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Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra

This paper classifies all infinitesimal commutative unipotent group schemes of order $p^n$ with a one-dimensional Lie algebra over an algebraically closed field of characteristic $p>0$ , explicitly describing them to characterize the subgroup schemes of supersingular elliptic curves and resolve a question posed by Brion regarding rational actions on curves.

Original authors: Bianca Gouthier

Published 2026-05-18

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Bianca Gouthier

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 master architect working in a very strange, magical world called Characteristic $p$ . In this world, the usual rules of geometry and algebra behave a bit differently than in our everyday reality.

The paper you are asking about is like a catalog of tiny, invisible building blocks. These blocks are called "infinitesimal commutative unipotent group schemes." That is a mouthful, so let's break it down using some simple metaphors:

Infinitesimal: These are not big, visible structures. They are so small they are almost invisible, like a single pixel that has been zoomed in infinitely. They exist only in this specific magical world (positive characteristic).
Unipotent: Think of these as blocks that only move in one direction, like a train that can only go forward and never backward or sideways. They are "one-way" structures.
Commutative: This means the order in which you combine them doesn't matter. If you stack Block A on top of Block B, it looks the same as stacking Block B on top of Block A.
Lie Algebra (One-dimensional): Every structure has a "core" or a "seed" that defines its shape. In this paper, the author is only interested in structures whose seed is a single, simple line.

The Big Discovery: Counting the Shapes

The main question the author, Bianca Gouthier, answers is: "If I want to build a structure of a specific size (specifically, size $p^n$ ) using these tiny, one-way, one-dimensional blocks, how many different unique shapes can I make?"

Before this paper, mathematicians knew the answer for very small sizes, but it was a mystery for larger sizes.

The Answer:
Gouthier proves that for any size $p^n$ , there are exactly $n$ different unique shapes.

She doesn't just count them; she gives you the blueprints for all of them. She describes them using a special mathematical tool called Witt vectors. You can think of Witt vectors as a very complex, magical Lego set.

Some of these shapes are simple, like a straight line of Legos (these are the ones that fit inside a "smooth" curve, like a circle or a line).
Others are more twisted and complex. These are the "weird" shapes that don't fit inside any smooth, simple curve.

The "Supersingular" Connection

The paper then takes these blueprints and applies them to a famous object in mathematics: the Supersingular Elliptic Curve.

Think of an elliptic curve as a fancy, twisted donut shape. In this magical world, some of these donuts are "supersingular," meaning they have special properties.

Gouthier uses her new catalog to list every single tiny piece (subgroup) that can be found inside these supersingular donuts.
She shows that the "twisted" shapes she discovered earlier are exactly the pieces that make up the "torsion" (the repeating patterns) of these special donuts.
This clears up a lot of confusion mathematicians had about how these donuts are built from the inside out.

The "Rational Action" Question

There was a question from another mathematician named Michel Brion: "Can these weird, tiny, invisible blocks spin or move around a curve in a way that feels natural, even if they aren't part of a smooth, simple shape?"

Gouthier's answer is yes.

She shows that most of the shapes in her catalog (the ones that aren't the simple straight lines or the supersingular donut pieces) can indeed "dance" on any curve.
They move in a way that is "generically free," which is a fancy way of saying they move around without getting stuck or overlapping in a messy way, even though they are invisible and tiny.
This proves that these strange, complex shapes are useful and active players in the mathematical world, not just theoretical curiosities.

The Non-Commutative Twist

Finally, the paper addresses a side question: "Are all these tiny blocks 'commutative' (order doesn't matter)?"

The answer is no.

Gouthier builds an example of a tiny block where the order does matter. If you stack A on B, it looks different than B on A.
This shows that the world of these tiny structures is even more diverse than previously thought, containing both "polite" (commutative) and "rude" (non-commutative) members.

Summary

In short, this paper is a comprehensive inventory and instruction manual for a specific family of tiny, invisible mathematical shapes.

It counts them: There are exactly $n$ types for a size of $p^n$ .
It draws them: It provides the exact formulas to build each one.
It connects them: It shows how these shapes are the hidden building blocks of supersingular elliptic curves.
It puts them to work: It proves these shapes can move freely on curves, solving a long-standing puzzle.

The author essentially turned a foggy, confusing landscape of tiny mathematical objects into a clear, organized map with a complete list of every possible shape.

Technical Summary: Infinitesimal Commutative Unipotent Group Schemes with One-Dimensional Lie Algebra

Problem Statement
The paper addresses the difficulty of explicitly describing infinitesimal commutative unipotent group schemes over an algebraically closed field $k$ of characteristic $p > 0$ . While such schemes are known to exist (e.g., as $p$ -torsion of abelian varieties with $p$ -rank zero), their explicit structure in terms of Hopf algebras is often opaque, particularly as dimension increases. The specific problem tackled is the classification of infinitesimal commutative unipotent $k$ -group schemes of order $p^n$ with a one-dimensional Lie algebra. This classification is motivated by a partial question raised by Fakhruddin regarding the existence of such schemes and a question by Brion concerning rational actions of these schemes on curves.

Methodology
The author employs the theory of Dieudonné modules over the ring of Witt vectors $W(k)$ to translate the geometric problem into an algebraic one involving modules over the non-commutative ring $E$ (generated by Frobenius $F$ and Verschiebung $V$ ).

Reduction to Dieudonné Modules: Using the anti-equivalence of categories between infinitesimal commutative unipotent group schemes and finite-length left $E$ -modules (where $F$ and $V$ are nilpotent), the problem is reduced to classifying specific $E$ -modules of length $n$ .
Inductive Classification: The proof proceeds by induction on $n$ . For a group scheme $G$ of order $p^n$ , the author analyzes the short exact sequence induced by the $(n-1)$ -th Frobenius morphism:
$0 \to \ker(F_G^{n-1}) \to G \to \alpha_p \to 0$
By the inductive hypothesis, the kernel is classified. The Dieudonné module $M(G)$ is then analyzed as an extension of the module corresponding to $\alpha_p$ .
Algebraic Simplification: A crucial step involves proving that certain parameters in the defining relations of the Dieudonné module can be normalized. Specifically, Proposition 4.3 demonstrates that for an algebraically closed field, terms involving coefficients in $W(k)$ can be eliminated via isomorphism, reducing the classification to a discrete set of parameters.
Explicit Construction: The paper explicitly constructs the Hopf algebras for the classified schemes, showing they arise as kernels of morphisms of the form $V - F^i$ on Witt vector group schemes $W_n$ .

Key Contributions and Results

Classification Theorem (Theorem 1.1): The paper proves that over an algebraically closed field $k$ of characteristic $p > 0$ , there are exactly $n$ isomorphism classes of infinitesimal commutative unipotent $k$ -group schemes of order $p^n$ with a one-dimensional Lie algebra.
These schemes are explicitly described as:
$W_n^n[V - F^i] := \ker(V - F^i : W_n^n \to W_n^n)$
for $i = 1, \dots, n$ . Here, $W_n^n$ denotes the group scheme of Witt vectors of length $\le n$ , and $F, V$ are the Frobenius and Verschiebung morphisms.
Cartier Duality (Corollary 4.5): The paper provides a parallel classification for schemes whose Cartier duals have a one-dimensional Lie algebra. These are given by $W_n^n[F - V^i]$ for $i = 1, \dots, n$ .
Subgroup Schemes of Supersingular Elliptic Curves (Proposition 5.1): The classification is applied to supersingular elliptic curves $E$ . The paper establishes that:
- The $p^n$ -torsion subgroup $E[p^n]$ is isomorphic to $W_{2n}^{2n}[V - F]$ .
- The $m$ -th Frobenius kernel $E[F^m]$ is isomorphic to $W_m^m[F - V]$ .
- Consequently, the infinitesimal subgroup schemes of a supersingular elliptic curve are exactly the self-dual schemes in the classified list (specifically the case $i=1$ in the dual classification).
Rational Actions on Curves (Proposition 5.3): Addressing Brion's question, the paper determines which of these $n$ schemes can be embedded into a smooth connected one-dimensional algebraic group.
- Only two cases are embeddable: $G \cong \alpha_{p^n}$ (subgroup of the additive group $\mathbb{G}_a$ ) and $G \cong W_n^n[F - V]$ (subgroup of a supersingular elliptic curve).
- All other $n-2$ schemes (where $1 < i < n$ ) are not subgroup schemes of any smooth connected one-dimensional algebraic group. However, by previous work cited in the paper, they still admit rational, generically free actions on any curve.
Non-Commutative Counterexamples (Example 5.4): The paper concludes by addressing a question regarding commutativity. It provides an explicit example of a non-commutative infinitesimal unipotent group scheme with a one-dimensional Lie algebra, demonstrating that commutativity is not guaranteed even with this dimensional constraint.

Significance
The paper provides a complete and explicit enumeration of a specific class of group schemes that were previously only understood via abstract Dieudonné module theory. By giving explicit descriptions in terms of Witt vectors and Hopf algebras, the work facilitates the construction of actions of these schemes on varieties. Furthermore, it resolves specific questions regarding the embedding of these schemes into smooth curves and clarifies the structure of torsion and Frobenius kernels on supersingular elliptic curves. The results serve to distinguish between group schemes that are "geometric" in the sense of being subgroups of smooth curves and those that are purely infinitesimal in nature yet still capable of acting on curves.