{"id":"db0d3653-171e-4b36-b86c-3ba6956a7e84","arxiv_id":"2412.14693","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Over F_2(t), the number of parameters y of height 2^M for which the conic x0^2+x0x1+yx1^2=t x2^2 has a rational point is asymptotically c 2^{2M}/M^{1/2}, with an explicit Euler product constant c.","lead":"This paper counts how many conics in a specific family over the function field F_2(t) have a rational point, and proves an exact asymptotic formula in which almost all fibres have no point. A smart generalist might read it because it is the first sharp result of this kind over function fields and shows a mechanism that does not appear over the rational numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Lemma 3.4 is load-bearing and its proof is sound upon inspection.","rationale":"The reader's weakest-assumption pick is exactly the place I would stress. I attempted to falsify Lemma 3.4 and could not: the anisotropic-form argument at t, the Hensel lift at infinity, and the unramified-unit argument elsewhere are all sound, and reciprocity closes the computation. The rest of the proof, including the branch-singularity Tauberian theorem, the convergence/positivity of the Euler product on Re(s)>3/2, and the character orthogonality in Lemma 4.6, is internally consistent. The paper's explicit statement that the direct symbol computation is hard is a verification risk, not a demonstrated error. Thus the ACCEPT verdict stands; an independent direct local computation would raise confidence from moderate to high.","tokens_in":797,"tokens_out":933,"duration_ms":324881,"concrete_test":"Perform a direct local solubility test for K = F2((s)) with s = t+1 and n=1. Determine whether there exist x0, x1, x2 in F2((s)) satisfying x0^2 + x0 x1 + s^{-1} x1^2 = (s+1) x2^2. Use truncated power series with Newton polygon and Hensel lifting up to precision s^20. Lemma 3.4 predicts no solution (equivalently [s^{-1},s+1)_s = 1). As a second check, compute the local Fourier factor [f_{t+1} H_{t+1}(2;1) by summation over truncated Laurent series; Lemma 4.2 predicts 9/8 for deg omega = 1 places. A mismatch in either check would invalidate Lemma 3.5 and require revising the leading constant, and possibly the exponent 1/2 in the logarithmic factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim hinges on Lemma 3.4, the computation [omega^{-n},t)_omega = 1 for omega not equal to t,t^{-1}. This value powers the half-measure solubility split in Lemma 3.5, which produces the branch order b=1/2 and hence the (log B)^{-1/2} factor, as well as the local constants c_omega. I examined the proof rather than relying on the admission that a direct computation is hard. At the place t, the contradiction argument is valid: since omega is a monic polynomial with omega(0)=1, reducing the scaled equation modulo t gives the norm form x0^2 + x0 x1 + x1^2, which is anisotropic over F2, forcing x0 = x1 = 0 mod t; a further valuation argument forces t dividing x2, contradicting the minimality of the chosen solution. At t^{-1}, the congruence equation x0^2 + x0 x1 = 0 has the nonsingular solution x0 = x1 = 1 mod t^{-1}, which Hensel lifts; thus the symbol vanishes there. At all other places v, the extension L_{n,v} is unramified and t is a unit, so Lemma 3.1(4) gives [omega^{-n},t)_v = 0. Reciprocity (3.2) then forces [omega^{-n},t)_omega = -[omega^{-n},t)_t = 1. I do not see a gap. The remaining steps, including the Tauberian theorem, the positive and nonvanishing Euler product on Re(s)>3/2, and the character decomposition in Lemma 4.6, are consistent. The residual risk is a hidden sign or valuation error in this indirect symbol computation; an independent direct local computation would eliminate that risk.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15943,"tokens_out":19712,"duration_ms":154873,"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":[{"comment":"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.","section":"Section 3 (Lemma 3.4)"}],"minor_comments":[{"comment":"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.","section":"Section 2.2 (Figure 2 caption)"},{"comment":"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.","section":"Section 3.4 (proof of Lemma 3.4)"},{"comment":"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.","section":"Section 4.1"},{"comment":"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.","section":"Section 4.2 (Lemma 4.2)"}],"recommendation":"accept","confidential_remarks":"I recommend acceptance. The only residual uncertainty is the indirect nature of the computation in Lemma 3.4; although I checked the proof and found no gap, an independent direct local computation by the authors would further reduce risk. The paper is within the scope of the journal, and the citation and attribution practices appear appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a solid paper and the main result is genuinely new: the first sharp asymptotic for rational points in a family of conics over a global function field, with the expected exponent 1/2 computed rather than bounded. The paper also proves a standalone Tauberian theorem for Dirichlet series with branch point singularities over function fields (Theorem 2.2), which is cleanly proven via Hankel contours and will be useful elsewhere.\n\nThe argument is coherent. The height zeta function is analyzed by harmonic analysis; Poisson summation gives a sum over automorphic characters, and the key Lemma 4.6 shows only the trivial character and the special character psi_{1/t} contribute, with equal Fourier transforms. That is a nice structural observation. The local solubility analysis is careful, and the load-bearing step is Lemma 3.4, the computation [omega^{-n}, t)_omega = 1 for omega not t, t^{-1}. The authors derive this indirectly via global reciprocity. I went through the proof in the stress test and it holds up: the contradiction at the place t works (the norm form x0^2 + x0 x1 + x1^2 is anisotropic over F2), the congruence at t^{-1} has a nonsingular solution that Hensel lifts, and the unramified places give zero, so reciprocity forces the symbol to be 1. I found no gap. A direct local computation would be a welcome sanity check, but the indirect argument is convincing.\n\nSoft spots are minor. The paper is dense, and the geometric discussion around the non-smooth model (Lemma 1.2) is terse; readers need to fill in some details about regular models. The constant's interpretation via the Loughran-Rome-Sofos conjecture is explicitly speculative, since the framework doesn't apply, but that is clearly flagged. The restriction to B = 2^M is natural over function fields.\n\nI think this deserves a serious referee. The result is important enough, the proof is complete enough, and the main risk (Lemma 3.4) has been checked. I would send it to peer review and expect acceptance after a careful reading.","headline":"New sharp asymptotic for rational points in a conic family over F_2(t); the load-bearing local symbol computation holds up.","tokens_in":16510,"tokens_out":2628,"would_cite":true,"duration_ms":18854,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G05","14F22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Almost all fibres in a conic family over F2(t) have no rational point","keywords":["rational points","conic bundles","function fields","height zeta function","Tauberian theorem","characteristic 2","local symbols","Brauer group"],"falsifier":"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 $2^{\\deg\\omega-1}-1$.","tokens_in":15401,"feed_emoji":"📐","tokens_out":8371,"duration_ms":65292,"temperature":0.7,"pith_summary":"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.","feed_headline":"Most conics in a function-field family have no rational point","feed_subtitle":"A new asymptotic with explicit constant proves the proportion of solvable fibres drops to zero.","key_machinery":"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 $1/\\Gamma(1/2)=1/\\sqrt{\\pi}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the characteristic-2 local symbol $[a,b)_\\omega$ and its norm interpretation, the basis for the solubility criterion in Proposition 3.2.","marker":"[Ser79]"},{"why":"Originates the counting problem for specialisations of Brauer group elements over $\\mathbb{Q}$, whose bounds the paper compares and refines over function fields.","marker":"[Ser90]"},{"why":"Provides the number-field conjecture and upper bound involving $\\Delta(\\pi)$ that the paper tests and shows does not apply at the fibre at infinity.","marker":"[LS16]"},{"why":"Supplies the Poisson summation formula on the adeles with the normalisation giving $\\mathrm{vol}(A_K/K)=1/2$ used in equation (4.3).","marker":"[Bou11]"},{"why":"Gives the zeta function of $\\mathbb{F}_q(t)$ and the Hankel contour method adapted in the Tauberian theorem.","marker":"[Ros02]"},{"why":"Is the classical Tauberian theorem for branch point singularities over number fields that the paper adapts to the function-field setting.","marker":"[Del54]"},{"why":"Gives the adelic volume statement $\\mathrm{vol}(A_K/K)=1/2$ and the generation of $A_K$ by $K$ and the product of local rings used in Lemma 4.5.","marker":"[Wei95]"},{"why":"Supplies the count of about $B^2$ points of height $B$ on $\\mathbb{P}^1$ over a global function field, used to conclude that $0\\%$ of fibres are solvable.","marker":"[Pey12]"}],"fun_headline_variants":["Solvable conics vanish at rate log^{-1/2} over F2(t)","Zero percent of conic fibres have rational points over F2(t)","Conic family over F2(t): only cB^2/√logB solvable","Function-field conics: solvable density decays like 1/√log B","Sharp log factor: conics over F2(t) almost never solvable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Solvable conics vanish at rate log^{-1/2} over F2(t)","Zero percent of conic fibres have rational points over F2(t)","Conic family over F2(t): only cB^2/√logB solvable","Function-field conics: solvable density decays like 1/√log B","Sharp log factor: conics over F2(t) almost never solvable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00098,"raw_usage":{"total_tokens":4124,"prompt_tokens":873,"completion_tokens":3251,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":3145}},"tokens_in":489,"tokens_out":3251,"duration_ms":21383,"temperature":1.0,"reasoning_tokens":3145,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:02:01.715913+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $2^{\\deg\\omega-1}-1$.","supporting_citations":[],"review_version":1}