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Formal Abel relations for curves in characteristic $p$

This working paper investigates the formal power series representation of Abel relations within the generalized Jacobians of reduced plane curves over an algebraically closed field of characteristic $p$ , serving as a precursor to a future study that aims to finalize the author's 1980 PhD thesis and establish a characteristic $p$ converse to Abel's theorem.

Original authors: John B. Little

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: John B. Little

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about how different paths on a map connect to each other. In the world of mathematics, there's a famous rule called Abel's Theorem. Think of it like a magical law of nature: if you have a specific shape (a curve) and you walk along it, adding up little steps you take, the total distance you "accumulate" follows a strict, predictable pattern. It's like saying, "If you walk this specific path, your steps will always add up to zero, no matter how you wiggle the path, as long as you stay on the curve."

For a long time, mathematicians knew this rule worked perfectly in a world where numbers behave nicely (what they call "characteristic 0," which is basically our normal world of real numbers). But there was a nagging question: Does this rule still hold if we switch to a different kind of number system, one where the rules of arithmetic are a bit weirder, like in a video game with a limited number system? This is called "characteristic $p$ ."

This paper, written by John B. Little, is a working investigation into that very question. It doesn't claim to have solved the whole mystery yet, but it builds a powerful new toolkit to study it.

The Detective's Toolkit: Formal Power Series

To investigate this, the author uses a special magnifying glass called formal power series. Imagine you have a curve, and you want to see what it looks like right up close, near a specific point. Instead of drawing the whole curve, you zoom in so far that the curve looks like a smooth, endless line made of tiny, tiny steps. You write down these steps as an infinite list of numbers (a series).

The paper shows that even in this weird "characteristic $p$ " world, you can still find these "Abel relations." It's like discovering that even if the rules of the game change, the hidden pattern of how the steps add up still exists. The author proves that for smooth curves (curves without sharp kinks), you can always find a way to combine these steps using a special "group law" (a mathematical rule for adding things together) so that they cancel out to zero. This is the main finding: the core idea of Abel's theorem survives in this strange new world, but it needs a new kind of "addition" to work.

The Twist: The "Addition" Rule Changes

Here is where it gets playful. In our normal world, adding things is simple: $1 + 1 = 2$ . But in this "characteristic $p$ " world, the way you add things together inside the mathematical "Jacobian" (a fancy machine that stores all the curve's secrets) is different.

The paper explains that in this world, the "addition" rule has a secret ingredient. If you try to add two numbers, sometimes the result isn't just the sum; it has extra terms that only appear when you reach a certain size (specifically, when you hit the number $p$ ). It's like a game where if you collect exactly 7 coins, the machine suddenly spits out a bonus coin you didn't expect. The author shows that this "bonus" is described by a special formula involving powers of $p$ .

The paper explicitly rules out the idea that the addition rule is always simple and straightforward in this world. It proves that for certain curves, the "addition" machine is much more complex than just $X + Y$ . It has to include these extra "bonus" terms to make the math work.

The Counter-Example: When the Rule Breaks

Now, here is the most exciting part. The author asks: "If I see these steps adding up to zero, does that guarantee I am looking at a real algebraic curve?" In the normal world, the answer is usually "Yes." But in this weird world, the paper presents a specific counter-example.

Imagine you have four lines that are all perfectly parallel. In the normal world, if you try to make them meet at a point, they would never touch. But in this specific mathematical setup, the author constructs a scenario where four "curves" (which are actually just straight lines with a tiny, weird twist added to them) seem to follow the "Abel relation" perfectly. They add up to zero just like a real curve should.

However, the paper explicitly states that these four lines do not come from a real, single algebraic curve of the expected degree. They are a "fake" solution. It's like finding a set of footprints that look exactly like a bear's, but when you check the DNA, you realize it's actually a very cleverly disguised dog. The paper demonstrates that in characteristic $p$ , there are specific situations (where tangent lines are concurrent) where you can have these "ghost" curves that mimic the behavior of real ones without actually being them. This is a crucial finding: the converse of Abel's theorem (if the steps add up, it must be a curve) is not always true in this world.

How Sure Are We?

The author is very careful about confidence.

Proven: The existence of the formal relations (the "steps adding up") for smooth curves is proven mathematically. The paper provides a rigorous proof that these relations exist and describes exactly how the "addition" machine works.
Proven: The existence of the specific counter-example (the "fake" curves in the concurrent tangent case) is proven. The author constructs specific equations and shows they satisfy the conditions but fail to be a real curve.
Suggested/To be addressed: The paper mentions that the ultimate goal is to establish a "general characteristic $p$ form of the converse of Abel's theorem." This means the author is suggesting that a modified version of the rule might still work, but that specific, complete proof is reserved for a future paper. The current work is a "working paper," meaning it's a snapshot of the investigation, not the final conclusion of the entire mystery.

The Takeaway

So, what's the big picture?

The Good News: The magical pattern of Abel's theorem survives in the weird world of characteristic $p$ . You can still find these hidden connections, but you have to use a more complex "addition" machine that includes special "bonus" terms.
The Bad News: You can't trust the pattern blindly. Just because the steps add up to zero doesn't guarantee you have a real curve. There are "impostors" in this world that look like curves but aren't.
The Future: The author has built the tools to see these things clearly, but the full story of how to distinguish the real curves from the impostors is a story for the next chapter.

It's a bit like realizing that in a new video game, the physics engine works differently. You can still jump and run, but sometimes you'll see a glitch where a character looks like a tree but isn't. The author has mapped out exactly how the physics engine works and found the glitch, but figuring out how to fix the game so the glitch never happens is the next level.

Technical Summary: Formal Abel Relations for Curves in Characteristic $p$

Problem Statement
This working paper investigates the structure of Abel relations for reduced plane curves over an algebraically closed field $k$ of characteristic $p$ . The author seeks to establish a formal power series analogue of Abel's theorem and its converse in positive characteristic. Specifically, the paper addresses how the classical relationship between the intersection of a curve with a moving line and the integration of Abelian differentials translates when the underlying field has characteristic $p$ . The ultimate goal, noted as a subject for future work, is to complete the author's 1980 PhD thesis by establishing a general characteristic $p$ form of the converse of Abel's theorem (as discussed by Griffiths), which relates addition laws on analytic curves to the Jacobian varieties of algebraic curves.

Methodology
The paper employs a formal power series approach to translate classical geometric statements into the local setting of formal completions.

Formal Setup: The author considers the formal completion of the projective plane $\mathbb{P}^2$ along a line $\Delta_0$ (the $y$ -axis). An algebraic curve $C$ of degree $d$ is locally represented by $d$ formal curves $C_i$ defined by $y = f_i(x) \in k[[x]]$ . Lines near $\Delta_0$ are parametrized as $\Delta_{u,v}: x = uy + v$ .
Formal Group Laws: Instead of standard integration (which is problematic in characteristic $p$ ), the "sum" of intersection points is defined via the formal group law $G$ of the (generalized) Jacobian $J(C)$ . The Abel relation is expressed as $\bigoplus_G t_i(x_i(u,v)) = 0$ , where $t_i$ are maps from the formal curves to the formal group.
Coefficient Analysis: The paper analyzes the coefficients of the power series expansions of the intersection points $x_i(u,v)$ and the maps $t_i$ . It utilizes the Lagrange reversion theorem and combinatorial properties of partitions to describe the structure of these coefficients.
Characteristic $p$ Specifics: The analysis explicitly incorporates the structure of formal group laws in characteristic $p$ , particularly the role of the Cartier operator and the $p$ -typical terms (involving $C_p(X,Y)$ ) that appear in the group law at degree $p$ .
Case Studies: The author provides explicit computations for smooth plane quartics ( $d=4, g=3$ ) over $\mathbb{F}_7$ , classifying them by their Cartier-Manin matrices, $p$ -ranks, and $a$ -numbers (ordinary, superspecial, and intermediate cases).

Key Contributions and Results

Formal Abel Theorem (Theorem 3): The paper establishes that for a smooth curve $C$ (or a reduced curve with a generalized Jacobian), there exist maps $t_i: C_i \to G$ such that the formal group sum of the images of the intersection points with a moving line is identically zero. The maps $t_i$ are local expansions of the canonical mapping to the Jacobian, and their differentials correspond to a basis of invariant differentials on the Jacobian.
Explicit Coefficient Relations (Theorem 4): For a sufficiently large prime $p > d+2$ , the paper proves that the "first part" of a maximal Abel relation (up to degree $d+2$ ) uniquely determines the local expansions of an algebraic curve of degree $d$ . Specifically, the coefficients of the formal curves $f_i(x)$ must satisfy specific polynomial relations (generalized Reiss relations) to admit an Abel relation.
Role of the Cartier Operator (Theorem 5 & Proposition 1): The paper clarifies the structure of the formal group law in characteristic $p$ . It demonstrates that the matrix $\Gamma$ appearing in the degree $p$ term of the group law ( $G(X,Y) \equiv X + Y + \Gamma C_p(X,Y)$ ) corresponds to the action of the Cartier operator on the invariant differentials of the Jacobian. This links the formal group structure directly to the Cartier-Manin matrix of the curve.
Non-Uniqueness of Mappings (Section 7): A significant finding is the non-uniqueness of the mappings $t_i$ in characteristic $p$ . Due to the Frobenius action, if $t_i$ defines an Abel relation, then a modified mapping $s_i$ (derived from the $p$ -th or $q$ -th power of the coefficients) also defines a valid Abel relation. This introduces a level of non-uniqueness absent in characteristic 0.
Counterexamples to the Converse (Theorem 7): The paper presents isolated counterexamples to the converse of Abel's theorem in characteristic $p$ . It constructs deformed formal curves (specifically $y = a_{i,0} + x^q$ ) that satisfy the formal Abel relation with the additive formal group law but do not arise from any algebraic curve of degree $d$ . This occurs when the tangent lines at the intersection points are concurrent (parallel in the affine chart).

Significance and Scope
The paper serves as a technical bridge between classical complex geometry (Abel's theorem, Lie-Wirtinger theorem) and arithmetic geometry in positive characteristic. It highlights that while the formal structure of Abel relations persists in characteristic $p$ , the underlying mechanisms differ significantly due to the behavior of the Cartier operator and the specific structure of formal group laws (e.g., the appearance of $p$ -typical terms).

The author explicitly states that this is a "working paper" intended to prepare for a subsequent paper that will address the general converse of Abel's theorem. The current work focuses on establishing the local formal statements, analyzing the explicit form of the relations for general $d$ , and identifying the specific constraints and anomalies (such as the counterexamples) that arise in characteristic $p$ . The results are framed as necessary steps toward a complete characteristic $p$ theory of double translation manifolds and web geometry, but the paper does not claim to have fully resolved the general converse theorem itself.

The Detective's Toolkit: Formal Power Series

The Twist: The "Addition" Rule Changes

The Counter-Example: When the Rule Breaks

How Sure Are We?

The Takeaway

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