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Rational points in a family of conics over $\mathbb{F}_2(t)$

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Pith's one-line read Almost all fibres in a conic family over F2(t) have no rational point

desk verdict New sharp asymptotic for rational points in a conic family over F_2(t); the load-bearing local symbol computation holds up. read the letter →

arxiv 2412.14693 v2 pith:4POWUUWT submitted 2024-12-19 math.NT

Daniel Loughran , Judith Ortmann This is my paper · ORCID
classification math.NT MSC 14G0514F22
keywords rationalpointsconicbundlesfunctionfieldsheightzetaTauberiantheoremcharacteristic2localsymbolsBrauergroup
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 establishes an exact asymptotic for the number of fibres $y\in \mathbb{F}_2(t)$ of height exactly $B$ for which the conic $x_0^2+x_0x_1+y x_1^2 = t x_2^2$ has a rational point. The count is $c B^2/(\log B)^{1/2} + O(B^2/(\log B)^{3/2})$ for $B=2^M$, with $c$ an explicit product of local factors. Since the parameter line has about $B^2$ points of height $B$, the result implies that the proportion of solvable fibres tends to zero, matching the qualitative rarity known over the rational numbers. The proof treats the height zeta function, whose singularities are branch points rather than poles, and supplies the needed Tauberian theorem over function fields.

What carries the argument

The height zeta function $Z(s) = \sum_{y\in K,\ C_y(K)\neq\emptyset} H(y)^{-s}$ is analysed by Poisson summation over additive adelic characters. Local solubility is detected through the characteristic-2 local symbol $[a,b)_\omega$ introduced in [Ser79], and the paper shows that every nontrivial automorphic character gives a vanishing Fourier transform except the special character $\psi_{1/t}$, whose contribution coincides with the trivial character; this doubling produces the factor $4$ in the constant. The zeta function then factors as $Z(s)=4\hat{f}_H(s;1)$, and the proof shows that $Z(s)\zeta_K(s-1)^{-1/2}$ is a nonzero absolutely convergent Euler product on $\mathrm{Re}(s)>3/2$, so the only singularities on the line $\mathrm{Re}(s)=2$ are branch points of order $1/2$. A Tauberian theorem for Dirichlet series in $q^{-s}$ converts this branch singularity into the asymptotic, with a Hankel contour calculation supplying the factor $1/\Gamma(1/2)=1/\sqrt{\pi}$.

What would settle it

Compute the symbol $[\omega^{-n},t)_\omega$ directly for one explicit place, say $\omega=t+1$ and $n=1$, by checking whether $t$ is a norm from the Artin-Schreier extension $x^2-x-(t+1)^{-1}$ over $\mathbb{F}_2(t+1)$ via a finite computation in that local field; the paper predicts the value $1$. A cheaper cross-check is to enumerate the finite residue classes of valuation $-k$ for $k=2$ at the place $t+1$ and count how many corresponding fibres are locally solvable, where Lemma 3.5 predicts exactly half, with the exceptional $k=1$ count $2^{\deg\omega-1}-1$.

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Extended reading notes

Core claim

On the global function field $K=\mathbb{F}_2(t)$, consider the conic bundle $x_0^2+x_0x_1+y x_1^2 = t x_2^2$ over the affine line, with height $H(y)=\prod_\omega \max\{1,|y|_\omega\}$. The paper proves that among $y$ of height $B=2^M$, the number with $C_y(K)\neq\emptyset$ is $c B^2/(\log B)^{1/2} + O(B^2/(\log B)^{3/2})$, where $c = 4(2\log 2/\pi)^{1/2} \prod_\omega (1-2^{-\deg \omega})^{1/2} c_\omega$, with $c_\omega=3/4$ for the two places $t$ and $t^{-1}$ and an explicit closed form for every other place. This gives $0\%$ of fibres solvable, and the exponent $1/2$ in the logarithmic factor is sharp. The paper also shows that this family has no smooth proper model and that the relevant Brauer group element is not tame, so the earlier number-field approach does not apply; the logarithmic factor instead arises from a characteristic-2 local symbol computation at the non-reduced fibre at infinity.

Load-bearing premise

The count rests on the value of a characteristic-2 local symbol, an analogue of the Hilbert symbol: the claim that $[\omega^{-n},t)_\omega = 1$ for every place $\omega$ other than $t$ and $t^{-1}$, a value derived indirectly from global reciprocity and an explicit nonsolubility argument at the place $t$; if this value were different, the leading constant and possibly the logarithmic exponent would change.

Editorial extensions

If this is right

  • Since there are about $B^2$ parameters of height $B$, the theorem implies that $0\%$ of the fibres have a rational point, in the same qualitative sense as the number-field result.
  • The logarithmic exponent $1/2$ is sharp: the error term is half a power of logarithm smaller, so the leading term is not an artifact of the counting method.
  • The height zeta function has a branch point rather than a pole at $s=2$; the Tauberian theorem proved here gives a template for other function-field families with square-root singularities.
  • At the places $t$ and $t^{-1}$, exactly half of the residue classes are solvable, while at all other places, fibres of negative valuation split into equal solvable and unsolvable halves for $k>1$.
  • The family admits no smooth proper model over $K$, so the result lies outside the prior number-field framework; the naively computed exponent $1/2$ from the non-split fibre at infinity still agrees with the theorem.

Reading between the lines

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

  • Beyond the paper: the same single-extra-character mechanism, in which only $\psi_{1/t}$ contributes to the Poisson sum, should occur for other families over $\mathbb{F}_2(t)$ defined by an Artin-Schreier norm form, predicting a universal factor of $2$ in the leading constant.
  • Beyond the paper: the explicit constant can be checked numerically at small $M$, since $B=2^M$ predicts $N \sim c\,2^{2M}/(M\log 2)^{1/2}$; the local factors $c_\omega$ can also be verified independently by finite-field residue counts.
  • Beyond the paper: replacing the coefficient $t$ by another element of $K$ likely changes only the local factors and the special character, suggesting a family of asymptotic formulas governed by the same branch-point Tauberian theorem.
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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

1 major / 4 minor

Summary. This paper studies the distribution of fibres y in K = F2(t), ordered by the naive height, for which the conic C_y: x_0^2 + x_0 x_1 + y x_1^2 = t x_2^2 has a K-point. Theorem 1.1 gives N(A1_K, pi, B) = c B^2 / (log B)^{1/2} + O(B^2 / (log B)^{3/2}) for B = 2^M, with an explicit Euler-product constant c. The proof has four parts: a Tauberian theorem for Dirichlet series over function fields with branch-point singularities (Theorem 2.2), proved by Hankel contours; a complete local-solubility analysis using Serre's characteristic-2 local symbol (Section 3), including the delicate computation in Lemma 3.4; an adelic Poisson-summation computation of the height zeta function (Section 4), in which only the trivial character and one nontrivial automorphic character contribute; and an application of the Tauberian theorem. The paper also constructs a regular proper model, shows that no smooth proper model exists, and discusses the relation to the Loughran-Smeets framework.

Significance. If correct, Theorem 1.1 is a substantial contribution: it gives the first sharp asymptotic over global function fields for a conic-bundle family in which the proportion of soluble fibres tends to zero, with a logarithmic exponent that cannot be obtained from the existing Loughran-Smeets framework because the fibre at infinity is not geometrically reduced. The leading constant is fully explicit and parameter-free, an Euler product of local Fourier transforms, and the half-power of the logarithm is traced to a half-measure local solubility set rather than to the splitting of a separable quadratic extension. The paper is unusually complete: the Tauberian theorem is proved rather than quoted, the local-symbol computation in Lemma 3.4 is checked place by place and via reciprocity, and the character decomposition in Lemma 4.6 is clean. I specifically stress-tested Lemma 3.4, the admitted hard step: at the place t the mod-t norm form is anisotropic, at t^{-1} Hensel's lemma applies, and at the remaining places Lemma 3.1(4) gives vanishing, so the reciprocity-determined value is correct. I found no fitted parameter and no circularity in the derivation.

major comments (1)
  1. [Section 3 (Lemma 3.4)] I have no major mathematical objections. The load-bearing local-symbol computation is sound as written: at the place t the contradiction argument uses the anisotropicity of x_0^2 + x_0 x_1 + x_1^2 over F_2 to force all variables to be divisible by t; at t^{-1} the residue equation x_0(x_0 + x_1) = 0 has the nonsingular solution x_0 = x_1 = 1, which lifts by Hensel's lemma; and at all remaining places the extension is unramified with t a unit, so Lemma 3.1(4) applies. Reciprocity (3.2) then forces the claimed value [omega^{-n}, t)_omega = 1. The only residual risk is a hidden sign or valuation error in this indirect computation; an independent direct local computation would eliminate that risk, but I see no defect in the present proof.
minor comments (4)
  1. [Section 2.2 (Figure 2 caption)] The caption of Figure 2 refers to the proof of Theorem 1.1, but the statement being proved at that point is Theorem 2.2; please correct the cross-reference.
  2. [Section 3.4 (proof of Lemma 3.4)] The displayed equation after 'But then' is typeset awkwardly and is hard to parse; it should read t x_2^2 = t^2(x_0^2 + omega^n x_0 x_1 + omega^n x_1^2), from which t | x_2 follows.
  3. [Section 4.1] In the displayed product for the local Fourier transform, the bracket notation is not closed (the expression '[f_omega H_omega(s; psi_omega)' appears without its matching parenthesis); please clean up the notation.
  4. [Section 4.2 (Lemma 4.2)] Lemma 4.2 displays two equivalent forms of the same local Fourier transform without saying that they are equivalent; since both forms are used later, a short sentence making this explicit would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the leading asymptotic is obtained from a self-contained Tauberian argument applied to an explicitly computed height zeta function, with no fitted parameters and no load-bearing self-citations.

full rationale

The paper's central derivation is not circular. The height zeta function Z(s) is defined in (1.2) as the generating series of the very counting function N(A1_K, pi, B), but this is the standard Tauberian method: one analyzes the analytic behaviour of the generating series and then recovers coefficient asymptotics via Theorem 2.2. The substantive work is the computation of the analytic continuation and singularities of Z(s) in Sections 3 and 4, which is independent of the final asymptotic. The leading constant c is an explicit Euler product obtained from local Fourier transforms (Lemma 4.4 and Theorem 4.7), not fitted to the counting function. The only load-bearing local computation is Lemma 3.4, the symbol value [omega^{-n}, t)_omega = 1; this is established by computing all other local symbols and applying global reciprocity, and its proof does not presuppose the asymptotic formula. An error in that lemma would affect correctness of the constant, but not constitute circularity. Self-citations such as [LS16] and [LRS] appear only in the introduction as context and comparison; they are not used in the proof of Theorem 1.1. External references for the Hasse principle, Serre's local symbol, and the zeta function of F_2(t) provide independent support. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is imported from the authors' prior work. The paper explicitly proves its own Tauberian theorem rather than relying on an unverified cited theorem. Overall, the derivation chain is self-contained apart from standard external inputs, and no circular step is present.

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

The central claim is self-contained given standard tools: the Hasse principle for conics, Poisson summation on the adeles, Serre's local symbols, and the zeta function of F_q(t). There are no parameters fitted to data and no invented entities. The only choices are standard normalizations of Haar measures and the naive height.

assumptions (5)
  • standard math Hasse principle for conics over global function fields: C_y(K) is nonempty iff C_y(K_omega) is nonempty for all omega.
    Used in Section 4.1 to write the global indicator f as a product of local indicators f_omega, citing Lam.
  • standard math Poisson summation on the adeles with 1/vol(A_K/K)=2 for K=F_2(t).
    Used in equation (4.3) to express the height zeta function as a sum over automorphic characters, citing Bourqui and Weil.
  • standard math Properties of Serre's local symbol in characteristic 2 from Serre, Local Fields, Chapter XIV Section 5.
    Used throughout Section 3 to translate local solubility of the conic into vanishing of [y,t)_omega, via Lemma 3.1 and Proposition 3.2.
  • standard math Zeta function of F_q(t): zeta_{F_q(t)}(s) = 1/((1-q^{1-s})(1-q^{-s})), with simple poles at s=1+2 pi i n/log q.
    Stated in Lemma 2.1 and used to locate the branch points in Theorem 4.7 and to compute the leading constant.
  • standard math The adelic volume vol(A_K/K)=q^{g-1}=1/2 for K=F_2(t).
    Used as the Poisson summation constant in equation (4.3), citing Weil, Basic Number Theory.

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Cite this review

Pith. "Pith review of Rational points in a family of conics over $\mathbb{F}_2(t)$." pith.science (2026). https://pith.science/paper/4POWUUWT

@misc{pith2026241214693,
  author       = {Pith},
  title        = {Pith review of: Rational points in a family of conics over $\mathbbF_2(t)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4POWUUWT}},
  note         = {Machine review of arXiv:2412.14693}
}
abstract

Serre famously showed that almost all plane conics over $\mathbb{Q}$ have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over $\mathbb{F}_2(t)$ which illustrates new behaviour. We obtain an asymptotic formula using harmonic analysis, which requires a Tauberian theorem over function fields for Dirichlet series with branch point singularities.

Figures

Figures reproduced from arXiv: 2412.14693 by the authors.

Figure 1
Figure 1. The Hankel contour H appearing in Lemma 2.3. By performing the variable change u = q −s , the region D transforms to C × and the region Re(s) ≥ a transforms to the punctured disk {u ∈ C × : |u| ≤ q −a}. We obtain the power series F(u) = X∞ M=0 φ(q M)u M, |u| < q−a . (2.3) A priori F(u) is only holomorphic for u ̸= 0, however holomorphicity at u = 0 is obtained from a simple application of the dominated convergence t… view at source ↗
Figure 2
Figure 2. The contours in the proof of Theorem 1.1. around q −a , with two line segments parallel to the ray from 0 to q −a joining this small circle to the bigger black incomplete circle C along |u| = q −aη . We apply Cauchy’s integral formula to the arrangement in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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