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Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves

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Pith's one-line read This paper uses finite-field sampling to conjecture that the non-ordinary locus of the hyperelliptic moduli space is geometrically irreducible for all genus at least 3.

desk verdict A reproducible computational heuristics paper with plausible conjectures about p-rank strata, but the sampling bias (and a false assertion about rational branch points) keeps the evidence from being decisive. read the letter →

arxiv 2506.06457 v1 pith:7MWSWTYQ submitted 2025-06-06 math.AG math.NT

Thomas Bouchet , Erik Davis , Steven R. Groen , Zachary Porat , Benjamin York This is my paper
classification math.AGmath.NT MSC 14Q0514H1014G1514G17
keywords p-rankstratahyperellipticcurvesmodulispacenon-ordinarylocusLang-WeilboundsHasse-Wittmatricesgeometricirreducibilityfinitefieldcomputations
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

This paper uses computation over finite fields to estimate the number of irreducible components of the $p$-rank strata of the moduli space of hyperelliptic curves, where the $p$-rank $f$ measures the $p$-torsion of the Jacobian and the stratum $H_g^f$ collects curves with $p$-rank at most $f$. Sampling curves over $\mathbb{F}_{p^r}$, computing their $p$-ranks via Hasse-Witt matrices, and applying Lang-Weil bounds, the authors estimate how many components are defined over each finite field. The data supports the conjecture that $H_g^{g-1}$ and $H_g^{g-2}$ are geometrically irreducible (irreducible over the algebraic closure) for all $g\ge3$ and $p\ge3$, and leaves open whether $H_g^f$ is irreducible for all $1\le f\le g$. In contrast, the $p$-rank $0$ locus $H_3^0$ appears to split into more geometric components as $p$ grows. These statements would answer several component-count questions raised in [AP11] and generalize the known cases $g\le2$ and $f=g$.

What carries the argument

The engine is the Lang-Weil heuristic: for a sample of size $s$ from $H_g(\mathbb{F}_{p^r})$, the fraction of curves with $p$-rank at most $f$ should be roughly $c_{g,f,p,r}p^{-r(g-f)}$, so multiplying the observed fraction by $p^{r(g-f)}$ estimates $c_{g,f,p,r}$. The $p$-ranks are computed from a Hasse-Witt matrix: for a model $y^2=f(x)$ with $f^{(p-1)/2}=\sum_m c_m x^m$, the matrix entries are $c_{jp-i}$, following the formulas in [Yui78] and [AH19]. Two sampling constructions supply the curves: a quasi-affine family $\mathcal{F}_{g,q}$ of pointed curves that are unique up to geometric isomorphism, and an enumeration of curves whose branch points all lie in the ground field (Galois type $(1,\ldots,1)$), adapted from [How25]. A technical fact from [GV08], that extra automorphisms form a locus of codimension $g-1$, justifies removing such curves from samples.

What would settle it

Enumerate all genus-3 hyperelliptic curves over a small prime field such as $\mathbb{F}_{13}$, compute their $p$-ranks, and compare the empirical ratio $M_{3,2,13,1}$ with the actual number of geometric components of $H_3^2$; a deviation beyond the Lang-Weil error, or the existence of two $\mathbb{F}_{13}$-points in different geometric components of $H_3^2$, would refute Conjecture 4.3.

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Extended reading notes

Core claim

The paper's central claim is a pair of conjectures backed by finite-field evidence: for every $g\ge3$ and every odd prime $p$, both the non-ordinary locus $H_g^{g-1}$ and the next stratum $H_g^{g-2}$ are geometrically irreducible, and for every $1\le f\le g$ the stratum $H_g^f$ is probably irreducible as well. The main evidence is the estimate $M_{g,f,p,r}=p^{r(g-f)}N/s$, where $N$ counts sampled curves over $\mathbb{F}_{p^r}$ with $p$-rank at most $f$ among $s$ total curves; the Lang-Weil bounds justify treating this estimate as the number of components defined over $\mathbb{F}_{p^r}$. Taking the maximum of these estimates over several values of $r$ approximates the number of geometric components, since every geometric component appears over some finite field. The data also point in a different direction for $p$-rank $0$: the number of geometric components of $H_3^0$ appears to grow with $p$, generalizing the known behavior for genera $1$ and $2$.

Load-bearing premise

The load-bearing premise is that curves drawn from the restricted families are representative of the full moduli space, so the fraction of the sample with $p$-rank at most $f$ agrees with the true fraction in $H_g(\mathbb{F}_{p^r})$ up to the Lang-Weil error; if the sampling is biased, the estimated component counts do not reflect the true numbers.

Editorial extensions

If this is right

  • If Conjecture 4.3 holds, the unique geometric component of $H_g^{g-1}$ would contain the moduli point of a chain of elliptic curves, realizing the clutching-locus corollary from [AP11, Section 3.7].
  • If Conjecture 5.1 holds, the stratum $H_g^{g-2}$ is geometrically irreducible for all $g\ge3$ and odd $p$, extending the irreducibility pattern one step further down.
  • An affirmative answer to Question 5.2 would give a complete list, $H_g^f$ irreducible for every $1\le f\le g$, and would answer the motivating question from [AP11].
  • If the number of components of $H_3^0$ grows without bound with $p$, it would match the known behavior for $H_1^0$ and $H_2^0$ and would support the analogous Question 5.4 for all genera.
  • Together these statements would generalize the solved cases $g\le2$ and the ordinary case $f=g$, yielding a conjectural full description of component counts for all $p$-rank strata of $\mathcal{H}_g$.

Reading between the lines

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

  • The anomalous excess of non-ordinary curves over $\mathbb{F}_p$ for $p\equiv3\pmod4$ reported in Remark 4.1 suggests that restricted sampling can become unrepresentative, so a natural robustness check is to run the family method on those primes and compare with full-model counts.
  • If Question 5.2 has an affirmative answer, the hyperelliptic locus would meet every component of the abelian-variety stratum $\mathcal{A}_g^f$, which is known irreducible for $g\ge3$ by [Cha05], strengthening what the Torelli map alone gives for hyperelliptic Jacobians.
  • The heuristic could be calibrated against strata with known component counts, such as $H_1^0$ and $H_2^0$, to see how quickly the estimates converge before they are trusted in unknown higher-genus cases.
  • A sharper test of the geometric-component interpretation is to fix a small genus and a large prime and compute $M_{g,f,p,r}$ for increasing $r$, checking that the maximum over $r$ stabilizes.
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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

3 major / 5 minor

Summary. The paper proposes a computational heuristic for the number of geometric irreducible components of the p-rank strata H_g^f of the moduli space of smooth hyperelliptic curves of genus g in characteristic p ≥ 3. The authors sample hyperelliptic curves over finite fields F_{p^r} using two methods: a 'family method' based on pointed curves with a rational branch point, and a 'Galois type method' that enumerates curves whose branch points all split completely. For each sampled curve they compute the p-rank via the Hasse-Witt matrix, and then use the Lang-Weil bound to convert the observed proportion of curves with p-rank at most f into an estimate M_{g,p,r} for the number of components c_{g,f,p,r}. On this basis they conjecture that H_g^{g-1} is geometrically irreducible for g ≥ 3 and p ≥ 3, that H_g^{g-2} is irreducible, that H_g^f may be irreducible for all 1 ≤ f ≤ g, and that the number of components of H_3^0 grows with p.

Significance. If the conjectures hold, they would answer questions raised by Achter and Pries and would generalize the known cases g ≤ 2. The computational core—Hasse-Witt matrix computation and Lang-Weil counting—is standard and appears to be implemented correctly, and the authors provide public Magma scripts and data, which is a strength for reproducibility. However, the significance of the paper as evidence for the conjectures depends entirely on whether the sampled families are representative of the full moduli space H_g(F_q). The paper itself contains a documented failure of the analogous assumption in Remark 4.1, and the family method samples only curves with a rational branch point. These issues are load-bearing for the central claims.

major comments (3)
  1. [§3.1, Eq. (2)] The assertion in §3.1 that "Over F_p, every hyperelliptic curve possesses a rational branch point" is false. A curve y^2 = f(x) with deg f = 2g+2 and f squarefree with no F_q-rational root has no F_q-rational branch point; such curves exist for all g ≥ 2 and all q. For large q, the number of effective degree-(2g+2) divisors on P^1 with no F_q-point is asymptotically e^{-1} times the total, so the omitted part is a positive proportion of the relevant moduli points, not a negligible locus. The family F_{g,q} therefore samples only curves whose branch divisor has an F_q-point, not a sample S ⊂ H_g(F_q) as required for the Lang-Weil ratio in Eq. (2). Consequently M_{g,p,r} estimates the component count of the p-rank stratum in the rational-branch-point locus (more precisely, in the marked-branch-point cover), not the integer c_{g,f,p,r} used in Conjectures 4.3 and 5.1. To use Eq. (2) as stated, the authors need a validation experiment comparing the p-rank distribution on F_{g,q} with the full distribution on H_g(F_q) for small q where full enumeration is feasible.
  2. [§3.2, Remark 4.1] The claim that Lang-Weil bounds apply to samples of Galois type (1,...,1) is not justified. The subset of (Sym^{2g+2}P^1 \ Δ_W)/PGL2 corresponding to branch points all defined over F_q is not the F_q-point set of an algebraic subvariety; Lang-Weil counts the F_q-points of an algebraic variety, and the map to H_g does not identify the F_q-points of the quotient with the F_q-points of this subset, as the text itself acknowledges when it says "the converse does not hold in general." Remark 4.1 is direct evidence that this is not a harmless technicality: over F_p with p ≡ 3 mod 4, the method produces M_{g,p,1} between 1.5 and 4 instead of near 1, and the data are then discarded without explanation. Since the same mechanism could distort the family method or the data for r > 1, the evidence for Conjecture 4.3 cannot be called strong without a mechanism or a validation experiment.
  3. [§4.3, §5.1] The central inferences for the non-ordinary locus and for H_g^{g-2} rely on the assumption that the sample ratio is stable across p and r. The data in Table 5, however, show very large fluctuations for g ≥ 6 (minima as low as 0.000–0.2 and maxima up to 2.37, with standard deviations of order 0.25–0.31) for the g−2 stratum. While the authors attribute this to rarity of low p-rank curves, the fluctuations are also consistent with the sampling bias described above; in particular, the medians hovering near 1 do not distinguish "one component of H_g^f" from "one component of a biased subfamily." The paper should either provide a direct comparison with an unbiased sample or substantially weaken the claim that these data provide strong evidence for Conjectures 4.3 and 5.1.
minor comments (5)
  1. [§2.1] "hyperellipitic" should be "hyperelliptic".
  2. [§3.1, Proposition 3.2] The scaling parameter u should lie in F_q, not F_p, since the polynomials f_i are defined over F_q and the isomorphism must be defined over F_q.
  3. [§2.2, Algorithm 3.8] The indices for the Hasse-Witt matrix are given as 0 ≤ i,j ≤ g, but the matrix is g×g; the indices should be 1 ≤ i,j ≤ g. This is presumably a typo, but it should be corrected to avoid ambiguity in the implementation.
  4. [Introduction] The expressions "Conjecture A (Theorem 4.3)" and similar should be replaced by "Conjecture 4.3" or "Conjecture 5.1," since these are not theorems in the paper.
  5. [§3.1, Proposition 3.4] The proof refers to "Theorem 3.2" and "Theorem 3.3" for statements that are labeled Propositions 3.2 and 3.3; the cross-references should be unified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: component-count estimates are Lang-Weil outputs, not fitted inputs; no target result is fed back into the derivation.

full rationale

The paper's derivation chain is not circular. The component-count heuristic (Eq. (2)) is obtained by applying the external Lang-Weil theorem (Thm. 2.4) to the ratio of sampled p-rank counts; the target integer c_{g,f,p} appears only as the coefficient being estimated, never as an input. The Hasse-Witt computation (Prop. 2.3) is an independent standard result, and the conjectures in Sec. 4.3 and Sec. 5 are explicitly inductive ('the data warrants', 'we feel confident'), not logical consequences of a fitted parameter. No self-citation is load-bearing: [BDGPY25] is the authors' own code/data repository, but it is reproducible and parameter-free, and the tables reproduce the evidence. The main caveats are correctness risks, not circularity: the sample is drawn from a restricted family (rational branch points or Galois type (1,...,1)), and Remark 4.1 documents a concrete failure of the Galois-type restriction over F_p; the claim that Lang-Weil applies to such restricted samples is not proved, and the assertion that every hyperelliptic curve over F_p possesses a rational branch point is false in general. These issues affect the reliability of the heuristic, but no equation reduces the target to its own input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim depends on Lang-Weil counting (standard), the Hasse-Witt computation (standard), and two ad hoc assumptions about sampling representativeness. The paper introduces no new mathematical entities; the family F_{g,q} is a subset of an existing moduli space. The free parameter R=4 is an arbitrary truncation of the field extension degree.

free parameters (1)
  • R (max extension degree) = 4
    The heuristic c_{g,f,p} = max_{r <= R} c_{g,f,p,r} requires R large enough so every geometric component is defined over F_{p^r}. The authors stop at r=4 due to computational limits, an ad hoc cutoff with no evidence that it is sufficient for all g and p.
assumptions (5)
  • standard math Lang-Weil bound: for an irreducible variety V of dimension d over F_q, #V(F_q) = c q^d + O(q^{d-1/2})
    Theorem 2.4 is the basis for equation (1).
  • standard math The p-rank of C equals the rank of the g-th semilinear power of the Hasse-Witt matrix
    Invoked in Section 2.2 via [Oda69]; used in Algorithm 3.8.
  • domain assumption H_f^g is pure of dimension g+f-1 and dense in its Deligne-Mumford compactification
    Quoted from [GP05, Prop 2.1] and [AP11, Lemma 3.2]; needed to know that the Lang-Weil ratio counts components of the open stratum.
  • ad hoc to paper The sampled families F_{g,q} and Galois-type (1,...,1) subsets are equidistributed across p-rank strata
    Section 3.1 and 3.2 assume the sample ratio #(S cap H_f^g)/#S approximates the full moduli ratio. No proof is given; Remark 4.1 shows this can fail for the Galois-type method over F_p.
  • ad hoc to paper Every hyperelliptic curve over F_p has a rational branch point
    Section 3.1 states this to justify the family method. It is false for g >= 2: e.g., a genus-2 curve y^2 = f(x) with irreducible degree-6 f over F_p has no F_p-rational branch points. This axiom is wrong.

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Pith. "Pith review of Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves." pith.science (2026). https://pith.science/paper/7MWSWTYQ

@misc{pith2026250606457,
  author       = {Pith},
  title        = {Pith review of: Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7MWSWTYQ}},
  note         = {Machine review of arXiv:2506.06457}
}
abstract

Let $\mathcal{H}_g$ denote the moduli space of smooth hyperelliptic curves of genus $g$ in characteristic $p\geq 3$, and let $\mathcal{H}_g^f$ denote the $p$-rank $f$ stratum of $\mathcal{H}_g$ for $0 \leq f \leq g$. Achter and Pries note in their 2011 work that determining the number of irreducible components of $\mathcal{H}_g^f$ would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various $p$-rank strata. Our strategy involves sampling curves over finite fields and calculating their $p$-ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera $g> 1$. The data also leads us to conjecture that the moduli space $\mathcal{H}^{g-2}_g$ is irreducible and suggests that $\mathcal{H}^f_g$ is irreducible for all $1 \leq f \leq g$. We conclude with a brief discussion on $\mathcal{H}^0_g$.

Figures

Figures reproduced from arXiv: 2506.06457 by the authors.

Figure 1
Figure 1. Values of M3,p,r for each prime p [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. demonstrates how the error of M3,1,p,1 behaves. The spread of the values of M3,1,p,1 widens as p grows, simply because the curves of p-rank at most g − 2 become increasingly rare. For example, in the g = 3 case, there are 1001 curves with p-rank f ≤ 1 when p = 103, but only 10 such curves when p = 883. Given this explanation along with the data available, we feel confident making the following conjecture. Conjecture… view at source ↗
Figure 3
Figure 3. Values of M3,0,p,1 for each prime p, collected using both sampling methods [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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Cited by 1 Pith paper

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

  1. The Torelli locus and Newton polygons

    math.AG 2025-08 conditional novelty 3.0 of 10

    A survey of the Torelli locus and Newton polygons that corrects a previously erroneous genus computation and thereby proves new infinite families of supersingular curves of genus δp(p−1)/2.

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Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 1 Pith paper

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    Yui,On the Jacobian varieties of hyperelliptic curves over fields of characteristic p >2, J

    (MR2514094) [Yui78] N. Yui,On the Jacobian varieties of hyperelliptic curves over fields of characteristic p >2, J. Algebra52(1978), no. 2, 378–410. (MR491717) (IR)REDUCIBILITY OFp-RANK STRATA 19 Thomas Bouchet | Laboratoire J.A. Dieudonné, Université Côte d’Azur, France Email address:thomas.bouchet@univ-cotedazur.fr URL:https://sites.google.com/view/thom...

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