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Formal Abel relations for curves in characteristic $p$

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Pith's one-line read Formal Abel relations in characteristic p recover algebraic plane curves from low-degree power series on their branches.

desk verdict Solid formal-power-series infrastructure for Abel relations in char p, with clean low-degree matching and honest counterexamples; the full converse is deferred. read the letter →

arxiv 2607.09471 v1 pith:67XD263B submitted 2026-07-10 math.AG

John B. Little This is my paper · ORCID
classification math.AG MSC 14H4014L0514H0514H20
keywords AbeltheoremformalgrouplawsJacobianscharacteristicpCartieroperatorplanecurveswebgeometrydualizingdifferentials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This working paper shows how Abel's classical addition laws on the intersections of a plane curve with nearby lines survive in characteristic p when rewritten in formal power series. The maps from the formal branches of the curve into the formal group law of its (generalized) Jacobian still sum to zero, and the pulled-back invariant differentials recover the usual Abelian (or dualizing) forms. For primes larger than the degree plus two, the equations coming from terms of total degree at most d+2 already force those formal branches to be the local expansions of a single algebraic curve of degree d. The same apparatus produces explicit examples of ordinary, superspecial and intermediate Jacobians, exhibits extra non-uniqueness of the maps coming from p-power periodicity, and isolates counter-examples to the unrestricted converse when the tangent lines are concurrent. The ultimate aim is a characteristic-p form of Griffiths' converse of Abel's theorem, usable for translation manifolds and web geometry.

What carries the argument

The maximal formal Abel relation: the product, under the formal group law G of the Jacobian, of the maps ti evaluated on the power-series points of intersection with the pencil of lines x=uy+v is identically zero. Coefficient comparison of that identity, after normalizing G so that G(X,Y)≡X+Y mod degree p, produces the polynomial equations that characterize algebraic curves.

What would settle it

Exhibit four formal power series y=fi(x) in characteristic p>6 whose degree-6 Abel equations (with the normalized group law) are satisfied, yet the product (y-f1) o(y-f4) is not a polynomial of total degree 4.

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Extended reading notes

Core claim

For a reduced plane curve of degree d over an algebraically closed field of characteristic p, the formal branches admit maps into the formal group law of the (generalized) Jacobian whose group-law sum vanishes identically, and when p>d+2 the degree-(d+2) truncation of any maximal such relation already forces the branches to arise from an algebraic curve of degree d.

Load-bearing premise

The formal group law of the Jacobian can always be normalized so that it looks like ordinary vector addition until degree p; if that normalization distorts the low-degree Abel equations for some curves, the identification of truncated relations with algebraic curves fails.

Editorial extensions

If this is right

  • For p>d+2 the space of maximal formal Abel relations truncated at degree d+2 coincides with the space of algebraic plane curves of degree d.
  • The matrix of degree-p terms in the formal group law is precisely the Cartier–Manin matrix of the curve, linking Abel relations to the p-rank and a-number.
  • Formal Abel relations continue to exist for singular curves, now taking values in the formal group of the generalized Jacobian (products of additive and multiplicative factors).
  • Extra solutions generated by p-power maps on the coefficients yield infinitely many distinct maps ti giving the same formal Abel relation on a fixed set of branches.
  • A subsequent paper can complete the characteristic-p converse of Abel’s theorem needed for double-translation manifolds and web geometry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same coefficient-matching technique should extend without change to the generalized Jacobian of a nodal or cuspidal curve, giving a purely formal characterization of dualizing differentials.
  • The counter-examples with concurrent tangents suggest that a correct converse statement must impose a non-concurrency (or transverse) hypothesis already visible in the degree-2 terms of the Abel relation.
  • Because the low-degree equations are defined over Z, reduction modulo p of a characteristic-0 Abel relation automatically yields a characteristic-p relation, furnishing a direct comparison between the two theories.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies formal Abel relations for reduced plane curves of degree d over algebraically closed fields of characteristic p. Theorem 3 asserts that the formal branches Ci of such a curve admit maps ti into the formal group law G of the (generalized) Jacobian so that the G-sum of the ti(xi(u,v)) vanishes identically, with the pull-backs of invariant differentials recovering a basis of dualizing differentials. Theorem 4 shows that when p>d+2 the degree-(d+2) truncation of a maximal Abel relation already forces the branches to arise from an algebraic curve of degree d; the argument is carried out in detail for d=4 via explicit coefficient equations and a Gröbner-basis comparison, and sketched for general d. Subsequent sections describe the first appearance of formal-group corrections at degree p (via the Cartier–Manin matrix), give concrete examples for ordinary, superspecial and intermediate genus-3 curves over F7, exhibit non-uniqueness of the maps ti coming from p-power operations, and construct counter-examples to a naïve converse when the tangent lines are concurrent.

Significance. If the low-degree matching of Theorem 4 extends cleanly to a full characteristic-p converse of Abel’s theorem, the work would complete a programme begun in the author’s 1980 thesis and supply the missing algebraic ingredient for a characteristic-p version of the Lie–Wirtinger theorem and related results in web geometry. The paper already contributes a careful formal-group analysis of Abel relations, an explicit link between the degree-p terms and the Cartier–Manin matrix, and a collection of worked examples that illustrate the range of possible formal Jacobian structures. These ingredients are of independent interest for the arithmetic geometry of curves and their Jacobians.

major comments (2)
  1. Theorem 4 is stated for general d≥4, yet the only complete verification is the d=4 Gröbner-basis computation that equates the ideal of Abel coefficients of total degree ≤5 with the ideal of algebraic Reiss-type relations (13)–(14). The phrase “mutatis mutandis” does not supply the corresponding elimination for higher d; either a uniform argument (e.g., via generating functions or the Lagrange reversion formula) or an explicit statement that the general case is deferred is needed before the identification X_max_{d+2}=C_{d+2} can be regarded as proved.
  2. The counter-examples of §8 (deformations y=ai,0+x^q with concurrent tangents) show that a naïve converse fails, but the manuscript never formulates the precise additional hypotheses under which a converse is expected to hold. Without such a statement the reader cannot judge how much of the classical converse survives in characteristic p, nor how the subsequent paper is intended to close the gap.
minor comments (4)
  1. The abstract and introduction repeatedly call the manuscript a “working paper”; for journal submission this language should be removed and the relation to the announced sequel clarified.
  2. Lemma 2.1 of the author’s earlier paper [10] is cited for the coefficient patterns of xi(u,v), yet a missing term of weight 4 is noted; a self-contained statement of the corrected formulae would improve readability.
  3. In the ordinary-curve example of §6 the Newton polygon is described as having only slopes 0 and 1, but the intermediate points (2,1) and (5,2) are said to “lie above” the relevant segments; a short clarification of the convex-hull construction would avoid confusion.
  4. Several typographical inconsistencies appear (e.g., “Grifffiths”, “charact´eristique”, “Fa` a di Bruno”); a careful copy-edit is required.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: formal Abel relations and low-degree algebraic forcing rest on standard Jacobian/formal-group facts, not on self-definition or fitted inputs.

full rationale

The paper's central claims (Theorem 3 existence of formal Abel relations via the Jacobian formal group law; Theorem 4 that the degree-(d+2) truncation forces algebraic curves of degree d when p>d+2) are derived from classical isomorphisms (Milne on Jacobians, Serre duality, formal implicit-function theorem, Lagrange reversion) and from Hazewinkel's structure theory of formal groups (Lemma 1, Theorem 5). The coefficient equations that define A_max_n and X_max_n are obtained by expanding the group-law sum and setting coefficients to zero; they are not inserted by definition of the target algebraic curves. The explicit d=4 Gröbner-basis verification that those equations recover the Reiss-type relations of an algebraic quartic is a direct algebraic computation, not a fit. Self-citations to the author's 1980 thesis and 1984 paper supply background for the d=3 case and the intended converse; they are not load-bearing for the new formal-group statements, the Cartier-Manin extraction, or the §8 counter-examples. The non-uniqueness constructions of §7 follow from the p-power map on coefficients already present in the Abel equations and do not redefine the relations. Consequently the derivation chain is self-contained against external mathematical facts and exhibits no circular reduction of a claimed prediction to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

Pure algebraic geometry over algebraically closed fields; the load-bearing background consists of standard facts about Jacobians, formal group laws and the Cartier operator. No numerical free parameters are fitted. The only ad-hoc choices are normalizations of bases and the restriction to plane curves of degree d with distinct intersections on the y-axis.

assumptions (4)
  • standard math The formal completion of the Jacobian of a smooth (or reduced) curve carries a commutative formal group law whose invariant differentials pull back to the (dualizing) differentials on the curve (Milne, Storrs notes).
    Used in the proof of Theorem 3 to produce the maps ti and the Abel relation.
  • standard math Every formal group law over an algebraically closed field of characteristic p is strictly isomorphic to one of the form X+Y+Γ Cp(X,Y) mod degree p+1 (Hazewinkel Prop. 20.1.7).
    Invoked as Theorem 5; controls the first non-additive terms that appear in Abel relations.
  • domain assumption The Cartier operator on differentials of a curve coincides with the action of the matrix Γ on the invariant differentials of the formal Jacobian.
    Proposition 1; links the formal-group coefficients to the classical Cartier-Manin matrix of the curve.
  • ad hoc to paper When p>d+2 the truncated Abel equations of total degree ≤d+2 already force the power-series branches to satisfy the algebraic relations of a degree-d plane curve.
    Theorem 4; verified by direct elimination for d=4 and claimed for general d by the same pattern.

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Pith. "Pith review of Formal Abel relations for curves in characteristic $p$." pith.science (2026). https://pith.science/paper/67XD263B

@misc{pith2026260709471,
  author       = {Pith},
  title        = {Pith review of: Formal Abel relations for curves in characteristic $p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67XD263B}},
  note         = {Machine review of arXiv:2607.09471}
}
abstract

In this working paper, we report recent work studying the form of Abel relations in the (generalized) Jacobians of reduced plane curves over an algebraically closed field of characteristic $p$ from a formal power series point of view. The ultimate goal (to be addressed in a subsequent paper) is to complete the work done in the author's PhD thesis from 1980 and to establish a general characteristic $p$ form of the converse of Abel's theorem discussed by Griffths and used in the Lie-Wirtinger theorem on double translation manifolds and the geometry of webs.

Figures

Figures reproduced from arXiv: 2607.09471 by the authors.

Figure 1
Figure 1. The Newton polygon of this ordinary curve. the corresponding logarithmic forms. The corresponding Ekedahl-Oort type (see [3]) would be the vector (1, 2, 3), with f = 3 and a = 0. Using the ideas from the proof of Theorem 3—in particular the iso￾morphisms (f P ) ∗ : H 0 (J(C), Ω 1 (J(C))) −→ H 0 (C, Ω 1 (C)), and the “monomial” basis dx ∂f/∂y , xdx ∂f/∂y , ydx ∂f/∂y for H0 (C, Ω 1 (C)), we take the local expansions t… view at source ↗
Figure 2
Figure 2. The Newton polygon for this superspecial curve. point counts are N1 = 8, N2 = 92, and N3 = 344. Using the formulas above, the zeta function is 343t 6 + 147t 4 + 21t 2 + 1 (1 − t)(1 − 7t) . The Newton polygon comes from the points (0, 0),(2, 1),(4, 2),(6, 3) since the zero coefficients of the odd powers of t can be visualized as corresponding to points (1, ∞),(3, ∞),(5,∞) which lie above the lines connecting the poin… view at source ↗
Figure 3
Figure 3. The Newton polygon of this curve with f = 2. with the comments above about logarithmic and exact forms on this curve. The Ekedahl-Oort type of the Jacobian is (1, 2, 2) in this case. Finally, using [17] again, we find that the curve 0 = f(x, y) =y 4 + (2x + 3)y 3 + 2x 2 y 2 + (x 3 + 4x 2 + 3x + 1)y + 4x 4 + 5x 3 + x 2 + 2x + 3 has Cartier-Manin matrix with respect to the basis (19) C =   5 6 6 3 5 5 6 3 3   , wh… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The Newton polygon of this curve with f = 1. the statements above about exact and logarithmic forms on this curve. The corresponding Ekedahl-Oort type is (1, 1, 1). Finally, although we have been focusing on the case of smooth quar￾tic curves, for completeness we discu…

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