Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra
Pith reviewed 2026-05-23 21:03 UTC · model grok-4.3
The pith
Over algebraically closed fields of characteristic p there are exactly n isomorphism classes of infinitesimal commutative unipotent group schemes of order p^n with one-dimensional Lie algebra.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Over an algebraically closed field k of characteristic p>0 there are exactly n infinitesimal commutative unipotent k-group schemes of order p^n with one-dimensional Lie algebra, up to isomorphism, and the paper gives explicit descriptions of them. This yields explicit descriptions of all infinitesimal subgroup schemes of supersingular elliptic curves and answers a question of Brion on rational actions of these group schemes on curves.
What carries the argument
The enumeration of the n isomorphism classes of infinitesimal commutative unipotent group schemes of order p^n with one-dimensional Lie algebra.
If this is right
- All infinitesimal subgroup schemes of any supersingular elliptic curve over an algebraically closed field receive explicit descriptions.
- The p^n-torsions of supersingular elliptic curves are recovered explicitly.
- A question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves is answered.
Where Pith is reading between the lines
- The explicit forms may support further study of group scheme actions on varieties in positive characteristic.
- Similar enumeration results could be sought for group schemes with Lie algebras of dimension greater than one.
Load-bearing premise
The base field must be algebraically closed of characteristic p>0 and the group schemes must be infinitesimal, commutative, unipotent with one-dimensional Lie algebra.
What would settle it
Exhibiting more than n or fewer than n non-isomorphic examples of infinitesimal commutative unipotent group schemes of order p^n with one-dimensional Lie algebra over an algebraically closed field of characteristic p would disprove the classification.
read the original abstract
We prove that over an algebraically closed field of characteristic $p>0$ there are exactly, up to isomorphism, $n$ infinitesimal commutative unipotent $k$-group schemes of order $p^n$ with one-dimensional Lie algebra, and we explicitly describe them. We consequently obtain an explicit description of all infinitesimal subgroup schemes of any supersingular elliptic curve over an algebraically closed field, recovering all their $p^n$-torsions as well. Finally, we use these results to answer a question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that over an algebraically closed field k of characteristic p>0 there exist exactly n isomorphism classes of infinitesimal commutative unipotent k-group schemes of order p^n with one-dimensional Lie algebra, and supplies explicit descriptions of all of them. As corollaries it gives explicit descriptions of all infinitesimal subgroup schemes of any supersingular elliptic curve (including their p^n-torsion) and answers a question of Brion concerning rational actions of such group schemes on curves.
Significance. The result supplies a complete, explicit classification in a concrete but previously unsettled case of commutative unipotent group schemes in positive characteristic. The explicit models immediately yield concrete descriptions of p-power torsion on supersingular elliptic curves and resolve an open question on curve actions; both are useful for further work in arithmetic geometry. The manuscript ships a self-contained classification together with the two applications, which strengthens its utility.
minor comments (2)
- The introduction would benefit from a short paragraph recalling the standard dictionary between infinitesimal group schemes and their Dieudonné modules or Hopf algebras, to make the explicit descriptions immediately usable by readers outside the immediate subfield.
- Notation for the n distinct group schemes (e.g., G_{n,i} or similar) should be introduced once in §2 or §3 and used consistently in the statements of the main theorem and the corollaries on elliptic curves.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, the clear summary of its contributions, and the recommendation to accept.
Circularity Check
No significant circularity identified
full rationale
The paper states a direct classification theorem: exactly n isomorphism classes of infinitesimal commutative unipotent k-group schemes of order p^n with 1-dimensional Lie algebra over algebraically closed fields of char p>0, with explicit descriptions. No equations, parameters, or uniqueness claims reduce to fitted inputs, self-definitions, or self-citation chains. The argument relies on standard algebraic geometry tools for group schemes without load-bearing self-references or ansatzes smuggled via prior work. The derivation is self-contained against external benchmarks in the field.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard properties of infinitesimal commutative unipotent group schemes and their Lie algebras over algebraically closed fields of char p hold.
Reference graph
Works this paper leans on
-
[1]
Actions of finite group schemes on curves, 2022
Michel Brion. Actions of finite group schemes on curves, 2022
work page 2022
-
[2]
Michel Demazure and Pierre Gabriel. Groupes alg\' e briques. T ome I : G \' e om\' e trie alg\' e brique, g\' e n\' e ralit\' e s, groupes commutatifs. Masson & Cie, \' E diteurs, Paris and North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice Corps de classes local par Michiel Hazewinkel
work page 1970
-
[3]
Finite group schemes of essential dimension one
Najmuddin Fakhruddin. Finite group schemes of essential dimension one. Doc. Math. , 25:55--64, 2020
work page 2020
-
[4]
Infinitesimal rational actions, 2023
Bianca Gouthier. Infinitesimal rational actions, 2023
work page 2023
-
[5]
Unexpected subgroup schemes of PGL_ 2,k in characteristic 2, 2024
Bianca Gouthier and Dajano Tossici. Unexpected subgroup schemes of PGL_ 2,k in characteristic 2, 2024
work page 2024
-
[6]
J. S. Milne. Algebraic groups , volume 170 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field
work page 2017
-
[7]
Susan Montgomery. Hopf algebras and their actions on rings , volume 82 of CBMS Regional Conference Series in Mathematics . Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993
work page 1993
-
[8]
Abelian varieties , volume 5 of Tata Institute of Fundamental Research Studies in Mathematics
David Mumford. Abelian varieties , volume 5 of Tata Institute of Fundamental Research Studies in Mathematics . Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition
work page 2008
-
[9]
Nguyen, Linhong Wang, and Xingting Wang
Van C. Nguyen, Linhong Wang, and Xingting Wang. Classification of connected H opf algebras of dimension p^3 I . J. Algebra , 424:473--505, 2015
work page 2015
-
[10]
F. Oort. Commutative group schemes. Springer-Verlag, Berlin-New York, 1966
work page 1966
-
[11]
Richard Pink. Finite group schemes, 2005. Lecture course in WS 2004/05, ETH Z\"urich
work page 2005
-
[12]
A short guide to p -torsion of abelian varieties in characteristic p
Rachel Pries. A short guide to p -torsion of abelian varieties in characteristic p . In Computational arithmetic geometry , volume 463 of Contemp. Math. , pages 121--129. Amer. Math. Soc., Providence, RI, 2008
work page 2008
- [13]
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.