Maximal curves over finite fields and a modular isogeny

Claudio Stirpe; Guido Lido; Pietro Mercuri; Valerio Dose

REVIEW 2 major objections 5 minor 1 cited by

Maximal curves over finite fields and a modular isogeny

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves that quotient modular curves of genus 7 and 12 over $\mathbb{F}_{11^5}$ attain the Hasse-Weil-Serre bound, giving explicit equations and a generalized Chen isogeny.

desk verdict The existence results for maximal curves of genus 7 and 12 over F_{11^5} are solid, but the genus-7 equation identification has an unproven numerical-to-exact rounding step. read the letter →

arxiv 2504.18894 v2 pith:KDBWUEQN submitted 2025-04-26 math.NT

Valerio Dose , Guido Lido , Pietro Mercuri , Claudio Stirpe This is my paper · ORCID

classification math.NT MSC 11G2011G1811T7114G35

keywords Hasse-Weil-SerreboundmaximalcurvesmanypointsmodularAtkin-LehnerinvolutionsChenisogenyCartansubgroupscanonicalequations

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 proves that two smooth projective curves, of genus 7 and genus 12, exist over the field with $11^5$ elements and have exactly the number of points allowed by the Hasse-Weil-Serre bound: 166666 and 170676 respectively. Both curves are quotients of modular curves of Borel-Cartan type by Atkin-Lehner type involutions, and the paper gives explicit equations for them. The genus-7 curve is described by a canonical model in $\mathbb{P}^6$ that is smooth over $\mathbb{Z}[1/42]$, while the genus-12 curve is $X_0(156)/\langle W_{13}\rangle$. To compute the point counts, the paper generalizes Chen's isogeny theorem to all such quotients, expressing each Jacobian as a product of newform abelian varieties with explicit multiplicities. The same computation yields 36 improved lower bounds and several new or non-isomorphic maximal curves, and the appendices tabulate all results.

What carries the argument

The load-bearing object is Theorem 2.2, a Chen-type isogeny for the whole family: for $Y=X(n_0,n_{ns})$ and $K$ a subgroup of its Atkin-Lehner type involutions, $\operatorname{Jac}(Y/K)$ is isogenous over $\mathbb{Q}$ to $\prod_f A_f^{m_f}$, where the $f$ run over weight-2 newforms of levels $d_0 d_{ns}^2$ and $m_f$ is a product of explicit factors involving the Atkin-Lehner eigenvalues $\varepsilon_{f,p^e}$ and the characters of $K$. This reduces every point count to modular-form data. For the equations, the method is Zywina's computation of a basis of $S_2(\Gamma_H)$ followed by extracting the $W_6$-invariant subspace of $\Omega^1$, applying Mercuri's canonical-embedding technique to find quadrics, and completing the identification by proving, through endomorphism rings of the Jacobian factors and absence of small-order automorphisms, that the hidden quotient involution is exactly $W_6$.

What would settle it

Recompute the $W_6$-invariant subspace of $\Omega^1(X(6,7)/\langle W_7\rangle)$ using exact modular symbols instead of $10^{-10}$ numerical evaluation; if its dimension is not 7 or the displayed quadrics are not exactly satisfied by the exact forms, the canonical-model identification fails. Independently, an enumeration of the $\mathbb{F}_{11^5}$-points on the reduction of the displayed model must return exactly 166666 for genus 7 and 170676 for genus 12.

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Extended reading notes

Core claim

The central claim is that the quotient $X(6,7)/\langle W_6,W_7\rangle$ has genus 7 and $166666$ points over $\mathbb{F}_{11^5}$, and that $X_0(156)/\langle W_{13}\rangle$ has genus 12 and $170676$ points, both numbers equal to the Hasse-Weil-Serre upper bound $q+1+g\lfloor 2\sqrt{q}\rfloor$ for $q=11^5$. The paper presents explicit equations for both: a system of quadrics giving the canonical model of the genus-7 curve, smooth over $\mathbb{Z}[1/42]$, plus singular planar models for both curves. The existence proof is arithmetic: the generalized isogeny determines the Jacobian decomposition, the Hecke eigenvalues at 11 give the Frobenius eigenvalues, and the point counts follow. The equation proof is geometric: the canonically embedded model is computed from invariant differential forms and then certified by showing the candidate quotient automorphism equals $W_6$ on the Jacobian.

Load-bearing premise

The load-bearing premise is that the numerically computed invariant differential forms are exact and exactly satisfy the displayed quadrics; if that numerical-to-exact bridge failed, the equations would not necessarily describe the intended quotient curve.

Editorial extensions

If this is right

For both new pairs $(g,q)=(7,11^5)$ and $(12,11^5)$, the Hasse-Weil-Serre bound is attained by curves with explicit equations; the genus-12 curve was previously known only as a point-counted quotient, and the genus-7 curve is new.
The generalized isogeny gives a uniform, equation-free method to compute $\#X(\mathbb{F}_q)$ for every quotient by Atkin-Lehner type involutions from Hecke eigenvalues alone.
The systematic search with $n_0 n_{ns}^2\leq 10^4$ and genus $\leq 50$ produces 36 new 'nice' lower bounds and several further maximal curves, including examples over odd-degree extensions such as $\mathbb{F}_{11^5}$.
Some maximal curves with the same point count have different real Weil polynomials, so the family contains non-isomorphic maximal curves of the same genus and field.

Reading between the lines

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

The numerical-to-exact certification used for the genus-7 model could be reused: any of the 36 record curves in Table 1 whose equations are not yet written down is a candidate for the same pipeline, turning counted quotients into explicit curves.
The appearance of maximality over an odd-degree extension suggests the point-producing mechanism is not simply supersingular elliptic curves rationalizing over even extensions; the Frobenius eigenvalue patterns of the specific newforms are doing the work, so similar searches over other odd-degree fields may yield more records.
The Appendix A lattice identities imply that point counts and genera of all quotients by subgroups of $(\mathbb{Z}/2\mathbb{Z})^r$ are determined by a small set of quotient curves; this could be used as a cheap pre-filter before running the isogeny computation on a large family.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know up front. First, the main existence claims hold up: the quotient modular curves X(6,7)/ and X(156,1)/ do have the stated genera and do meet the Serre bound over F_{11^5}, and the point counts come from modular form eigenvalues, not from the equations. Theorem 2.2 genuinely generalizes Chen-type isogeny decompositions to all quotient groups by Atkin-Lehner type involutions, and the 36 new table records are a concrete payoff. Second, the explicit equations for the genus-7 curve are shakier than the rest of the paper. The rounding from numerical linear algebra (precision 10^-10, denominators <40) to exact differential forms on Y is asserted rather than certified. The genus-12 equations have a real certificate: the q^180 vanishing plus the degree 4g-4=44 bound forces exact zero. The genus-7 section has no comparable Sturm-bound or exact modular-symbol argument, so the identification of those quadrics with X(6,7)/ is genuinely conditional. The stress-test note landed on the right spot here.

The paper does several things well besides the theorem. The worked examples make the point-count algorithm transparent, and the genus-12 planar model is backed by a rigorous vanishing argument. The automorphism-based strategy for proving the canonical model is a good idea, and the authors are honest about the approximation process. I would also credit the clean separation between existence, which relies on the isogeny theorem and known eigenvalue data, and the explicit-equation claims, which are more fragile.

Soft spots, in proportion. The main one is the missing rigorous bridge for the s_i in §4.1. This is fixable, likely by replacing the numerical linear algebra with exact modular-symbol arithmetic and a Sturm-bound check, or by publishing a machine-checked certificate. Reproducibility is a smaller but real concern: the full enumeration code is not supplied as a single commit-hashed artifact, and the LMFDB data dependency is not pinned to a version. Those are workflow issues, not mathematical errors. The paper acknowledges the genus-12 curve already appeared numerically in [DLMS23]; the new content there is the explicit model and the maximality observation, and that is handled honestly.

Who is this for: anyone working on curves with many points or explicit modular curves. The existence results and the isogeny theorem merit attention even if the genus-7 equation display is treated as provisional. With the §4.1 bridge tightened, the paper would be fully convincing. I would send it to a serious referee and ask for that tightening, or for a clear statement that the displayed equations are only conjectural models pending exact verification. The core mathematics deserves the referee time.

Referee Report

2 major / 5 minor

Summary. The paper studies quotient modular curves of Borel-Cartan type and their quotients by Atkin-Lehner type involutions. Its main theoretical contribution is Theorem 2.2, a generalization of Chen's isogeny theorem giving an explicit isogeny decomposition of the Jacobian of each quotient. Using this theorem together with LMFDB eigenvalue data, the authors compute point counts for many such curves over finite fields and identify 36 new records. The central arithmetic claims are that the genus-7 curve X(6,7)/<W6,W7> and the genus-12 curve X(156,1)/<W13> over F_{11^5} attain the Hasse-Weil-Serre bound, with explicit equations for both. The genus-12 identification is backed by an exact degree-bound argument, while the genus-7 identification relies on a numerical-to-exact rounding step in Section 4.1 that is not fully certified.

Significance. If the results are correct, this is a meaningful contribution to the curves-with-many-points literature: the two curves would be the first known genus-7 and genus-12 curves over F_{11^5} reaching the Serre bound, and the paper provides explicit equations for both. The generalization of Chen's isogeny theorem is independently useful, and the large tables in Appendices B and C give many new record lower bounds. A clear strength is that the computations are supported by a public Magma repository, and the genus-12 proof contains a genuine exact certificate. The main caveat is that the genus-7 identification rests on an unverified numerical bridge in Section 4.1; this is a correctness-risk concern rather than a demonstrated error, and it appears repairable by adding an exact verification step.

major comments (2)

[§4.1, paragraphs 'For the computation of S' and 'To turn clues into a proof'] The identification of the displayed genus-7 canonical model with X(6,7)/<W6,W7> rests on an unproven numerical-to-exact step. The forms s1,...,s7 are obtained by solving a linear system to precision 10^-10 and rounding coefficients to rationals with denominators less than 40; the text then asserts, without proof, that these forms lie exactly in Omega^1(Y) and are exactly W6-invariant. The subsequent Riemann-Hurwitz degree argument and the automorphism argument proving u=W6 use this exactness as a hypothesis, so an approximate result would leave the displayed equations unconnected to the intended modular curve. This contrasts with Section 4.3, where F_j(g)=O(q^180) is forced to vanish exactly by the degree bound 4g-4=44. Please provide an exact certificate for each s_i, for example by verifying the linear relation br + sum_j c_j b_j = 0 on Fourier coefficients through an exact or provably sufficient computation, and state that certificate in the text.
[§4.1, paragraph after the ten displayed quadrics] Even if the s_i were known to lie in Omega^1(Y), the claim that the ten quadratic equations are exactly satisfied by them is not proved in the text; the numerical computation with 5800 Fourier coefficients and precision 10^-10 does not by itself establish exact vanishing, and no degree-bound argument like the one in Section 4.3 is supplied. The same paragraph also asserts smoothness over Z[1/42] without describing the verification. Please add a certificate for the exact vanishing of each quadratic relation and a clear description of the smoothness check, or replace the numerical part of the proof by an exact computation.

minor comments (5)

[Example 3.5] The phrase 'no cusp forms of weight 2 and level 14 or less than 10' is confused; the intended statement is likely 'no cusp forms of weight 2 and level at most 14'.
[§4.2, planar equation] The coefficient '(4x^3+4x^2+2−1)y^8' contains '2−1', which should presumably be '1'; please correct this typo.
[§4.1] The operator |2m6 is defined only by reference to [Miy05]; a one-line definition in the text would improve readability.
[§4.1, final paragraph] The injectivity of the reduction map Aut(X_Q) -> Aut(X_F5) is asserted in a single sentence; a short justification would help the reader trust that the Magma bound #Aut(X_F5) <= 2 indeed bounds the rational automorphism group.
[Data availability] The GitHub repository is cited without a commit hash or version; since the paper relies on it for equations and for the proofs of several computational claims, please include a persistent version identifier.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the point counts are computed from modular form eigenvalues, and the equation computations are checked against independent data; the genus-7 numerical-to-exact step is a correctness gap, not a circular fit.

full rationale

The paper's central claims are not equivalent to their inputs by construction. The point counts, including the record values 166666 and 170676 over F_{11^5}, are derived from Hecke eigenvalues read from LMFDB together with the multiplicity formula in Theorem 2.2; no parameter is fitted to the target point count and then relabeled as a prediction. Theorem 2.2 itself relies on [DLMS23, Theorem 3.8], a previously published theorem by overlapping authors, but that citation is transparent, independent, and is real evidence rather than a self-referential uniqueness assertion. The genus-12 equation computation has an exact certificate: the invariant forms are computed exactly in Magma, and the check F_j(g)=O(q^180) forces exact vanishing because the relevant section has degree 4g-4=44. The genus-7 identification in Section 4.1 contains a genuine numerical-to-exact gap: forms are obtained with precision 10^-10 and denominators sought below 40, and the paper then asserts these forms lie in Omega^1(Y) and satisfy the displayed quadrics exactly. This is a potential correctness or reproducibility gap, but it is not circularity: the equations are not defined in terms of the claimed point count, and the matching point count is used only as a consistency check and as part of a geometric identification argument. No equation in the paper reduces to its own conclusion, no fitted parameter is renamed as a prediction, and the self-citations do not carry an unverified load-bearing premise. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No physical or mathematical constant is fitted; the only numerical inputs are LMFDB Hecke eigenvalues and Magma computations, treated as external oracle data. The paper introduces no new particles, forces, or unobserved entities.

assumptions (4)

domain assumption The modular curves X(n0,nns) have good reduction outside primes dividing n0*nns, and the quotients by Atkin-Lehner type involutions also have good reduction there.
Used throughout Section 3 to reduce point counts modulo p; cited to Deligne-Rapoport [DR73] and [LL99], [You93], [NS96].
domain assumption The Jacobian of X(n0,nns) decomposes over Q into simple abelian varieties A_f attached to newforms, with the multiplicities given by [DLMS23, Theorem 3.8] for quotients by subgroups generated by a subset of Atkin-Lehner involutions.
Theorem 2.2 reduces to this cited result and uses its multiplicities for subsets Z to determine the character multiplicities d_chi; this is prior independent work by the same authors.
domain assumption The endomorphism algebra of the Jacobian of Y = X(6,7)/<W7> has the product form from Kani's theorem [Kan08, Theorem 6], with matrix algebras over the coefficient fields of the newforms.
Used in Section 4.1 to compare the actions of u, W2, and W6 on the Jacobian and to prove the automorphism identification.
domain assumption The Hecke eigenvalue data in the LMFDB for newforms of weight 2 and level up to 10000 are correct, and the cited Magma computations in [MerCode] are correct.
All point counts and the automorphism checks in Sections 3 and 4 depend on these computational artifacts; no machine-checked certificates or commit hashes are provided.

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Pith. "Pith review of Maximal curves over finite fields and a modular isogeny." pith.science (2026). https://pith.science/paper/KDBWUEQN

@misc{pith2026250418894,
  author       = {Pith},
  title        = {Pith review of: Maximal curves over finite fields and a modular isogeny},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KDBWUEQN}},
  note         = {Machine review of arXiv:2504.18894}
}

abstract

We prove the existence of curves of genus $7$ and $12$ over the field with $11^5$ elements, reaching the Hasse-Weil-Serre upper bound. These curves are quotients of modular curves and we give explicit equations. We compute the number of points of many quotient modular curves in the same family without providing equations. For various pairs (genus, finite field) we find new records for the largest known number of points. In other instances we find quotient modular curves that are maximal, matching already known results. To perform these computations, we provide a generalization of Chen's isogeny result.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Point counts, automorphisms, and gonalities of Shimura curves
math.NT 2025-07 conditional novelty 7.0 of 10

An algorithm based on Ribet's isogeny computes point counts for Shimura curves, yielding 116 record curves, automorphism results for 9288 of 10609 curves, and a tetragonal classification up to 32 exceptions each.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

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[AAL23] S. Anni, E. Assaf, and E. Lorenzo Garcıéa. “On smooth plane models for modular curves of Shimura type”. In: Research in Number Theory 9.2 (2023), p

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https://web.archive.org/web/20240913142835/https://manypoints.org//Details.aspx?g=4&q=1 [GMQ23] R. Gupta, E. A. Mendoza, and L. Quoos. “Reciprocal po lynomials and curves with many points over a ﬁnite ﬁeld”. In: Research in Number Theory 9.3 (2023), p

arXiv 2023

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[AL70] A. O. L. Atkin and J. Lehner. “Hecke operators on Γ 0pmq”. In: Math. Ann. 185 (1970), pp. 134–160. [AS90] M. Akbas and D. Singerman. “The normalizer of Γ 0pN q in PSLp2,Rq”. In: Glasgow Math. J. 32.3 (1990), pp. 317–327. [Bal+19] J. Balakrishnan et al. “Explicit Chabauty-Kim for the split Cartan modular curve of level 13”. In: Ann. of Math. (2) 189...

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[Gon17] J. González. “Constraints on the automorphism grou p of a curve”. In: J. Théor. Nombres Bordeaux 29.2 (2017), pp. 535–548. [How17] E. W. Howe. “Curves of medium genus with many points” . In: Finite Fields and Their Applications 47 (2017), pp. 145–160. [Kan08] E. Kani. “Endomorphisms of Jacobians of modular cur ves”. In: Archiv der Mathematik 91.3 ...

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[BGGP05] M. H. Baker, E. González-Jiménez, J. González, and B. Poonen. “Finiteness results for modular curves of genus at least 2”. In: Amer. J. Math. 127.6 (2005), pp. 1325–1387. [BGKM20] D. Bartoli, M. Giulietti, M. Kawakita, and M. Monta nucci. “New examples of maximal curves with low genus”. In: Finite Fields and Their Applications 68 (2020), p. 10174...

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