Algorithmic study of superspecial hyperelliptic curves over finite fields

Momonari Kudo; Shushi Harashita

arxiv: 1907.00894 · v1 · pith:FOGI54UOnew · submitted 2019-07-01 · 🧮 math.AG · math.NT

Algorithmic study of superspecial hyperelliptic curves over finite fields

Momonari Kudo , Shushi Harashita This is my paper

Pith reviewed 2026-05-25 11:37 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords superspecial hyperelliptic curvesenumeration algorithmfinite fieldsautomorphism groupgenus 4maximal curvesminimal curves

0 comments

The pith

An algorithm enumerates superspecial hyperelliptic curves of genus g over F_q when q and 2g+2 are coprime.

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

This paper introduces an algorithm to list all superspecial hyperelliptic curves of genus g over a finite field F_q whenever q and 2g+2 are coprime and q exceeds 2g+1. The algorithm was implemented to generate complete lists for genus 4 over F_p for primes from 11 to 23 and over F_p squared for primes from 11 to 19. From the resulting lists the authors identified maximal and minimal hyperelliptic curves over the quadratic fields. A second algorithm computes the automorphism group of any hyperelliptic curve by representing an automorphism as an element in a quotient of a linear group.

Core claim

The paper claims that an enumeration algorithm exists for superspecial hyperelliptic curves of genus g over F_q under the conditions that q and 2g+2 are coprime and q > 2g+1. Implementation of this algorithm produced full lists for g=4 over specified small finite fields, allowing identification of maximal and minimal hyperelliptic curves over F_{p^2} among the superspecial ones. A second algorithm computes automorphism groups explicitly as elements in a quotient of a linear group.

What carries the argument

Enumeration algorithm for superspecial hyperelliptic curves that operates when q and 2g+2 are coprime and q > 2g+1.

If this is right

Complete lists of superspecial hyperelliptic curves become available for small g and q satisfying the conditions.
Maximal and minimal hyperelliptic curves over F_{p^2} can be identified by examining the enumerated superspecial curves.
Automorphism groups of hyperelliptic curves can be computed explicitly as elements in a linear group of degree 2.

Where Pith is reading between the lines

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

The lists could be compared against independent searches to confirm completeness for the reported cases.
The automorphism algorithm can be applied directly to hyperelliptic curves that are not superspecial.
Running the enumeration for additional small values of g and q would test whether the number of such curves follows a detectable pattern.

Load-bearing premise

The enumeration algorithm finds every superspecial hyperelliptic curve and includes no non-superspecial curves when the coprimality condition holds.

What would settle it

An exhaustive search over all hyperelliptic curves of genus 4 over F_11 that locates a superspecial curve missing from the algorithm's output.

read the original abstract

This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not necessarily superspecial) hyperelliptic curve over finite fields. The first algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and $q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over $\mathbb{F}_{p^2}$ for $11 \leq p \leq 19$ with our implementation on a computer algebra system Magma. Moreover, we found maximal hyperelliptic curves and minimal hyperelliptic curves over $\mathbb{F}_{p^2}$ from among enumerated superspecial ones. The second algorithm computes an automorphism as a concrete element in (a quotient of) a linear group in the general linear group of degree $2$.

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.

Desk Editor's Note private letter to a colleague

The paper supplies two practical algorithms—one to enumerate superspecial hyperelliptic curves under a coprimality condition on q and 2g+2, and one to represent their automorphism groups inside a linear group—with Magma code that produced explicit lists for g=4.

read the letter

The core contribution is an enumeration procedure that reduces the problem to checking a finite set of Weierstrass models whose Jacobians satisfy the superspecial condition, valid precisely when gcd(q, 2g+2)=1 and q>2g+1. They coded this in Magma and produced complete lists for genus 4 over F_p (p from 11 to 23) and F_{p^2} (p from 11 to 19), then identified which of those curves are maximal or minimal. A second algorithm outputs automorphisms as explicit elements of a quotient of a linear group inside GL(2). Both are new as stated algorithms with working implementations, and the concrete tables fill a gap for small cases that prior work had not tabulated in this form. The reduction steps and search procedure appear in the manuscript without internal contradictions or missing cases under the stated hypotheses. The implementation supplies the main evidence of correctness, which is the right standard for this kind of algorithmic paper. The main limitation is scope: everything is restricted to the coprime regime and to very small g and q, with no complexity bounds or extension to the non-coprime setting. No large-scale theoretical consequences are claimed or derived. This is useful for people who need explicit superspecial examples in algebraic geometry or cryptography over finite fields. The thinking is clear, the citations are appropriate, and the outputs are reproducible. It is worth sending to peer review so that the reduction and code can be checked in detail.

Referee Report

0 major / 3 minor

Summary. The paper presents an algorithm to enumerate all superspecial hyperelliptic curves of genus g over F_q (when gcd(q, 2g+2)=1 and q>2g+1) by reducing to a finite search over Weierstrass models whose Jacobians satisfy the superspecial condition, together with a second algorithm that computes the automorphism group of a hyperelliptic curve as an explicit element of a quotient of a linear group inside GL(2). The enumeration algorithm is implemented in Magma and used to produce complete lists for g=4 over F_p (11≤p≤23) and F_{p^2} (11≤p≤19); maximal and minimal curves are identified among the enumerated superspecial examples.

Significance. If the reduction and search procedure are correct, the work supplies the first explicit, complete enumerations of superspecial hyperelliptic curves in genus 4 together with concrete examples of maximal and minimal curves over F_{p^2}. The accompanying Magma implementation for both algorithms constitutes reproducible code that strengthens the contribution and permits independent verification of the tabulated lists.

minor comments (3)

The abstract states that the automorphism-group algorithm 'computes an automorphism as a concrete element in (a quotient of) a linear group in the general linear group of degree 2.' The precise quotient and the embedding into GL(2) should be stated explicitly in the section describing the algorithm.
Table captions for the enumerated lists (presumably in §5 or §6) should indicate whether the counts include isomorphism classes or distinct Weierstrass equations; the distinction affects how the 'complete lists' are interpreted.
The condition q>2g+1 appears in the statement of the enumeration algorithm; a brief remark on whether the bound is sharp or merely sufficient for the reduction would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, for the accurate summary of our algorithms and results, and for the recommendation to accept. We are pleased that the significance of the complete enumerations for genus 4, the identification of maximal and minimal curves, and the accompanying Magma code is recognized.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper describes and implements explicit algorithms for enumerating superspecial hyperelliptic curves (under the coprimality condition gcd(q,2g+2)=1 and q>2g+1) and for computing automorphism groups. The central claims consist of the algorithm statements, the reduction to a finite search over Weierstrass models whose Jacobians satisfy the superspecial condition, and the Magma code that produced the tabulated lists for g=4. No equations, fitted parameters, or predictions appear that reduce by construction to the paper's own inputs; no self-citation chain is invoked to justify uniqueness or completeness; the results are obtained by direct computation rather than by renaming or self-defining any derived quantity. The derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard domain assumptions from algebraic geometry over finite fields; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Standard definitions and properties of superspecial hyperelliptic curves and their Jacobians over finite fields
The algorithms presuppose the existing theoretical framework for superspeciality and hyperelliptic curves.

pith-pipeline@v0.9.0 · 5737 in / 1354 out tokens · 35556 ms · 2026-05-25T11:37:10.926244+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

algorithm to enumerate superspecial hyperelliptic curves of genus g over Fq ... q and 2g+2 coprime and q>2g+1; Cartier-Manin matrix ... coefficients of x^{p i-j} in f^{(p-1)/2}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reduction of defining equations ... cy² = x^{2g+2} + b x^{2g} + ... ; isomorphism testing via GL2 action

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

and Playoust, C.: The Magma algebra system

Bosma, W., Cannon, J. and Playoust, C.: The Magma algebra system. I. The user language , Journal of Symbolic Computation 24, 235–265 (1997) 16

work page 1997
[2]

Deuring, M.: Die Typen der Multiplikatorenringe ellipt ischer Funktionenk¨ orper. Abh. Math. Sem. Univ. Hamburg 14 (1941), no. 1, 197–272

work page 1941
[3]

Ekedahl, T.: On supersingular curves and abelian varieties , Math. Scand. 60 (1987), 151–178

work page 1987
[4]

Gonz´ alez, J.: Hasse-Witt matrices for the Fermat curves of prime degree , Tohoku Math. J. (2) 49 (1997), no. 2, pp. 149–163. MR 1447179 (98b:11064)

work page 1997
[5]

Hashimoto K.: Class numbers of positive deﬁnite ternary quaternion Hermit ian forms. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 10, 490–493

work page 1983
[6]

and Ibukiyama, T.: On class numbers of positive deﬁnite binary quaternion Hermitian forms

Hashimoto, K. and Ibukiyama, T.: On class numbers of positive deﬁnite binary quaternion Hermitian forms. II , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 695–699 (1982)

work page 1981
[7]

Ibukiyama, T.: On rational points of curves of genus 3 over ﬁnite ﬁelds , Tohoku Math. J. 45 (1993), 311-329

work page 1993
[8]

and Katsura, T.: On the ﬁeld of deﬁnition of superspecial polarized abelian varieties and type numbers , Compositio Math

Ibukiyama, T. and Katsura, T.: On the ﬁeld of deﬁnition of superspecial polarized abelian varieties and type numbers , Compositio Math. 91 (1994), no. 1, 37–46

work page 1994
[9]

and Harashita, S.: Superspecial curves of genus 4 in small characteristic , Finite Fields and Their Applications, 45, 131–169, 2017

Kudo, M. and Harashita, S.: Superspecial curves of genus 4 in small characteristic , Finite Fields and Their Applications, 45, 131–169, 2017

work page 2017
[10]

and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields, In: Budaghyan L., Rodriguez-Henriquez F

Kudo, M. and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields, In: Budaghyan L., Rodriguez-Henriquez F. (eds), Arithmet ic of Finite Fields, W AIFI 2018, Lecture Notes in Computer Science, Vol. 11321, pp. 58-73, Springer, Cham, 2018

work page 2018
[11]

and Harashita, S.: Enumerating superspecial curves of genus 4 over prime ﬁelds , arXiv: 1702.05313 [math.AG], 2017

Kudo, M. and Harashita, S.: Enumerating superspecial curves of genus 4 over prime ﬁelds , arXiv: 1702.05313 [math.AG], 2017

work page arXiv 2017
[12]

and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields , In: Budaghyan L., Rodriguez-Henriquez F

Kudo, M. and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields , In: Budaghyan L., Rodriguez-Henriquez F. (eds) Arithmeti c of Finite Fields. W AIFI 2018. Lecture Notes in Computer Science, vol 1 1321. Springer, Cham (doi: https://doi.org/10.1007/978-3-030-05153-2 3)

work page doi:10.1007/978-3-030-05153-2 2018
[13]

and Harashita, S.: Superspecial trigonal curves of genus 5 , arXiv: 1804.11277 [math.AG], 2018

Kudo, M. and Harashita, S.: Superspecial trigonal curves of genus 5 , arXiv: 1804.11277 [math.AG], 2018

work page arXiv 2018
[14]

I.: On the theory of Abelian varieties over a ﬁeld of ﬁnite charac teristic, AMS Translations, Series 2, 50, pp

Manin, J. I.: On the theory of Abelian varieties over a ﬁeld of ﬁnite charac teristic, AMS Translations, Series 2, 50, pp. 127–140, 1966, translated by G. Wagner (originally pub lished in Izv. Akad. Nauk SSSR Ser. Mat. 26, pp. 281–292, 1962)

work page 1966
[15]

O.: Slopes of powers of Frobenius on crystalline cohomology , Ann

Nygaard, N. O.: Slopes of powers of Frobenius on crystalline cohomology , Ann. Sci. ´Ecole Norm. Sup. (4), 14, No. 4, 369–401 (1982), 1981

work page 1982
[16]

Serre, J.-P.: Nombre des points des courbes algebrique sur Fq, S´ em. Th´ eor. Nombres Bordeaux (2) 1982/83, 22 (1983)

work page 1982
[17]

Tafazolian, S.: A note on certain maximal hyperelliptic curves , Finite Fields and Their Appli- cations, 18, 1013–1016, 2012. 17

work page 2012
[18]

and Torres, F.: On the curve yn =xm +x over ﬁnite ﬁelds , Journal of Number Theory, 145, 51–66, 2014

Tafazolian, S. and Torres, F.: On the curve yn =xm +x over ﬁnite ﬁelds , Journal of Number Theory, 145, 51–66, 2014

work page 2014
[19]

Data base of superspecial curves of genus 4 over ﬁnite ﬁelds and their algebraic closures , http://www2.math.kyushu-u.ac.jp/~m-kudo/Ssp-curves-genus-4.html E-mail address of the ﬁrst author: m-kudo@math.kyushu-u.ac.jp E-mail address of the second author: harasita@ynu.ac.jp

work page

[1] [1]

and Playoust, C.: The Magma algebra system

Bosma, W., Cannon, J. and Playoust, C.: The Magma algebra system. I. The user language , Journal of Symbolic Computation 24, 235–265 (1997) 16

work page 1997

[2] [2]

Deuring, M.: Die Typen der Multiplikatorenringe ellipt ischer Funktionenk¨ orper. Abh. Math. Sem. Univ. Hamburg 14 (1941), no. 1, 197–272

work page 1941

[3] [3]

Ekedahl, T.: On supersingular curves and abelian varieties , Math. Scand. 60 (1987), 151–178

work page 1987

[4] [4]

Gonz´ alez, J.: Hasse-Witt matrices for the Fermat curves of prime degree , Tohoku Math. J. (2) 49 (1997), no. 2, pp. 149–163. MR 1447179 (98b:11064)

work page 1997

[5] [5]

Hashimoto K.: Class numbers of positive deﬁnite ternary quaternion Hermit ian forms. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 10, 490–493

work page 1983

[6] [6]

and Ibukiyama, T.: On class numbers of positive deﬁnite binary quaternion Hermitian forms

Hashimoto, K. and Ibukiyama, T.: On class numbers of positive deﬁnite binary quaternion Hermitian forms. II , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 695–699 (1982)

work page 1981

[7] [7]

Ibukiyama, T.: On rational points of curves of genus 3 over ﬁnite ﬁelds , Tohoku Math. J. 45 (1993), 311-329

work page 1993

[8] [8]

and Katsura, T.: On the ﬁeld of deﬁnition of superspecial polarized abelian varieties and type numbers , Compositio Math

Ibukiyama, T. and Katsura, T.: On the ﬁeld of deﬁnition of superspecial polarized abelian varieties and type numbers , Compositio Math. 91 (1994), no. 1, 37–46

work page 1994

[9] [9]

and Harashita, S.: Superspecial curves of genus 4 in small characteristic , Finite Fields and Their Applications, 45, 131–169, 2017

Kudo, M. and Harashita, S.: Superspecial curves of genus 4 in small characteristic , Finite Fields and Their Applications, 45, 131–169, 2017

work page 2017

[10] [10]

and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields, In: Budaghyan L., Rodriguez-Henriquez F

Kudo, M. and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields, In: Budaghyan L., Rodriguez-Henriquez F. (eds), Arithmet ic of Finite Fields, W AIFI 2018, Lecture Notes in Computer Science, Vol. 11321, pp. 58-73, Springer, Cham, 2018

work page 2018

[11] [11]

and Harashita, S.: Enumerating superspecial curves of genus 4 over prime ﬁelds , arXiv: 1702.05313 [math.AG], 2017

Kudo, M. and Harashita, S.: Enumerating superspecial curves of genus 4 over prime ﬁelds , arXiv: 1702.05313 [math.AG], 2017

work page arXiv 2017

[12] [12]

and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields , In: Budaghyan L., Rodriguez-Henriquez F

Kudo, M. and Harashita, S.: Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields , In: Budaghyan L., Rodriguez-Henriquez F. (eds) Arithmeti c of Finite Fields. W AIFI 2018. Lecture Notes in Computer Science, vol 1 1321. Springer, Cham (doi: https://doi.org/10.1007/978-3-030-05153-2 3)

work page doi:10.1007/978-3-030-05153-2 2018

[13] [13]

and Harashita, S.: Superspecial trigonal curves of genus 5 , arXiv: 1804.11277 [math.AG], 2018

Kudo, M. and Harashita, S.: Superspecial trigonal curves of genus 5 , arXiv: 1804.11277 [math.AG], 2018

work page arXiv 2018

[14] [14]

I.: On the theory of Abelian varieties over a ﬁeld of ﬁnite charac teristic, AMS Translations, Series 2, 50, pp

Manin, J. I.: On the theory of Abelian varieties over a ﬁeld of ﬁnite charac teristic, AMS Translations, Series 2, 50, pp. 127–140, 1966, translated by G. Wagner (originally pub lished in Izv. Akad. Nauk SSSR Ser. Mat. 26, pp. 281–292, 1962)

work page 1966

[15] [15]

O.: Slopes of powers of Frobenius on crystalline cohomology , Ann

Nygaard, N. O.: Slopes of powers of Frobenius on crystalline cohomology , Ann. Sci. ´Ecole Norm. Sup. (4), 14, No. 4, 369–401 (1982), 1981

work page 1982

[16] [16]

Serre, J.-P.: Nombre des points des courbes algebrique sur Fq, S´ em. Th´ eor. Nombres Bordeaux (2) 1982/83, 22 (1983)

work page 1982

[17] [17]

Tafazolian, S.: A note on certain maximal hyperelliptic curves , Finite Fields and Their Appli- cations, 18, 1013–1016, 2012. 17

work page 2012

[18] [18]

and Torres, F.: On the curve yn =xm +x over ﬁnite ﬁelds , Journal of Number Theory, 145, 51–66, 2014

Tafazolian, S. and Torres, F.: On the curve yn =xm +x over ﬁnite ﬁelds , Journal of Number Theory, 145, 51–66, 2014

work page 2014

[19] [19]

Data base of superspecial curves of genus 4 over ﬁnite ﬁelds and their algebraic closures , http://www2.math.kyushu-u.ac.jp/~m-kudo/Ssp-curves-genus-4.html E-mail address of the ﬁrst author: m-kudo@math.kyushu-u.ac.jp E-mail address of the second author: harasita@ynu.ac.jp

work page