether the algorithm's linear algebra in Step 5 still selects the correct section $F-hG$ or instead lands on a nonzero section vanishing at $P$ to order greater than $8g+24$. The sharp threshold in Lemma 6.2 is exactly the claim that no nonzero section of the relevant line bundle can vanish that deeply, so any explicit section with order $8g+25$ at $P$ would falsify the bound.","tokens_in":9496,"feed_emoji":"📐","tokens_out":13040,"duration_ms":108401,"temperature":0.7,"pith_summary":"This paper proves a recovery theorem: if $X$ is a smooth projective curve of genus $g\\ge 2$ over $\\mathbb{Q}$ and $P$ is a point defined over a number field $K$, then the first $B=19g+48$ coefficients of the power-series expansions of a $\\mathbb{Q}$-basis of the regular 1-forms at $P$ are enough to determine equations for $X$ over $\\mathbb{Q}$. In the nonhyperelliptic case the output is the canonical model; in the hyperelliptic even-genus case it is a separable equation $y^2=f(x)$; in the hyperelliptic odd-genus case it is a double cover $y^2=H$ of a smooth plane conic $Q=0$. This extends an earlier rational-point construction, and the new work is making the hyperelliptic case work when the point's field of definition is larger than $\\mathbb{Q}$, including the situation where the conic has no $\\mathbb{Q}$-point. The paper also gives a full precision analysis showing that the cut-off $B=19g+48$ is sufficient, and reports that the method has been run on thousands of hyperelliptic modular curves without rational cusps.","feed_headline":"19g+48 terms at one nonrational point recover the curve's equations","feed_subtitle":"Extends the rational-point method to nonrational points, covering hyperelliptic and nonhyperelliptic curves.","key_machinery":"The load-bearing object is the precision ledger in Table 1, together with the trace construction. Table 1 tracks, for each Laurent series produced along the way (the 1-forms $\\omega_j$, the function $t$, its differential $dt$, the tangent-bundle sections $t^i\\,d/dt$, the traced operators $\\partial_i$, monomials $M(\\partial_0,\\partial_1,\\partial_2)$, and the functions $f$, $df$, $y$, $h$, $hG$, $F-hG$), both an absolute error $\\mathcal{O}(q^n)$ and a relative error $1+\\mathcal{O}(q^n)$, together with the possible orders of vanishing at $P$. The decisive bounds are $\\operatorname{ord}_P(M(\\partial_0,\\partial_1,\\partial_2))\\le 3d$ for a monomial of degree $d$, and Riemann\\textendash Roch degree counts implying that a nonzero section of a certain line bundle of degree at most $8g+24$ cannot vanish at $P$ to order greater than $8g+24$. The $(L\\otimes K)/K$-trace is what converts objects expanded over the larger field $L$ back into objects defined over $\\mathbb{Q}$, which is exactly what the rational-point version of the algorithm could not do.","core_discovery":"The central claim, Theorem 3.1, is a deterministic algorithm with the following specification. Given $g$, a number field $K$, and truncated expansions to order $B=19g+48$ of $g$ regular 1-forms forming a $\\mathbb{Q}$-basis, the algorithm outputs: homogeneous polynomials over $\\mathbb{Q}$ cutting out a curve in $\\mathbf{P}^{g-1}$ linearly isomorphic to the canonical model, if $X$ is nonhyperelliptic; a separable $f\\in\\mathbb{Q}[x]$ of degree $2g+1$ or $2g+2$ with $X$ birational to $y^2=f(x)$, if $X$ is hyperelliptic of even genus; or a quadratic form $Q$ and a form $H$ of degree $g+1$ over $\\mathbb{Q}$ presenting $X$ as the double cover $y^2=H$ of the conic $Q=0$, if the genus is odd. The proof constructs an isomorphic copy $L$ of $K$ so that the point becomes an $L$-point; orders the forms by vanishing order; forms the rational function $t$ as the ratio of the last two forms; takes $(L\\otimes K)/K$-traces to produce three sections $\\partial_0,\\partial_1,\\partial_2$ of the tangent bundle of the conic; finds the unique quadratic relation $Q(\\partial_0,\\partial_1,\\partial_2)=0$; and defines $h=(d(a/b)/\\omega_1)^2$, whose square root generates the hyperelliptic cover. A precision table tracks the order of vanishing and the absolute and relative errors of every series, and the condition $B\\ge 19g+48$ is derived from worst-case degree counts.","pith_inferences":["The bound $B=19g+48$ is probably not optimal: Step 6 only requires $B\\ge 17g/2+1$, and several losses in Table 1 are worst-case, so a sharper ledger could reduce the required input length or accelerate the linear algebra.","The trace construction that turns expansions over $L$ into objects defined over $\\mathbb{Q}$ is a transfer principle that should work for other situations where a genus-0 quotient is visible in local data, for example quotients of curves by finite groups.","A numerical variant could feed floating-point expansions, use Table 1's relative-error bounds to decide when to stop, guess $F,G,H,J$ by approximate linear algebra, and then certify the output exactly; the paper's analysis provides the stopping-criterion evidence such a variant would need.","The same bounds should carry over to expansions at several points, trading fewer terms per point for better conditioning in cases where the canonical image has high degree."],"forward_implications":["For nonhyperelliptic curves, the canonical model over $\\mathbb{Q}$ is computed from local expansions alone; the only global input is the genus and the truncated series.","For hyperelliptic curves of even genus, the output is a separable $f\\in\\mathbb{Q}[x]$ of degree $2g+1$ or $2g+2$, so the curve is exhibited as an explicit equation $y^2=f(x)$ over $\\mathbb{Q}$.","For hyperelliptic curves of odd genus, the curve is exhibited as a double cover $y^2=H$ of the conic $Q=0$; if a $\\mathbb{Q}$-point of the conic is also known, an affine equation $y^2=f(x)$ follows.","The same proof works over any computable characteristic-$0$ field, so the theorem is not specific to number fields.","For hyperelliptic modular curves without rational cusps, whose geometric gonality is 2 and whose genus is at most 17, the algorithm has produced equations for over 4700 curves; over 1500 of these are double covers of pointless genus-0 conics."],"supporting_citations":[{"why":"The rational-point algorithm this paper extends; its Section 2.1 is the baseline for the nonhyperelliptic case and its Remark 4.5 supplies the gonality and genus bound used in Remark 6.4.","marker":"[BGGP05]"},{"why":"Supplies the hyperelliptic basis of regular 1-forms and the dimension count for quadrics in the canonical ring used in Lemma 4.3 to distinguish the hyperelliptic case.","marker":"[ACGH85]"},{"why":"Supplies the theorem that quadrics, cubics, and quartics cut out the canonical model of a nonhyperelliptic curve, so that computing the degree-2, degree-3, and degree-4 parts of the ideal suffices.","marker":"[Pet23]"},{"why":"Identifies the motivating input: expansions at nonrational points of regular 1-forms arise naturally in explicit computations of open images for elliptic curves.","marker":"[Zyw24]"},{"why":"Supplies the local-global criterion for quadratic forms used in Remark 3.3 to decide whether the conic is isomorphic to the projective line, which is needed for the odd-genus affine model.","marker":"[Shi10]"}],"fun_headline_variants":["Nonrational point curve recovery via 19g+48 terms","Curve equations from one nonrational point expansion","Extending rational-point algorithm to nonrational points","19g+48 terms from one nonrational point yield curve equations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithm is only as strong as its power-series error ledger: if the bound $\\operatorname{ord}_P(M(\\partial_0,\\partial_1,\\partial_2))\\le 3d$ for monomials of degree $d$ is wrong, or if the absolute and relative errors in Table 1 are understated, then the linear-algebra steps can select the wrong forms even with $B=19g+48$ terms supplied.","fun_headline_variants_meta":{"raw":{"variants":["Nonrational point curve recovery via 19g+48 terms","Curve equations from one nonrational point expansion","Extending rational-point algorithm to nonrational points","19g+48 terms from one nonrational point yield curve equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001031,"raw_usage":{"total_tokens":4399,"prompt_tokens":1055,"completion_tokens":3344,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":671,"completion_tokens_details":{"reasoning_tokens":3278}},"tokens_in":671,"tokens_out":3344,"duration_ms":22080,"temperature":1.0,"reasoning_tokens":3278,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:56:24.912448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to take a known hyperelliptic curve over $\\mathbb{Q}$, choose a nonrational point $P$ defined over a quadratic field, compute the first $19g+47$ terms instead of the advertised $19g+48$, and see whether the algorithm's linear algebra in Step 5 still selects the correct section $F-hG$ or instead lands on a nonzero section vanishing at $P$ to order greater than $8g+24$. The sharp threshold in Lemma 6.2 is exactly the claim that no nonzero section of the relevant line bundle can vanish that deeply, so any explicit section with order $8g+25$ at $P$ would falsify the bound.","supporting_citations":[],"review_version":2}