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Maximal curves with respect to quadratic extensions over finite fields

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Pith's one-line read A curve hits the Hallouin-Perret bound exactly when the real parts of its Frobenius eigenvalues are all equal, and this single condition organizes new coefficient bounds, a low-genus classification, and a characterization of Ihara-maximal…

desk verdict A sound, useful paper on Hallouin-Perret-maximal curves whose only real weakness is the unshipped enumeration behind the low-genus completeness claim. read the letter →

arxiv 2506.08603 v3 pith:3CO3KS33 submitted 2025-06-10 math.AG

Yves Aubry (IMATH , I2M , UTLN) , Fabien Herbaut (UCA INSPE , UniCA , IMATH) , Julien Monaldi (IMATH) This is my paper
classification math.AG MSC 11G2011G2511G1014G1514H25
keywords algebraiccurvesfinitefieldsHallouin-Perret-maximalIharaboundDiophantinestabilityWeilpolynomialsabelianvarietieszetafunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the Hallouin-Perret bound, which compares the number of rational points of a curve over $\mathbb{F}_q$ with its number over $\mathbb{F}_{q^2}$: the second count is bounded by $2gq - (1/g)(\#X(\mathbb{F}_q)-(q+1))^2$ above $q^2+1$. The main claim is that equality in this bound holds exactly when the real parts $\alpha_1,\dots,\alpha_g$ of the curve's Frobenius eigenvalues are all equal, so that the zeta function has the one-factor form $(1-2\alpha T+qT^2)^g/((1-T)(1-qT))$. The paper writes the gap between the two sides as a variance, which turns the inequality into the non-negativity of a variance and yields new upper and lower bounds for the coefficient $a_2$ of any $q$-Weil polynomial in every genus. Because equality is characterized by one condition, the paper can identify Ihara-maximal curves with HP-maximal curves that are Diophantine-stable, list all Ihara-maximal curves of genus at most $18$, prove that the Suzuki curves are Ihara-maximal in their genus, and decide which polynomials $(T^2+aT+q)^2$ occur as Jacobians of genus-$2$ HP-maximal curves.

What carries the argument

The carrying object is the list of real parts $\alpha_1,\dots,\alpha_g$ of the Frobenius eigenvalues of the curve. The key identity is the variance formula $\delta_{\mathrm{HP}}(X)=4g^2V(\alpha)=4\sum_{1\le j<k\le g}(\alpha_j-\alpha_k)^2$, where $\delta_{\mathrm{HP}}$ is the HP-defect, the gap between the two sides of the Hallouin-Perret inequality; the inequality is therefore the positivity of a variance, and equality means all $\alpha_j$ coincide. This degenerate case is exactly a curve whose L-polynomial, the numerator polynomial of the zeta function, is a single quadratic factor raised to the $g$-th power, which is the form that makes the Jacobian a power of a simple abelian variety and the Frobenius-angle set one element. A Hallouin-Perret-maximal curve is defined as a curve reaching equality in the Hallouin-Perret bound, and this machinery turns that geometric extremality into an arithmetic one-line shape of the zeta function.

What would settle it

Carry out an independent exhaustive enumeration of curves of genus at most $18$ over the fields appearing in Table 1 and check whether every curve attaining the Ihara bound appears in the table; finding one missing isomorphism class would refute the completeness claim in Proposition 4.5. A smaller check is to settle the two outstanding cases the paper names: whether the genus-$7$ curve over $\mathbb{F}_7$ with $36$ rational points is unique, and whether a genus-$19$ curve over $\mathbb{F}_{19}$ with $153$ rational points exists.

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Extended reading notes

Core claim

The paper's central discovery is that the Hallouin-Perret-maximal condition is a single-number condition. For a curve $X$ of genus $g$ over $\mathbb{F}_q$ with Frobenius eigenvalues of real parts $\alpha_1,\dots,\alpha_g$, the equality $\#X(\mathbb{F}_{q^2})-(q^2+1)=2gq-\frac{1}{g}(\#X(\mathbb{F}_q)-(q+1))^2$ holds if and only if $\alpha_1=\cdots=\alpha_g$. The zeta function is then $Z_X(T)=(1-2\alpha T+qT^2)^g/((1-T)(1-qT))$ with $2\alpha=(q+1-\#X(\mathbb{F}_q))/g$ an integer, and the Jacobian of $X$ has characteristic polynomial $(T^2-2\alpha T+q)^g$. From this equivalence the paper derives the refined inequalities on the point counts and on the coefficient $a_2$, the characterization of Ihara-maximal curves as HP-maximal and Diophantine-stable, and the classification statements for low genus and for abelian surfaces.

Load-bearing premise

The most fragile premise is that the previously published upper bounds on the maximum number of rational points of curves over small finite fields, together with the database searches that the paper uses, are complete and correct; if one of those bounds missed a case, the claimed complete list of Ihara-maximal curves of genus at most $18$ could be missing an entry.

Editorial extensions

If this is right

  • A curve that is Hallouin-Perret-maximal has a zeta function of the form $(1-2\alpha T+qT^2)^g/((1-T)(1-qT))$, so all its arithmetic over $\mathbb{F}_q$ and $\mathbb{F}_{q^2}$ is governed by a single integer $2\alpha$.
  • Ihara-maximal curves are precisely the HP-maximal curves with $\#X(\mathbb{F}_{q^2})=\#X(\mathbb{F}_q)$, so the known Ihara-bound extremal curves form the intersection of two explicit conditions.
  • Every $q$-Weil polynomial of degree $2g$ satisfies $a_2\le a_1^2(g-1)/(2g)+gq$ and the stated lower bounds, extending the previously known small-genus inequalities to all $g$.
  • The complete list of Ihara-maximal curves with genus at most $18$ provides explicit equations and L-polynomials for all cases except the unresolved uniqueness question at $(g,q)=(7,7)$.
  • The Suzuki curve over $\mathbb{F}_{2^{2t+1}}$ is the unique Ihara-maximal curve of genus $g=\sqrt{q}(q-1)/\sqrt{2}$, settling maximality in that genus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Since the HP-defect is a computable variance, screening curves with a fixed $\#X(\mathbb{F}_q)$ by minimal defect is a practical way to search for curves with many points over $\mathbb{F}_{q^2}$ without solving the full optimization problem.
  • The covering stability of HP-maximality suggests that subcovers of curves with a single Frobenius angle, such as Suzuki and Hermitian curves, form a rich supply of HP-maximal curves; this could be tested by explicitly analyzing the covers of these families.
  • The new bounds on $a_2$ are necessary conditions for a $q$-Weil polynomial but are not sufficient; combining them with the complete small-genus classifications of abelian surfaces could sharpen the search for polynomials that actually arise from Jacobians.
  • The open cases the paper flags, such as the genus-$7$ uniqueness question and the potential genus-$19$ curve over $\mathbb{F}_{19}$, are concrete next targets: the discriminant criterion already predicts their point counts, so a direct search would confirm or break the pattern.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the Hallouin-Perret inequality relating the numbers of rational points of a curve over F_q and over its quadratic extension. The main results are: (1) a characterization of equality (HP-maximality) via the equality of the real parts of the Frobenius eigenvalues, with an explicit zeta-function form; (2) a variance interpretation of the defect and refined bounds on it; (3) upper and lower bounds on the coefficient a_2 of q-Weil polynomials, generalizing results of Rück, Haloui, Haloui-Singh, and Sohn; (4) a characterization of Ihara-maximal curves as HP-maximal and Diophantine-stable, leading to a low-genus classification (g ≤ 18, except g = 7) that relies on a Python enumeration and on cited point-count upper bounds; (5) a Suzuki-curve analogue of the Rück-Stichtenoth theorem; and (6) a classification of genus-2 HP-maximal curves via the isogeny classes of their Jacobians.

Significance. The algebraic core is sound and useful: Proposition 2.1 reduces HP-maximality to a Cauchy-Schwarz equality, and the variance viewpoint gives a transparent proof of the Hallouin-Perret bound and sharp lower bounds. The bounds on a_2 in Theorem 3.9 are new in the stated generality and correctly specialize to the known low-genus inequalities. The genus-2 classification is a solid contribution built on a careful use of the cited classifications of abelian surfaces. However, the low-genus classification of Ihara-maximal curves is not independently verifiable as presented: the enumeration is not shipped, the tables are typeset in a garbled way, and the completeness statement relies on several external bounds and database lookups. The paper does not ship machine-checked proofs or executable enumeration code, so the strongest contribution is the conceptual characterization rather than the computational completeness claims.

major comments (2)
  1. [§4.3, Proposition 4.5, Table 2] The completeness claim for Table 1 is not reproducible. The proof states that 'with the help of a Python program we list all the couples (g,q)' for which the discriminant (8q+1)g^2 + 4qg(q-1) is a square, but neither the code nor the generated list of candidates is provided. Table 2, which is supposed to record the discarded pairs together with Ihara bounds, upper bounds, and references, is misaligned in the manuscript: the same q=11 appears twice and the rows cannot be matched unambiguously to the stated Ihara and upper-bound columns. Please provide the enumeration output or an archive containing the code, correct Table 2, and for every discarded pair a precise pointer to the specific upper bound (theorem or page) in [14], [18], [28], [29], or [30] that eliminates it.
  2. [§4.3, Table 1 and Remark 4.6] The final list in Table 1 is asserted 'up to isomorphism', but the proof does not supply isomorphism certificates for the retained curves, and several rows depend on database lookups (LMFDB, ManyPoints) or on classifications in [4], [20], [25] without explicitly stating which external result establishes completeness in each row. As typeset in the provided text, Table 1 mixes column headers with data rows: for example, the row '2 5 1 + 2T + 2T^2 y^2+y=x^3+x' cannot be read as a genus-2 Ihara-maximal curve, since the displayed polynomial and equation do not match the stated N1=N2 value. The table should be re-typeset with one pair per row and the data aligned with the headers, and each row should be accompanied by a verifiable reference or a stated computational check establishing uniqueness.
minor comments (4)
  1. [Abstract and Introduction] There are several typographical errors: 'hightlighted' should be 'highlighted', 'developped' should be 'developed', and 'straighforward' should be 'straightforward'.
  2. [Proposition 2.10] The bound 'g ≤ 23q2 log q' appears to be a typographical garble of what should be g ≤ 2^3 q^2 log q (or the correct constant from Theorem 1.1 of [5]); please use unambiguous exponent notation throughout.
  3. [Remark 3.10] The comparison with Haloui's lower bound a_2 ≥ 4√q|a_1| − 9q states that the new bound is better if and only if the displayed inequality holds, but the subsequent reduction to |a_1| ≤ 2√q is terse; adding one line of algebra would make the argument easier to follow.
  4. [Tables 1 and 2] Both tables need to be re-typeset from the LaTeX source. The extracted text shows misaligned columns and repeated entries, which prevents the reader from checking the numerical data even when the cited sources are available.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central HP-maximal characterization is a variance identity, and the low-genus completeness claim depends on external bounds and an unshipped enumeration rather than on circular reasoning.

full rationale

The paper's central equivalence (Proposition 2.1) is derived from Weil's expressions for N1 and N2: substituting the HP-maximal equality (6) into those expressions reduces, by algebraic expansion, to g*sum(α_j^2) = (sum α_j)^2, which is exactly Cauchy-Schwarz equality. This is a genuine algebraic reduction, with no fitted constants and no use of the target statement as an input. The HP-defect identity δHP = 4g^2 V(α) (Theorem 3.2) is an elementary expansion in terms of the real parts of Frobenius eigenvalues, and the bounds in Theorem 3.9 follow from the non-negativity and range of the variance, plus the relation between a1, a2 and the α_j. That derivation does not presuppose any of the bounds it proves. Proposition 4.1 is likewise an equivalence obtained from equality conditions in Ihara's classical inequalities and the HP-maximal characterization, not a renaming of the Ihara bound. The low-genus classification (Proposition 4.5) does rely on a Python enumeration whose code and output are not shipped and on cited upper bounds from [14], [18], [28], [29], [30] and LMFDB/ManyPoints; if any of those were wrong or incomplete, a curve could be missing from Table 1. That is a completeness and auditability risk, not a circularity risk. The only self-citation with author overlap is the use of Aubry-Haloui-Lachaud [1] in Proposition 2.9, a published theorem on equality in the abelian-variety point bound; it appears in a supplementary characterization and is not used to force the main bounds or the classification. Accordingly, no equation or claim in the paper reduces to its own input by construction, and the paper is not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivations are self-contained algebraic identities; the free-parameter count is zero. The axioms listed are standard theorems or citations to the literature. The paper introduces the terms HP-maximal and HP-defect as names for equality in an existing bound and for the gap in that bound; these are definitions, not new mathematical entities.

assumptions (5)
  • standard math Riemann Hypothesis for curves over finite fields (Weil): all reciprocal roots of the L-polynomial have absolute value sqrt(q).
    Used throughout to write |alpha_j| <= sqrt(q), giving the variance bounds in Proposition 3.4 and the genus bounds in Proposition 2.10.
  • standard math Honda-Tate theory and Deuring-Waterhouse classification of isogeny classes of abelian varieties over finite fields.
    Underpins Proposition 2.5, Remark 2.8, and the genus-2 classification in Theorem 5.1.
  • domain assumption Completeness of the classifications in Maisner-Nart [20], Howe [13], Howe-Nart-Ritzenthaler [15], and Maisner-Nart [21] for abelian surfaces and their Jacobians.
    The proof of Theorem 5.1 relies on these tables to list all possible a-values for simple and split abelian surfaces.
  • domain assumption Correctness and exhaustiveness of known upper bounds for N_q(g) and of the LMFDB and ManyPoints databases (Savitt [28], Serre [29],[30], Howe-Lauter [14], Kohnlein [18]).
    Used in the proof of Proposition 4.5 to discard candidate (g,q) pairs and to assert that Table 1 is the complete list of Ihara-maximal curves for g <= 18 and q <= 13.
  • standard math Fuhrmann-Torres Theorem 2 [6]: a curve over F_{2^{2t+1}} of genus 2^t(2^{2t+1}-1) with q^2+1 rational points is the Suzuki curve.
    Reformulated as Theorem 4.4 to characterize Ihara-maximal curves of Suzuki genus.

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Pith. "Pith review of Maximal curves with respect to quadratic extensions over finite fields." pith.science (2026). https://pith.science/paper/3CO3KS33

@misc{pith2026250608603,
  author       = {Pith},
  title        = {Pith review of: Maximal curves with respect to quadratic extensions over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CO3KS33}},
  note         = {Machine review of arXiv:2506.08603}
}
read the original abstract

We propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which make a connection with the variance enable to obtain optimal refinements. We focus on the curves reaching the bound, which we call Hallouin-Perret-maximal curves. We provide different characterizations and stress natural links with the curves which attain the Ihara bound. As consequences, we establish the list of such curves with low genus and we outline a maximality result which involves the Suzuki curves. At last we determine which polynomials correspond to the Jacobian of a Hallouin-Perret-maximal curve of genus 2.

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