REVIEW 2 major objections 4 minor 1 references
Curve equations from expansions of 1-forms at a nonrational point
T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read From power-series expansions of regular 1-forms at a single nonrational point, the algorithm returns defining equations over ℚ for the curve.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [5, Step 7] 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.
- [5, Step 6] 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.
minor comments (4)
- [5, Step 2] 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.
- [5, Step 7] 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.
- [4, Lemma 4.4] 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.
- [5, Steps 3 and 5] 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.
Circularity Check
No circularity: the algorithm recovers curve equations by linear algebra from the input expansions, with no fitted or self-cited quantity renamed as a prediction.
full rationale
The derivation is self-contained. The algorithm takes only the truncated expansions w̄_i and reconstructs the ideals I_d as kernels of explicit Q-linear maps via Corollary 4.2, with hyperellipticity decided by the dimension count in Lemma 4.3. The intermediate objects t, ∂_i, Q, h, F, G, H, and J are obtained by solving linear systems whose uniqueness is proved by vanishing-order bounds in Lemmas 6.1 and 6.2; no target equation is used to fit a parameter. The relation h = y^2 with y = df/ω_1 is a structural identity used to identify the double cover, and the final H is solved from FG ≡ HJ^2 mod Q with existence proved via Riemann-Roch, not by imposing the output curve. The only self-citation, [BGGP05], is used for context and for the genus bound in Remark 6.4, and the main theorem does not rest on that paper's output. The skeptic's Step 7 concern about deg U = 2g+2 being asserted without proof is a possible proof gap in the odd-genus hyperelliptic case, but it is not a circularity: the asserted degree is not an input to the algorithm and is not used to define the object being predicted.
Assumptions & free parameters
assumptions (6)
- standard math Riemann-Roch theorem for curves, including the genus 0 case used for conics and line bundle dimensions.
- standard math Petri's theorem: for nonhyperelliptic canonical curves, I2, I3, I4 generate the ideal.
- standard math Max Noether's theorem on syzygies of canonical curves.
- domain assumption Field operations in the base field are computable.
- domain assumption The input expansions arise from a nice curve X over Q and a point P with K = Q(P).
- standard math Hasse-Minkowski and Hilbert symbols can decide whether a conic over a number field is split.
Cite this review
Pith. "Pith review of Curve equations from expansions of 1-forms at a nonrational point." pith.science (2026). https://pith.science/paper/OBOWTVXL
@misc{pith2026250614026,
author = {Pith},
title = {Pith review of: Curve equations from expansions of 1-forms at a nonrational point},
year = {2026},
howpublished = {\url{https://pith.science/paper/OBOWTVXL}},
note = {Machine review of arXiv:2506.14026}
}
abstract
We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker, Gonz\'alez-Jim\'enez, Gonz\'alez, and Poonen for expansions at a rational point. If the curve is hyperelliptic, the equations present it as an explicit double cover of a smooth plane conic, or as a double cover of the projective line when possible. If the curve is nonhyperelliptic, the equations cut out the canonical model. The algorithm has been used to compute equations over $\mathbb{Q}$ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database.
Reference graph
Works this paper leans on
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[1]
Geometry of alge- braic curves. Vol. I
[ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris. “Geometry of alge- braic curves. Vol. I”. Vol. 267. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985, pp. xvi+386 (↑3). [BGGP05] Matthew H. Baker, Enrique González-Jiménez, Josep González, and Bjorn Poo- nen. ...
arXiv 2005
Reviewed August 15, 2026 · model on record in the stance chip above.
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