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arxiv: 2604.17901 · v1 · submitted 2026-04-20 · 🧮 math.AG

Automorphism groups of hyperelliptic curves of 2-rank zero

Kohtaro Yamaguchi , Shushi Harashita This is my paper

Pith reviewed 2026-05-10 04:16 UTC · model grok-4.3

classification 🧮 math.AG
keywords hyperelliptic curvesautomorphism groups2-rank zerocharacteristic 2Artin-Schreier curvesreduced automorphism groupsOort conjecture
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The pith

Hyperelliptic curves of 2-rank zero in characteristic 2 have their reduced automorphism groups determined explicitly for small genera as specific semidirect products.

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

The paper studies hyperelliptic curves in characteristic 2 with 2-rank zero, written in Artin-Schreier form as y squared minus y equals a polynomial f of x. It first shows that the full automorphism group always decomposes as a semidirect product of a translation subgroup and a stabilizer subgroup for any genus. For small genera the paper then runs explicit computations in the Magma algebra system to list the concrete reduced automorphism groups that arise. These lists support two new conjectures that restrict which groups can appear, in direct parallel with the Oort conjecture for generic supersingular abelian varieties.

Core claim

For an Artin-Schreier curve y squared minus y equals f of x of 2-rank zero, the automorphism group is always a semidirect product whose structure is determined by the additive translations and the linear maps that preserve f; when the genus is small, explicit Magma enumeration yields the precise reduced groups that occur, and these results motivate two conjectures limiting the possible reduced groups for generic curves of this type.

What carries the argument

The semidirect-product decomposition of the automorphism group into the normal subgroup of translations by elements of the additive group and the complementary stabilizer of the defining polynomial f.

If this is right

  • The full automorphism group of any such curve decomposes as a semidirect product for arbitrary genus.
  • Concrete lists of possible reduced automorphism groups are obtained for all curves of small genus.
  • Only certain groups appear as reduced automorphism groups, in analogy with the Oort conjecture.
  • The conjectures predict that generic curves in this family realize only a restricted set of the possible groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the same Magma search to one or two larger genera would provide an immediate test of the two conjectures.
  • The same translation-plus-stabilizer decomposition may apply to other families of curves in positive characteristic that admit an additive equation.
  • The results supply concrete data that could be compared with existing tables of automorphism groups of curves in characteristic two.

Load-bearing premise

The Magma computations for small genera correctly enumerate every possible reduced automorphism group without omissions or software errors.

What would settle it

An explicit example of a 2-rank-zero hyperelliptic curve of genus three or four over a field of characteristic two whose reduced automorphism group is not one of the groups obtained in the computations would disprove the claimed determination.

read the original abstract

In this paper, we determine the reduced automorphism groups of hyperelliptic curves of a small genus in characteristic $2$, when they are of $2$-rank $0$. Such a curve is an Artin-Schreier curve defined in the form $y^2-y=f(x)$ for a polynomial $f(x)$. After we clarify semidirect-product structures of the automorphism groups for an arbitrary genus, we derive the detailed group structures for the reduced automorphism groups of the curves of a small genus, through computations using the computational algebra system Magma. With these experiments, we formulate two conjectures, which are analogues for our curves of the Oort conjecture on automorphism groups of generic principally polarized supersingular abelian varieties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper determines the reduced automorphism groups of hyperelliptic curves of small genus in characteristic 2 with 2-rank zero, realized as Artin-Schreier curves y² - y = f(x). It first clarifies semidirect-product structures of the full automorphism groups for arbitrary genus, then uses Magma computations to obtain explicit reduced group structures for small genera, and formulates two conjectures analogous to the Oort conjecture on automorphism groups of generic principally polarized supersingular abelian varieties.

Significance. If the enumeration is exhaustive and the structures hold, the work supplies concrete data on possible reduced automorphism groups in the 2-rank zero case, which is relevant to the geometry of superspecial curves and their Jacobians in characteristic 2. The computational approach for small genera and the formulation of Oort-type conjectures could help constrain the possible automorphism groups of generic supersingular abelian varieties, provided the results are independently verifiable.

major comments (2)
  1. [section on semidirect-product structures] The semidirect-product decomposition of the automorphism group for arbitrary genus is asserted as a preliminary clarification but without a visible general derivation or proof; this structure is load-bearing for separating the hyperelliptic involution from the reduced group and for extending the small-genus computations.
  2. [computational results for small genera] The Magma computations that enumerate all possible reduced automorphism groups for small genera lack any description of the input polynomials f(x), the procedure used to generate all degree-d polynomials yielding 2-rank zero, or the verification steps confirming that every automorphism group was found without omissions; this directly affects the completeness of the explicit lists that support the conjectures.
minor comments (2)
  1. [introduction] The abstract and introduction refer to 'small genus' without stating the precise range of genera for which computations were performed; this information should appear explicitly early in the paper.
  2. [preliminaries] Notation for the reduced automorphism group and the precise meaning of '2-rank zero' in the Artin-Schreier model could be clarified with a short preliminary subsection to aid readers unfamiliar with characteristic-2 curve theory.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight areas where additional clarity and documentation will strengthen the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [section on semidirect-product structures] The semidirect-product decomposition of the automorphism group for arbitrary genus is asserted as a preliminary clarification but without a visible general derivation or proof; this structure is load-bearing for separating the hyperelliptic involution from the reduced group and for extending the small-genus computations.

    Authors: We agree that the semidirect-product structure, while standard in the literature on hyperelliptic curves in characteristic 2, requires an explicit general derivation to be fully rigorous. The decomposition follows from the fact that the hyperelliptic involution is central in the automorphism group and commutes with the reduced automorphisms arising from the Artin-Schreier equation. In the revised manuscript we will insert a short lemma (with proof) establishing this semidirect-product decomposition for arbitrary genus before the computational sections, thereby making the separation between the full group and the reduced group transparent. revision: yes

  2. Referee: [computational results for small genera] The Magma computations that enumerate all possible reduced automorphism groups for small genera lack any description of the input polynomials f(x), the procedure used to generate all degree-d polynomials yielding 2-rank zero, or the verification steps confirming that every automorphism group was found without omissions; this directly affects the completeness of the explicit lists that support the conjectures.

    Authors: We accept that the computational methodology must be documented in detail for reproducibility and to substantiate the conjectures. The revised version will include an expanded computational section that (i) lists the explicit families of polynomials f(x) of each degree d that were tested, (ii) describes the coefficient conditions used to enforce 2-rank zero, and (iii) outlines the Magma verification procedure (exhaustive search over the automorphism group of the function field together with cross-checks against known group orders). These additions will confirm that the enumeration is exhaustive within the stated genus range. revision: yes

Circularity Check

0 steps flagged

No circularity: general semidirect-product clarification precedes independent Magma enumeration on explicit equations; conjectures are extrapolations from data.

full rationale

The paper first states a general clarification of semidirect-product structures of automorphism groups for arbitrary genus (a theoretical step independent of specific computations), then uses Magma to enumerate reduced groups on concrete Artin-Schreier equations y²-y=f(x) of 2-rank 0 for small genera. These computations operate directly on curve equations rather than fitting parameters to prior outputs or self-citations. The two conjectures are formulated from the resulting data as analogues of the Oort conjecture and do not redefine or tautologically reproduce the inputs. No load-bearing step reduces by construction to a fitted value, self-citation chain, or ansatz smuggled from the authors' prior work; the derivation chain remains self-contained against external software verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard facts about Artin-Schreier covers and 2-rank in characteristic 2 but introduces no new free parameters, invented entities, or ad-hoc axioms beyond domain conventions in algebraic geometry.

axioms (1)
  • domain assumption Hyperelliptic curves of 2-rank 0 in characteristic 2 are precisely the Artin-Schreier curves given by y^2 - y = f(x) for a suitable polynomial f.
    Directly stated in the abstract as the form used for all subsequent analysis.

pith-pipeline@v0.9.0 · 5416 in / 1347 out tokens · 48528 ms · 2026-05-10T04:16:38.450693+00:00 · methodology

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Reference graph

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