REVIEW 3 major objections 5 minor 26 references
Large automorphism groups of ordinary curves of even genus in odd characteristic
T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Every even-genus ordinary curve has fewer than $821.37g^{7/4}$ automorphisms
desk verdict New bound with a heavy but explicit caveat: Theorem 3.10 lives or dies with the quoted classification from [4]. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the classification quoted in Lemma 3.4: an odd core-free non-solvable automorphism group of a curve of even genus in odd characteristic has a simple minimal normal subgroup $N$ of one of five types, $\operatorname{PSL}(2,q)$, $\operatorname{PSL}(3,q)$, $\operatorname{PSU}(3,q)$, $\operatorname{Alt}_7$, or $M_{11}$, and $G$ is contained in the corresponding projective semilinear automorphism group. Around this classification the proof uses two quantitative mechanisms. First, ordinary curves inherit Nakajima's bound $|\operatorname{Aut}(\mathcal{X})|\le 84g(g-1)$ and the vanishing of second ramification groups $G_P^{(2)}=1$ at every point $P$. Second, a structural lemma for large solvable groups, Lemma 3.8 and Proposition 3.9, shows that such a group is a semidirect product $Q\rtimes U$ with rational quotient curve, exactly two non-tame short orbits, and a cyclic complement of order $|U|<\sqrt{4g+4}$. The Hurwitz genus formula and the Deuring--Shafarevich formula convert these structural facts into the explicit inequalities that bound $q$, and hence $|G|$, in terms of $g$.
What would settle it
Run the ramification-data check of Theorem 3.13 for every group in Lemma 3.4, or equivalently search computationally for an ordinary curve of even genus over an algebraically closed field of odd characteristic whose automorphism group has order at least $821.37g^{7/4}$. A concrete target is the plane model $y^{10}(y+1)^9 = x^{22} - y(y+1)^4x^{11}(y^3+2y+1)$ in characteristic $3$: if it fails to be ordinary of genus $26$ with automorphism group $M_{11}$, the exceptional case would be empty; if a curve with $|G|\ge821.37g^{7/4}$ exists, the main theorem would be false.
Extended reading notes
Core claim
The central claim, Theorem 3.10, is that for an ordinary curve $\mathcal{X}$ of even genus $g\ge 2$ over an algebraically closed field of odd characteristic, every automorphism group $G$ has order below $821.37g^{7/4}$. The authors split the problem by group structure. A solvable group, or one with an elementary abelian minimal normal subgroup, already satisfies the stronger bound $|G|\le 34(g+1)^{3/2}$ by Theorem 3.1. The remaining odd core-free non-solvable groups are forced by the quoted classification to contain a simple minimal normal subgroup $N$ isomorphic to $\operatorname{PSL}(2,q)$, $\operatorname{PSL}(3,q)$, $\operatorname{PSU}(3,q)$, $\operatorname{Alt}_7$, or $M_{11}$, with $G$ lying inside the corresponding automorphism group. For the projective families the proof bounds $q$ in terms of $g$ using the Hurwitz and Deuring--Shafarevich formulas together with a structural lemma for large solvable subgroups, and obtains $|G|<821.37g^{7/4}$. The sporadic cases are eliminated by enumerating all ramification data that could violate $|G|<84(g-1)$; the only surviving data are $p=3$, $g=26$, $G\cong M_{11}$, and the paper exhibits $X(11)$ in characteristic $3$ as an ordinary curve of genus $26$ with this automorphism group.
Load-bearing premise
The proof assumes that the classification quoted as Lemma 3.4 is complete: every non-solvable automorphism group of a curve of even genus in odd characteristic, once reduced by its largest normal subgroup of odd order, has a simple minimal normal subgroup of one of the five listed types. If a group type is missing from that list, the case analysis could miss a curve that violates the bound.
Editorial extensions
If this is right
- For every ordinary curve of even genus $g\ge2$ over an algebraically closed field of odd characteristic, every automorphism group has order below $821.37\,g^{7/4}$.
- The Hurwitz bound $|G|<84(g-1)$ is restored for $G\cong \operatorname{Alt}_7$ and $G\cong M_{11}$, with exactly one exception: $p=3$, $g=26$, $G\cong M_{11}$.
- There is no ordinary curve of genus $10$ in characteristic $5$ with automorphism group $\operatorname{Alt}_7$, eliminating the only candidate that the ramification data allow.
- In the exceptional $M_{11}$ case, the quotient of the curve by its Sylow $3$-subgroup is the ordinary genus-$2$ hyperelliptic curve $y^2=x^5-x$.
- The exceptional case is nonempty: the modular curve $X(11)$ in characteristic $3$ is ordinary of genus $26$ and has automorphism group $M_{11}$.
Reading between the lines
- Beyond the paper, the same two-mechanism proof could be run with sharper constants: the $821.37$ arises from worst-case estimates, and an exhaustive search over the allowed ramification data for small $g$ would show how far the true maximum lies below the bound.
- Beyond the paper, the dichotomy between solvable and non-solvable cases is tied to Sylow $2$-subgroups of even-genus curves; for odd genus the same classification input would not be available, so a similar bound would need a new structural statement before the conjectured $O(g^{8/5})$ growth could be tested.
- Beyond the paper, the authors leave open whether $X(11)$ is the only ordinary genus-$26$ curve in characteristic $3$ with automorphism group $M_{11}$; a computation over plane models of that genus and characteristic could settle the uniqueness question.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies automorphism groups G of ordinary projective curves X of even genus g ≥ 2 over an algebraically closed field K of odd characteristic p. Its main result, Theorem 3.10, asserts that every such G satisfies |G| < 821.37·g^{7/4}, and that for the two sporadic possibilities G ≅ Alt_7 and G ≅ M_11 the classical Hurwitz bound |G| < 84(g−1) holds unless p = 3, g = 26, and G ≅ M_11. The proof splits the problem into solvable and non-solvable cases: solvable groups are handled by earlier work of Korchmáros–Montanucci (Theorem 3.1), while non-solvable odd core-free groups are classified by Lemma 3.4, quoted from the preprint [4], into cases with minimal normal subgroup PSL(2,q), PSL(3,q), PSU(3,q), Alt_7, or M_11. The PSL(2,q), PSL(3,q), and PSU(3,q) cases are treated in Propositions 3.11 and 3.12 using subgroup bounds and Lemma 3.3. The Alt_7 and M_11 cases are treated by a ramification analysis in Propositions 3.14 and 3.18, with a non-existence proof for the possible genus-10 Alt_7 curve (Propositions 3.15 and 3.16). Section 4 gives the modular curve X(11) in characteristic 3 as an ordinary genus-26 curve with M_11 automorphism group.
Significance. If the quoted classification is accepted, this is a strong and quantitatively explicit result: it improves Nakajima's general bound from O(g^2) to O(g^{7/4}) for ordinary curves of even genus in odd characteristic, and it gives a complete exceptional analysis for the two sporadic groups, including a concrete ordinary curve attaining the exceptional case. The case analysis in Propositions 3.9, 3.11, 3.12, 3.14, and 3.18 is detailed and largely self-contained beyond the external classification; the use of Deuring–Shafarevich and Hurwitz formulas is careful and appropriate. The paper also makes good use of structural restrictions imposed by ordinary even genus, such as Theorem 2.2, to narrow the possible ramification filtrations. The main weaknesses are the unproved completeness of Lemma 3.4, which is load-bearing, and several essential computer checks that are neither documented by code nor replaced by human-checkable proofs.
major comments (3)
- [§3, Lemma 3.4 and Remark 3.6] The proof of Theorem 3.10 is logically dependent on the completeness of the classification stated in Lemma 3.4 and summarized in Remark 3.6, which is quoted from the preprint [4]. This lemma asserts that every odd core-free non-solvable automorphism group of an even-genus curve in odd characteristic has a simple minimal normal subgroup among PSL(2,q), PSL(3,q), PSU(3,q), Alt_7, and M_11, and that G is contained in the corresponding PΓL-group. Every subsequent proposition begins with an 'according to Remark 3.5' assumption, so if the classification is incomplete or misstated, Theorem 3.10 has no support. Since [4] is an arXiv preprint, the completeness part is not verified by a peer-reviewed source. Please either cite a published version of [4], or include a proof of Lemma 3.4, or make explicit in Theorem 3.10 that the result is conditional on this classification; the current presentation gives no independent verification of the most load-bearing input.
- [§3, Propositions 3.14 and 3.15; §4] Several essential steps rely on MAGMA computations for which no code, certificates, or reproducible scripts are provided. Specifically: Proposition 3.14 uses 'it can be verified by MAGMA that the largest solvable subgroup of Alt_7 whose Sylow p-subgroup (p ∈ {3,5,7}) has a cyclic complement has size 36'; Proposition 3.15 uses the claim that for a ≠ −1 the curve Z_a is isomorphic to y^2 = a_5 x^5 + a_3 x^3 + a_0 with 5-rank 2; and Section 4 uses 'by direct MAGMA computation' the equality γ(X(11)) = g(X(11)) = 26. These facts are load-bearing: the first excludes the Alt_7 Hurwitz-bound violations, the second is needed to rule out the genus-10 Alt_7 case, and the third establishes that the constructed exceptional example is ordinary. The authors should supply the MAGMA scripts and outputs (or a precise algorithmic description sufficient for independent verification), or replace these computations by mathematical proofs.
- [§3, Proposition 3.12, PSL(3,q) case] In the PSL(3,q) case, the argument excludes the possibility |H| ≥ 30(g−1) by using Lemma 3.3 to conclude d = p and then Theorem 3.2 to obtain that the Sylow p-subgroup of H is elementary abelian, contradicting the structure of a Sylow p-subgroup of PSL(3,q) for the relevant q. This step is sound, but the presentation compresses the reason why the normalizer subgroup H is a solvable subgroup of the automorphism group of an ordinary curve to which Theorem 3.2 applies. Since Theorem 3.2 is stated for X ordinary and H solvable, and H is indeed a subgroup of Aut(X), the step is valid; however, the reader is left to fill in the verification that the normalizer N_{PSL(3,q)}(Q) described by [11, Theorem 2.4] is the subgroup H used in Lemma 3.3. The authors should spell out this identification, preferably with a sentence or a short lemma, so that the application of Lemma 3.3 is fully transparent.
minor comments (5)
- [§3, Proposition 3.11, Case 2] In Case 2 of Proposition 3.11, the text refers to 'Remark 3.9'; the intended reference is Proposition 3.9, where the bound |U| < sqrt(4g+4) is proved.
- [§3, Propositions 3.14 and 3.18] The cross-references 'Equation (3.14)' and 'Equation (3.18)' should be to the displayed Hurwitz formulas numbered (20) and (23), respectively; the current numbering is confusing.
- [§3, Proposition 3.11, Case 4] In Case 4 of Proposition 3.11, the expression 'PGL(2, 53) = 5859000' appears to be a typesetting or OCR error for 'PGL(2, 5^3)' (i.e., q = 125), since |PGL(2,125)| = 1,953,000 and three times that is 5,859,000; the printed 'q ≥ 53' likewise should read 'q ≥ 5^3'. Please correct this.
- [Throughout] There are several minor typos and stylistic inconsistencies that should be fixed in a revision: 'wich' for 'which' in Propositions 3.11 and 3.12; 'pagg 600-601' in the reference to [19]; and the notation 'G(2)_P' in the proof of Proposition 3.9 is used without definition (it presumably denotes the second ramification group of the stabilizer G_P).
- [References] Reference [4] is cited only as an arXiv preprint. If a peer-reviewed or final version of this classification paper has appeared, the authors should update the citation; if not, this strengthens the need for the major comment above.
Circularity Check
No significant circularity: the main bound is derived from an external classification and prior parameter-free bounds, not assumed as an input.
full rationale
The paper's central derivation chain does not reduce to its own inputs. The target bound |G| < 821.37 g^{7/4} is not assumed anywhere; it is obtained by excluding solvable and elementary-abelian cases through Theorem 3.1 from [12], a parameter-free bound with stated assumptions, and then analyzing non-solvable odd core-free groups according to Lemma 3.4 from [4]. Lemma 3.4 is an external classification by Giulietti and Korchmáros, not by the present authors, and it lists possible minimal normal subgroups without assuming the paper's bound. The subsequent case analyses in Propositions 3.11, 3.12, 3.14, and 3.18 use the Hurwitz genus formula, Deuring–Shafarevich, Nakajima's theorems, and elementary group-order estimates; constants such as 821.37 arise from crude inequalities, not from fitting. The Alt7 and M11 refinements are derived by case checking against Hurwitz data, and the exceptional p=3, g=26 case is supported by a MAGMA computation on a defining equation. Although the paper cites two works coauthored by one of its authors ([12] and [15]), those results are independent, parameter-free, and do not incorporate the g^{7/4} bound, so they do not create circularity. The only substantive external dependency is the completeness of Lemma 3.4 quoted from the arXiv preprint [4]; if that classification were incomplete, the proof would miss cases, but that is a correctness risk about an external premise, not a circular derivation.
Assumptions & free parameters
assumptions (7)
- domain assumption For an odd core-free automorphism group G of a curve of even genus in odd characteristic with a non-abelian minimal normal subgroup N, N is simple and belongs to the list PSL(2,q), PSL(3,q), PSU(3,q), Alt7, or M11 (Lemma 3.4 from [4]).
- standard math For an ordinary curve, the second ramification group at every point is trivial (Nakajima's Theorem 2.2).
- standard math The curve c T f(X) - T^{d+1} - 1 - T h(T) = 0 is irreducible and has genus 1/2(q-1)(d+1) (Hirschfeld, Korchmaros, Torres, Remark 12.12).
- ad hoc to paper The largest solvable subgroup of Alt7 whose Sylow p-subgroup (p in {3,5,7}) has a cyclic complement has order 36.
- ad hoc to paper For a not equal to -1, the curve Z_a is isomorphic to y^2 = a5 x^5 + a3 x^3 + a0 with 5-rank 2, and Z_{-1} has 5-rank 0.
- domain assumption The modular curve X(11) in characteristic 3 is an ordinary curve of genus 26 with automorphism group M11.
- standard math There is, up to isomorphism, a unique genus 2 curve over characteristic different from 2 with the relevant automorphism group C4 or SD16, as described in [22, Section 3.2].
Cite this review
Pith. "Pith review of Large automorphism groups of ordinary curves of even genus in odd characteristic." pith.science (2026). https://pith.science/paper/OW4DNWLV
@misc{pith2026190804684,
author = {Pith},
title = {Pith review of: Large automorphism groups of ordinary curves of even genus in odd characteristic},
year = {2026},
howpublished = {\url{https://pith.science/paper/OW4DNWLV}},
note = {Machine review of arXiv:1908.04684}
}
abstract
Let $\mathcal{X}$ be a (projective, non-singular, geometrically irreducible) curve of even genus $g(\mathcal{X}) \geq 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. If the $p$-rank $\gamma(\mathcal{X})$ equals $g(\mathcal{X})$, then $\mathcal{X}$ is \emph{ordinary}. In this paper, we deal with \emph{large} automorphism groups $G$ of ordinary curves of even genus. We prove that $|G| < 821.37g(\mathcal{X})^{7/4}$. The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic, see \cite{giulietti-korchmaros-2017}. According to this classification, for the exceptional cases ${\rm Aut}(\mathcal{X}) \cong {\rm Alt}_7$ and ${\rm Aut}(\mathcal{X}) \cong \rm{M}_{11}$ we show that the classical Hurwitz bound $|{\rm Aut}(\mathcal{X})| < 84(g(\mathcal{X})-1)$ holds, unless $p=3$, $g(\mathcal{X})=26$ and ${\rm Aut}(\mathcal{X}) \cong \rm{M}_{11}$; an example for the latter case being given by the modular curve $X(11)$ in characteristic $3$.
Reference graph
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