The geometry of secondary terms in arithmetic statistics

Michael Kural

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The geometry of secondary terms in arithmetic statistics

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Pith's one-line read The paper proves that the number of degree-3 extensions of $\mathbb{F}_q(t)$ with absolute norm of discriminant $q^{2N}$ is $c_1q^{2N}-c_2^i q^{5N/3}+O(N^4q^{3N/2}+1)$, with explicit $q$-dependent constants depending on $N\bmod 3$.

desk verdict First proven secondary term for cubic extensions over F_q(t), with explicit constants and a coherent but long geometric sieve; deserves a serious referee. read the letter →

arxiv 2504.17909 v1 pith:J6ARTDBD submitted 2025-04-24 math.NT math.AG

Michael Kural This is my paper · ORCID

classification math.NTmath.AG MSC 11R5811R4514H6014J26

keywords cubicextensionsfunctionfieldssecondarytermsarithmeticstatisticsdiscriminantsHirzebruchsurfaceselementarytransformationssievemethods

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 the number of degree-3 field extensions of the function field $\mathbb{F}_q(t)$ with absolute norm of discriminant $q^{2N}$ is $c_1 q^{2N} - c_2^i q^{5N/3} + O(N^4 q^{3N/2}+1)$, where $i\equiv N \pmod 3$ and $c_1,c_2^i$ are explicit constants. This is the first proven secondary term for cubic extensions of a function field, the analogue over $\mathbb{F}_q(t)$ of the long-conjectured and later proven secondary correction for cubic extensions of $\mathbb{Q}$. A sympathetic reader should care because it shows that a geometric sieve on ruled surfaces can produce a power-saving asymptotic with a visible correction term, not just a main term. The correction term is periodic in $N$ modulo $3$, so the answer depends on the arithmetic of the base field in a finer way than the heuristic main term alone.

What carries the argument

The argument is carried by a geometric parametrization of Gorenstein triple covers: a degree-3 cover $X\to C$ embeds into the ruled surface $\mathbb{P}(E)$ over its Tschirnhausen bundle $E$ as the zero scheme of a section of $\mathcal{O}_{\mathbb{P}(E)}(3)\otimes \pi^*(\wedge^2 E)^\vee$, i.e. a twisted binary cubic form. Over $C=\mathbb{P}^1$ every such surface is a Hirzebruch surface $F_k=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-k))$, fibered over $\mathbb{P}^1$, and sections become bi-homogeneous polynomials $A_0 x^3+A_1 x^2y+A_2xy^2+A_3y^3$. The count is converted to an inclusion-exclusion sieve over singular sections, and an elementary transformation (a blowup followed by a blowdown of ruled surfaces) turns sections singular at prescribed points into sections vanishing at points on a different Hirzebruch surface, the geometric analogue of a discriminant-reducing identity. The sieve splits into small, medium, and large ranges; the medium range is controlled by a dichotomy (Proposition 7.10) saying that the restriction map $H^0(F_k,\mathcal{O}(3,\ell))\to H^0(D,\mathcal{O}_D)$ is either surjective or has a kernel consisting only of horizontally reducible sections, meaning sections whose zero scheme has a component that is a section of the fibration. The model count is assembled into a rational generating function whose denominator factors reveal the secondary scale $q^{5m}$.

What would settle it

Exhibit one $D$-marking in the medium range with $\ell-2k\ge (\deg D+k)/2$ and $\deg D>\ell$ for which the map $H^0(F_k,\mathcal{O}(3,\ell))\to H^0(D,\mathcal{O}_D)$ is neither surjective nor has a kernel contained in the horizontally reducible sections; this would invalidate Proposition 7.10 and break the medium-range error bound, allowing the error to overwhelm the claimed $q^{5m}$ term.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 9.12, is an asymptotic with a secondary term for the geometric count of cubic covers of $\mathbb{P}^1$ over $\mathbb{F}_q$. Writing $N=3m+i$ with $i\in\{0,1,2\}$, the paper establishes $$\operatorname{Cov}_3(2N)=c_1 $q^{{2N}}$ - c_2^i $q^{{5m}}$ + O($N^{4}$ $q^{{3N/2}}$+1),$$ with $c_1,c_2^0,c_2^1,c_2^2$ explicit constants depending only on $q$, and the same formula governs the count of degree-3 field extensions of $\mathbb{F}_q(t)$ with norm of discriminant $q^{2N}$. The $q^{5m}$ term is the secondary term, and the error is a genuine power saving over it. The theorem holds for every $N\ge 0$ and makes no assumption on the characteristic of $\mathbb{F}_q$, after separately bounding cyclic and inseparable covers.

Load-bearing premise

The load-bearing premise is the medium-range dichotomy: for every choice of points lying one per fiber over a divisor $D$, restricting sections of $\mathcal{O}(3,\ell)$ to $D$ either is as surjective as the dimension predicts, or every section in the kernel is automatically reducible; if this failed for even a small proportion of point choices, the error estimate $O(N^7 q^{4N/3})$ could exceed the $q^{5N/3}$ secondary term.

Editorial extensions

If this is right

The number of degree-3 extensions with norm-of-discriminant $q^{2N}$ is known up to an error of size $O(N^4 q^{3N/2}+1)$, which is small compared with the secondary term $q^{5N/3}$.
The secondary coefficient is periodic in $N$ modulo $3$, so the counts in the three residue classes of $N$ have their own explicit constants.
The result holds in every characteristic of $\mathbb{F}_q$; cyclic and inseparable covers are controlled separately and contribute only $O(Nq^N+1)$.
The geometric sieve yields a power-saving error, not merely an asymptotic, for cubic extensions over $\mathbb{F}_q(t)$.
The leading constant agrees with the previously known main-term constant for cubic covers of a nice curve over $\mathbb{F}_q$ when specialized to $\mathbb{P}^1$.

Reading between the lines

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

If the medium-range dichotomy of Proposition 7.10 is as universal as claimed, the same elementary-transformation sieve should yield secondary terms for other fixed-degree covers over function fields, with the role of $F_k$ played by projective bundles of higher rank.
The period-3 behavior in $N$ likely reflects the arithmetic of cube roots of unity in $\mathbb{F}_q$; examining how the constants $c_2^i$ vary with $q$ modulo $3$ would clarify whether the correction has a purely local origin.
The size scale $q^{5N/3}$ gives a precise numerical prediction for the kind of 'secondary stability' one might look for in the homology of the associated Hurwitz spaces.
A direct computer enumeration for a moderate pair $(q,N)$ with $q^{5N/3}$ dominating the error bound could test the explicit constants, though the allowed error means the secondary term is visible only when $q$ is large relative to $N$.

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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

The headline result is what it claims to be: a secondary term with explicit periodic constants for the count of cubic extensions of F_q(t). That is genuinely new. Prior work over Q by Bhargava-Shankar-Tsimerman and Taniguchi-Thorne, and Gunther's main term over function fields, left this open. The paper also argues, in detail, that Zhao's thesis proof is flawed; I did not see a reason to dispute that, and the construction here is different enough to stand on its own.

What is actually done well: the parametrization via Casnati-Ekedahl and the elementary transformation package is carefully assembled, the sieve retains the necessary sheaf-theoretic data, and the generating function extraction of c1 and c2^i is explicit. No fitted constants appear, and the leading constant matches Gunther's main term—a consistency check that would catch any serious misnormalization. The root-counting estimates in the small, medium, and large ranges are tracked with explicit error terms, and the final error O(N^4 q^{3N/2}) is stronger than needed.

The soft spot is the one the reader identified: Proposition 7.10, the medium-range dichotomy for D-markings. That proposition is load-bearing: if it failed for a positive proportion of markings, the medium-range error E_medium could swallow the secondary term. The proof is structural and depends on the minimal (D,h)-curve bound from Lemma 7.9. I read through the argument and could not find a gap; the case split in Proposition 7.3 is sound, and the edge case on the directrix is handled. Still, this is the part most likely to have a hidden issue, simply because it is a global statement about all D-markings and is proven by geometric reduction rather than by a robust numerical estimate. The paper's own Section 7.1 acknowledges the narrowness of this type of argument, which is honest but not a fix.

Minor comment: the thesis is long and partially terse at the coefficient extraction step in Section 9. A referee would need to check the partial fraction algebra carefully, but it looks right from what the paper shows.

Who this is for: arithmetic statisticians working on discriminants over function fields, and anyone interested in geometric sieves with elementary transformations. It deserves a serious refereeing: the main theorem is important within the field, and the proof is substantively verifiable. I would engage with it.

Referee Report

1 major / 4 minor

Summary. The paper proves an asymptotic for the number of degree 3 field extensions K/F_q(t) with absolute norm of discriminant q^{2N}, namely c_1 q^{2N} - c_2^i q^{5m} + O(N^4 q^{3N/2}+1), where N=3m+i and the c_i are explicit q-dependent constants. The proof uses the Casnati--Ekedahl/Miranda parametrization of Gorenstein triple covers by twisted binary cubic forms, a sieve over singular sections on Hirzebruch surfaces organized into small, medium, and large ranges, and a generating-function extraction of the main and secondary terms. The critical medium-range step is Proposition 7.10, which asserts that in the medium range the restriction map to a D-marking is either surjective or has kernel consisting only of horizontally reducible sections. If correct, this is the first proven secondary term for cubic extensions of a function field.

Significance. The result is significant for arithmetic statistics over function fields: it supplies explicit, non-fitted constants and a power-saving error for a count where only the main term was previously known. The proof is coherent: I checked the medium-range dichotomy in Proposition 7.10, the error estimates in Proposition 9.2, and the partial fraction extraction around (9.11); the leading constant agrees with Gunther's main term as a consistency check, and no constant is fitted to Cov_3(2N). The stress-test concern about Proposition 7.10 does not land: Lemma 7.9 supplies a minimal (D,h)-curve with h <= (deg D + k)/2, and Proposition 7.3 yields exactly the required surjectivity-or-fixed-component dichotomy. The principal blemish is a misstatement of the secondary exponent in Theorem 1.1 and the abstract.

major comments (1)

[Theorem 1.1 and Abstract] The main theorem is stated with the secondary term c_2^i q^{5N/3}, but Theorem 9.12 and the coefficient extraction in Section 9 give c_2^i q^{5m} with N=3m+i (constants as in (9.11)). For i=1,2 these two expressions are not equal, and the constants in (9.11) do not absorb the fractional power q^{5i/3}. Since the abstract and Theorem 1.1 are what most readers will quote, this needs to be corrected to q^{5m} (or the theorem restated with the periodic secondary term) in both places.

minor comments (4)

[Section 5.1, Eq. (5.13)] The displayed definition of the zeta function has (1 - T^{-deg P})^{-1}, which is not a power series in T and contradicts the earlier convention Z_C(T)=prod_P (1-T^{deg P})^{-1}; the exponent should be +deg P.
[Section 9, after Proposition 9.8] The partial fraction decomposition determining the constants in (9.11) is stated without derivation; since these constants are the main output, adding a few lines or a reference for the decomposition would make the extraction easy to verify.
[Section 8.4] In the proof of Proposition 8.9 there is a typo: 'H0(FkL)n' should read 'H0(F_k,L)_{nz}' or similar.
[General notation] Theorem 1.1 introduces i but not m; the statement should explicitly define m by N=3m+i, as done in Theorem 9.12, to avoid ambiguity in the exponent q^{5m}.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the secondary coefficient is computed from an explicit generating function, not fitted to the count.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity audit. The main count Θ(C,N) is rewritten via the Casnati–Ekedahl parametrization as a weighted sum over sections and then converted, using elementary transforms, into sums of explicit root-counting functions Rir and Φir (Propositions 3.11, 5.7, and 5.14). The model function bΦir is obtained from an exact small-range count, not from Cov3(2N): in the small range Φir(ℓ,k,D) is explicitly q^{4ℓ−6k+4−degD} − q^{3ℓ−3k+3} when ℓ−3k ≥ degD and 0 otherwise (Proposition 8.6, Definition 9.1). The constants c1 and c2^i are then extracted from the rational generating function bG(T) = (1−T)(1−qT)bF(T), whose coefficients are algebraic functions of q alone (Propositions 9.8 and 9.10, equation (9.11)). No parameter is fitted to the target count, and no term of the final asymptotic is inserted as an input. The comparison with Gunther's main term is explicitly a consistency check, not a source of the constant. The medium-range dichotomy (Proposition 7.10) is load-bearing but is a genuine geometric statement about restriction maps on Hirzebruch surfaces, proved from minimal (D,h)-curves and cohomological vanishing; it is not an assumed version of the final statistic. The paper's use of Zhao's work is also not circular: the cited identities are either reproved in the text or are standard indicator-function decompositions, and no load-bearing claim depends on a uniqueness theorem asserted only by the same authors. Overall, the derivation of the secondary term does not reduce to its own inputs, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No constants are fitted to data, and no new physical entities are introduced. The elementary transformation Elm_D and the model functions bPhi, bPsi, bTheta are definitions and bookkeeping devices, not independent postulates.

assumptions (7)

standard math Casnati-Ekedahl-Miranda equivalence of primitive trisections and Gorenstein triple covers, including discriminant compatibility.
Invoked in Sections 2 and 3 to convert every smooth cover to a section s in H0(P(E), ME) and to identify discriminant degrees.
standard math Birkhoff-Grothendieck splitting: every rank 2 vector bundle on P1 splits, so every ruled surface is some Hirzebruch surface F_k.
Section 6 uses this to reduce the sum over vector bundles to pairs (ell,k) with 2ell - 3k = N.
standard math Classical discriminant formula for binary cubic forms and its identification with the triple-cover discriminant.
Appendix A computes this locally and it is used to translate branch degree into degrees of the vector bundle E.
domain assumption A smooth irreducible X over a nice curve implies f : X to C is Gorenstein, so X embeds in P(E).
Proposition 2.6 uses this so all covers of interest lie in the smooth irreducible part of the Casnati-Ekedahl correspondence.
domain assumption Assumption 5.10: either C is P1 or char F_q is not 3, so the generating function has only nonnegative powers.
This restricts the generating function step; the final theorem only needs C = P1, so the main result is unconditional.
standard math Finiteness of contributing vector bundles and divisors for fixed N.
Propositions 12.2 and 12.3 ensure the sieve sums and generating function manipulations are finite.
standard math Zeta function of P1 satisfies Z(T) = 1 divided by (1-T)(1-qT).
Used in Sections 5.1 and 6.6 to convert the normalized generating function back to the cover count.

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Cite this review

Pith. "Pith review of The geometry of secondary terms in arithmetic statistics." pith.science (2026). https://pith.science/paper/J6ARTDBD

@misc{pith2026250417909,
  author       = {Pith},
  title        = {Pith review of: The geometry of secondary terms in arithmetic statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6ARTDBD}},
  note         = {Machine review of arXiv:2504.17909}
}

abstract

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field $\mathbb{F}_q(t)$ of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of discriminant equal to $q^{2N}$ is $c_1 q^{2N} - c_2^{i} q^{5N/3} + O_{\varepsilon}\left(q^{(3/2+\varepsilon)N}\right)$, where $c_1$ and $c_2^{i}$ are explicit constants and $c_2^{i}$ only depends on $N\pmod{3}$. This builds on the work of Bhargava-Shankar-Tsimerman and Taniguchi-Thorne, who proved the existence of a secondary term for the count of cubic extensions of $\mathbb{Q}$ with bounded discriminant. Our approach uses a parametrization of Miranda and Casnati-Ekedahl, which can be seen as a geometric version of the classical parametrization by binary cubic forms used by Davenport-Heilbronn. This allows us to count and sieve for smooth curves embedded in Hirzebruch surfaces, in the same spirit as Zhao and Gunther.

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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

On the counting function of cubic function fields
math.NT 2025-04 conditional novelty 7.0 of 10

The number of cubic extensions of F_q(T) of discriminant q^M is C_1 q^M - C_2(M) q^(5M/6) + O(q^((2/3+epsilon)M)), with explicit constants and an unconditional omega-result for refined counts.

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Pith tools

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

[1] [1]

The distribution of class groups of function fields

Jeffrey D. Achter. “The distribution of class groups of function fields”. In: J. Pure Appl. Algebra 204.2 (2006), pp. 316–333. issn: 0022-4049,1873-1376. doi: 10 . 1016 / j . jpaa . 2005 . 04 . 003. url: https://doi.org/10.1016/j.jpaa.2005.04.003

doi:10.1016/j.jpaa.2005.04.003 2006

[2] [2]

On the Davenport-Heilbronn theorems and second order terms

Manjul Bhargava, Arul Shankar, and Jacob Tsimerman. “On the Davenport-Heilbronn theorems and second order terms”. In: Invent. Math. 193.2 (2013), pp. 439–499. issn: 0020-9910,1432-1297. doi: 10.1007/s00222-012-0433-0 . url: https://doi.org/10.1007/s00222-012-0433-0

work page doi:10.1007/s00222-012-0433-0 2013

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