On the counting function of cubic function fields

Victor Ahlquist

REVIEW 3 major objections 4 minor 32 references

On the counting function of cubic function fields

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Strong function-field analogue of Bhargava-Taniguchi-Thorne with a secondary term and O(X^{2/3+ε}) error; the advertised exponent rests on a single unproved 'function field analogue of [BTT, Prop 4.5]' quoted before (5.15), which is load-bearing and needs proof or precise citation. read the letter →

arxiv 2504.12160 v2 pith:DVOBFUAP submitted 2025-04-16 math.NT

Victor Ahlquist This is my paper · ORCID

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keywords functioncountingfieldstermcubicerrorformulamatches

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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 counts cubic extensions of the rational function field F_q(T), the function field analogue of counting cubic number fields. A cubic extension is selected by its discriminant, a power q^M. The main theorem states that the number of such extensions equals an explicit main term plus a secondary term of size Y^(5/6), with an error O(Y^(2/3+epsilon)). This error matches the best known error for cubic number fields, proved by Bhargava, Taniguchi and Thorne. A previous function field result by Kural gave error O(Y^(3/4+epsilon)), after an earlier claim by Zhao was found to contain a gap.

The proof uses the Levi-Delone-Faddeev correspondence, which turns cubic rings into orbits of binary cubic forms under GL_2. The author counts these orbits using geometry of numbers over the local field F_q((1/T)). A sieve removes nonmaximal rings, and finite Fourier transforms, with exact formulas from Taniguchi and Thorne, control the sieve. The paper also proves an exact formula for the number of orbits of cubic forms of fixed discriminant.

A second result specifies the splitting behaviour of finitely many primes, again with error O(Y^(2/3+epsilon)|P|^(2/3)). A third result, using the one-level density of Artin L-functions, shows that for refined counts no asymptotic of the form C_1 Y + C_2 Y^(5/6) + O(Y^(theta+epsilon)|P|^omega) can have theta+omega < 1/2. Because the Riemann Hypothesis is a theorem for function fields, this lower bound is unconditional, unlike the analogous number field result of Cho, Fiorilli, Lee and Sodergren.

Extended reading notes

Core claim

Theorem 1.1, equation (1.4): the number of cubic function field extensions of F_q(T) with discriminant Y = q^M, M >= 0 even, equals (q^2-1)(q^3-1)/(q^4(q-1)) Y - (q^2-1)/q C_2(M) Y^(5/6) + O(Y^(2/3+epsilon)), with C_2(M) explicitly given by a table depending on M mod 3. If this is correct, the error term matches the best-known number field result of Bhargava, Taniguchi and Thorne.

Load-bearing premise

The O(Y^(2/3+epsilon)) error estimate relies on a bound, quoted in Section 5.2 immediately before equation (5.15), called 'the function field analogue of [BTT, Proposition 4.5]': the number of GL_2(R)-orbits of binary cubic forms with r^2 | Disc(y) and |Disc(y)| = Y is << Y/|r|^(2-epsilon). The paper does not prove this bound nor cite a precise function-field source, and this bound is what controls the contribution of non-degenerate dual forms y in the sieve. If the analogue is weaker than claimed, the error term worsens.

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Desk Editor's Note

read the letter

Here's the short version. Ahlquist proves an asymptotic for cubic function fields with a secondary term and error O(X^{2/3+ε}), matching the best number-field result of Bhargava-Taniguchi-Thorne; he also handles splitting-type refinements and gives an unconditional omega-result via the one-level density. The geometry-of-numbers arguments are self-contained and clean, and the exact orbit-count formula (Thm 1.4) is a real new tool. Section 1.2's retraction of the Zhao attribution is honest.

The soft spot is exactly where the stress-test points: at (5.15) the paper invokes 'the function field analogue of [BTT, Proposition 4.5]' — the orbit count bound O(Y/r^{2-ε}) for forms with r^2|Disc — without proof or a precise citation. That bound is genuinely load-bearing. Without the r^{-2+ε} saving, the sieve error term would only come out as something like |fg|^3 times Y, which is far too large; the advertised exponent 2/3 depends on having that saving. The same unproved input is reused in Section 7.2.1, so Theorem 7.2 inherits the issue. It may well be provable by adapting [BST, Lemma 34] or [BTT], but as written it's an assertion at the exact point where the improvement over Kural's O(X^{3/4+ε}) is obtained.

There's also a smaller presentational bug: in the final summation of Section 5.2, the text says 'using δ=2/3', but the correctly tuned parameter is δ=1/3 (the value picked by an earlier exponent balance). With δ=1/3 the final error is still X^{2/3+ε}; with δ=2/3 the X^{2δ} term from the first case blows up. It looks like a typo, but it should be fixed.

The reader's skepticism is otherwise matched: the main and secondary constants are computed from volumes and integrals with no fitted parameters; the final C_2(M) table matches simpler cases; the number-field benchmarks are cited. The omega-result is a genuine unconditional analogue of CFLS. A couple of other bounds are imported without proof (e.g., (5.1)), but those are standard and less dangerous.

For whom: anyone working in arithmetic statistics over function fields, or on Davenport-Heilbronn-type theorems. It's a serious paper that deserves refereeing. If the referee forces a proof or explicit citation of the function-field analogue of [BTT, Prop 4.5], the paper should be accepted without too much further pain. I'd engage with it.

Referee Report

3 major / 4 minor

Summary. The paper studies the asymptotic counting function of cubic extensions of F_q(T), with q not divisible by 2 or 3. Theorem 1.1 gives #F(Y) = ((q^2-1)(q^3-1)/(q^4(q-1)))Y - ((q^2-1)/q) C_2(M)Y^{5/6} + O(Y^{2/3+epsilon}) for Y = q^M, M even, with C_2(M) depending on M mod 3; this matches the error-term quality of Bhargava-Taniguchi-Thorne over Q. Theorem 1.2 and Theorem 7.2 refine the count by imposing splitting types at finitely many primes, with error O(X^{2/3+epsilon}|P_1...P_n|^{2/3}), and Theorem 1.3 proves an omega-result omega+theta >= 1/2 for such refined counts, unconditional because the Riemann Hypothesis holds over function fields. The proofs use geometry-of-numbers methods: an exact orbit-count formula for binary cubic forms with fixed discriminant (Theorem 1.4), a sieve for maximality based on finite Fourier transforms, and a one-level density computation for Artin L-functions. The paper also acknowledges a gap in earlier work of Zhao and compares with independent work of Kural.

Significance. The results, if correct, are significant. The error term O(Y^{2/3+epsilon}) in Theorem 1.1 matches the best-known number-field result of Bhargava-Taniguchi-Thorne and improves Kural's O(Y^{3/4+epsilon}); the explicit secondary-term constants depending on M mod 3 and the refined counting theorem with splitting conditions are new in the function-field setting. The paper is careful in several respects: the main constants are computed from volumes and integrals rather than fitted; Theorem 1.4 is an exact orbit count; the one-level density computation yields an unconditional lower-bound result; and the author is transparent about the prior gap in Zhao's work and the relation to Kural's preprint. The main caveat is that the decisive r-saving estimate used in the sieve is currently asserted rather than proved, so the advertised error-term improvement is conditional until that input is supplied.

major comments (3)

[5.2, before (5.15); 7.2.1] Section 5.2, immediately before Eq. (5.15): the paper invokes 'the function field analogue of [BTT, Proposition 4.5]', asserting that the number of GL_2(R)-orbits of binary cubic forms with r^2|Disc(y) and |Disc(y)|=Y is << Y/|r|^{2-epsilon}. This estimate is neither proved nor traced to a precise function-field reference. It is load-bearing: after the change of variables, (5.15) bounds the non-degenerate contribution by a sum over r|fg of |r|^3 times this orbit count, and the r^{-2+epsilon} saving is what reduces the total to << |fg| and hence to the final O(X^{2/3+epsilon}) after summing over |F| <= X^delta. With only the trivial bound on the orbit count, the |r|^3 factor gives a contribution of size |fg|^3 per divisor, which is far larger than X^{2/3}. The same unproved input is reused in Section 7.2.1, so Theorem 7.2 inherits the gap. I recommend that the authors supply a full proof of this function-field analogue (for instance by adapting the argument in [BTT, Section 4] or [BST, Lemma 34]) or give a precise citation.
[5, Eq. (5.1)] Section 5, Eq. (5.1): the estimate N(W_F cap V(R)_sigma;X) << X/|F|^{2-epsilon} is stated as being 'proven in the same way as [BST, Lemma 34]', without a proof or a function-field citation. This bound is used to truncate the inclusion-exclusion sum at |F| <= X^delta and to produce the O(X^{1-delta+epsilon}) tail error, so it is part of the central error analysis. If the adaptation is straightforward, please provide a brief proof or a precise reference; as written, the reader cannot verify the tail truncation from the cited number-field result alone.
[5.2, choice of delta] Section 5.2, paragraph immediately before the non-degenerate case: the error terms are listed as having exponents 1-delta, 5/6-2delta/3, 2/3 and 2delta, and it is stated that these are minimized at delta = 1/3. However, the concluding sentence of the same section says the sum over |F| <= X^delta is evaluated 'using delta = 2/3'. If delta = 2/3 is used globally, the X^{2delta} term would become X^{4/3}, contradicting the claimed O(X^{2/3+epsilon}); if the intended value is delta = 1/3, the conclusion is consistent. Please correct the inconsistency and state the final choice of delta explicitly.

minor comments (4)

[4.3, end of Proposition 4.3] The proof says that the independence of I^sigma_1(lambda_0) from the choice of v_sigma is 'postponed to the end of the section', but Section 4.5 does not explicitly identify where this fact is proved; the argument in Proposition 4.5 shows an analogous invariance for I'_sigma, so please add a pointer or a short direct argument for I^sigma_1.
[7, around (7.25) and Theorem 7.1] The formal product symbol with phi_ell(n) and the subsequent identification of expressions with their evaluated values is not fully specified for infinite products; please clarify the definition of the infinite formal product and justify the interchanges used in the displayed identities.
[1, Theorem 1.4, Eq. (1.8)] The symbol C_2(ell) appears in Theorem 1.4 without a definition in the statement; it would help to state explicitly that C_2(ell) is the value I'_sigma(q^ell) computed in Proposition 4.5 and related to C_2(M) by the summation over sigma in Section 5.3.
[5.3 and 7, after Proposition 5.1] The removal of reducible maximal rings is summarized as 'cf. [BTT, Lemma 8.1]'; for the refined splitting-condition theorem the same reduction is asserted with 'Removing the reducible forms as before'. Please indicate how the splitting conditions are preserved under this reduction, or give the needed quadratic-field estimate.

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

No numbers are fitted to data or chosen by hand; the constants in Theorems 1.1 and 1.2 are derived from volumes, integrals, and explicit Fourier transforms. The choice delta = 2/3 balances error terms rather than fitting a result. No new particles, forces, or mathematical objects are postulated; the phi_M symbols and formal product in Section 7 are notation for explicit sequences, not invented entities.

assumptions (5)

standard math Riemann Hypothesis for global function fields (Weil's theorem): all roots of the L-function P_L(u) have absolute value q^(-1/2).
Invoked in Section 6 to bound the one-level density by O(log_q Y); this is a theorem over function fields, making Theorem 1.3 unconditional.
domain assumption The function field analogue of [BTT, Proposition 4.5]: the number of GL_2(R)-orbits of binary cubic forms with r^2 | Disc(y) and |Disc(y)| = Y is << Y/|r|^(2-epsilon).
Used in Section 5.2 to bound non-degenerate dual form contributions; quoted without proof or precise citation.
standard math The Fourier transform formulas for omega_P and for splitting-type indicator functions from [TT2, Proposition 1], [TT2, Theorem 11], and [M].
External theorems from the literature, used throughout Sections 5 and 7 to evaluate sieve sums.
domain assumption The bound N(W_F intersect V(R)_sigma; X) << X/|F|^(2-epsilon) for forms nonmaximal at F, 'proven in the same way as [BST, Lemma 34]'.
Used to truncate the inclusion-exclusion sieve in (5.1)-(5.2); the proof is asserted by analogy to BST rather than reproduced.
standard math Riemann-Hurwitz formula and the semilocal/global discriminant relation, including the factor q^a for ramification at P_infinity.
Used in Section 5.3 to relate counts at semilocal discriminant X to global discriminant Y and to connect genus with discriminant.

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Pith. "Pith review of On the counting function of cubic function fields." pith.science (2026). https://pith.science/paper/DVOBFUAP

@misc{pith2026250412160,
  author       = {Pith},
  title        = {Pith review of: On the counting function of cubic function fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DVOBFUAP}},
  note         = {Machine review of arXiv:2504.12160}
}

abstract

We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches the best-known result, due to Bhargava, Taniguchi and Thorne, over $\mathbb{Q}$. Furthermore, we obtain estimates for the refined counting function, where one specifies the splitting behaviour of finitely many primes. Also in this case, our error term matches what is known for number fields. However, in the function field setting, the secondary term becomes more difficult to write down explicitly. Our proof uses geometry of numbers methods, which are especially effective for function fields. In particular, we obtain an exact formula for the number of orbits of cubic forms with fixed absolute discriminant. Moreover, by studying the one-level density of a family of Artin $L$-functions associated to these cubic fields, we prove an unconditional lower bound on the error term in the estimate for the refined counting function. This generalises a conditional result over $\mathbb{Q}$, due to Cho, Fiorilli, Lee and S\"odergren.

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Pith tools

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

[1] [1]

Error estimates for the Davenport-Heilbronn theorems

K. Belabas, M. Bhargava, and C. Pomerance, “Error estimates for the Davenport-Heilbronn theorems”, Duke Math. J., vol. 153, no. 1, pp. 173–210, 2010

work page 2010

[2] [2]

On the Davenport–Heilbronn theorems and second order terms,

M. Bhargava, A. Shankar, and J. Tsimerman, “On the Davenport–Heilbronn theorems and second order terms,” Invent. Math., vol. 193, no. 2, pp. 439–499, 2013

work page 2013

[3] [3]

Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces,

M. Bhargava, A. Shankar, and X. Wang, “Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces,” Preprint, 2015, arXiv:1512.03035

arXiv 2015

[4] [4]

Improved error estimates for the Davenport–Heilbronn theorems,

M. Bhargava, T. Taniguchi, and F. Thorne, “Improved error estimates for the Davenport–Heilbronn theorems,” Math. Ann., vol. 389, no. 4, pp. 3471–3512, 2024

work page 2024

[5] [5]

Hurwitz spaces, Nichols algebras, and Igusa zeta functions,

K. Chang, “Hurwitz spaces, Nichols algebras, and Igusa zeta functions,” Preprint, 2023, arXiv:2306.10446

arXiv 2023

[6] [6]

Omega results for cubic field counts via lower-order terms in the one-level density,

P. J. Cho, D. Fiorilli, Y. Lee, and A. Södergren, “Omega results for cubic field counts via lower-order terms in the one-level density,” Forum Math. Sigma, vol. 10, no. e80, 33 pp., 2022

work page 2022

[7] [7]

Low lying zeros of Artin L -functions

P. J. Cho and H. H. Kim, "Low lying zeros of Artin L -functions", Math. Z., vol. 279, no. 3–4, pp. 669–688, 2015

work page 2015

[8] [8]

On the density of discriminants of cubic fields. II,

H. Davenport and H. Heilbronn, “On the density of discriminants of cubic fields. II,” Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., vol. 322, no. 1551, pp. 405–420, 1971

work page 1971