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Counting quadratic points on Fano varieties

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Quadratic pairs on quadrics match the predicted B log B count

desk verdict Serious, technically deep, probably right, but the main theorem rests on a uniform error bound that only just suffices—worth a careful referee, not a desk reject. read the letter →

arxiv 2505.17940 v2 pith:GMFGLSQZ submitted 2025-05-23 math.NT math.AG

Francesca Balestrieri , Kevin Destagnol , Julian Lyczak , Jennifer Park , Nick Rome This is my paper · ORCID
classification math.NTmath.AG MSC 11G5014G0514J4514C0511R42
keywords quadraticpointssymmetricsquaresManin–PeyreconjectureFanovarietiesHilbertschemeoftworestrictionscalarslatticepointcountingthinsets
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

Most counting in arithmetic geometry fixes one number field and counts rational points on a variety. This paper instead counts quadratic points on Fano varieties, that is, unordered pairs of points whose coordinates live in a quadratic extension, packaged as rational points on the symmetric square $\mathrm{Sym}^2 X$. The paper's main theorem is an exact asymptotic for this count for the infinite family of non-split quadric surfaces $X_d \subset \mathbb{P}^3$ given by $x^2 - d y^2 = zw$: outside a thin set $Z$ consisting of rational pairs and conjugate pairs from $L=\mathbb{Q}(\sqrt d)$, the number of points of anticanonical height at most $B$ is $c_{\mathrm{Hilb}^2 X_d,\mathbb{Q}} B \log B + O_d(B(\log B)^{3/4})$, where $c$ is the Peyre constant. The complement $Z$ contributes a comparable $c_Z B \log B$, so removing it is essential for the leading constant. Along the way the paper develops a general framework for $\mathrm{Sym}^2 X$ of any weak Fano variety (anticanonical bundle nef and big), summing Tamagawa numbers over quadratic fields, determining the cutoff that controls which fields can contain small quadratic points, and proving lattice-counting estimates over rings of integers.

What carries the argument

The engine of the proof is the restriction-of-scalars comparison between the symmetric square and quadratic twists. For a quadratic extension $K/\mathbb{Q}$, the morphism $\eta\colon \mathrm{Res}_{K/\mathbb{Q}} X_K \to \mathrm{Sym}^2 X$ sends $P \in X(K)$ to the pair $\{P,\bar P\}$; together with the rational-pair piece, this decomposes $\mathrm{Sym}^2 X(\mathbb{Q})$ into contributions labelled by quadratic fields. The asymptotic is assembled from four pieces: Theorem 6.10 evaluates $\sum_{|\Delta_K|\le Y} \tau(X_K)$ as $\tau(\mathrm{Hilb}^2 X)\log Y + O(1)$ for surfaces and as a convergent sum in dimension $\ge 3$; the cutoff theory of Section 7 determines the optimal value $\gamma = 1/2$ for $X_d$, which enters the leading constant through $\alpha(\mathrm{Hilb}^2 X) = \tfrac12\gamma\,\alpha(X)$; Proposition 8.11 is a basis-free Davenport lemma for $\mathcal{O}_F$-lattices, reducing the dimension of the lattice-counting problem by viewing ideals in the biquadratic field $M=KL$ as modules over $\mathcal{O}_L$; and Proposition 9.21 gives the sharp successive-minima estimate $\sum_i \lambda_1(\mathfrak d,i)^{-1} \ll N_{M/\mathbb{Q}}(\mathfrak d)^{-1/4} N(\mathcal{D}^{-1})$ that makes the summed error small enough. The analytic backbone is an on-average truncation of twisted Artin $L$-functions at $s=1$, showing that the values $L(1,\rho\otimes\chi_K)$ are constant on average over quadratic twist families.

What would settle it

Compute the uniform point count in Proposition 1.7 for a fixed squarefree $d$ and all quadratic $K$ with $|\Delta_K|\le B^{1/2}$: if the error term has exponent larger than $7/8$ in $B$, or if the summed quantity $\sum_K (h_M R_M)^{3/4}(1+\mathrm{res}_{s=1}\zeta_M(s))|\Delta_K|^{-3/2}$ grows like a positive power of $B$ rather than staying bounded up to logarithmic factors, the total error exceeds the $B\log B$ main term. A direct check of Proposition 9.21 — comparing $\sum_i 1/\lambda_1(\mathfrak d,i)$ with $N(\mathfrak d)^{-1/4}N(\mathcal{D}^{-1})$ over biquadratic fields with varying $K$ — would expose the same failure.

Watch

Extended reading notes

Core claim

The central discovery is that the Manin–Peyre conjecture for the crepant resolution $\mathrm{Hilb}^2 X_d \to \mathrm{Sym}^2 X_d$ holds, and is proved by counting quadratic points field by field and summing over fields, provided a thin set of type II (the image of a generically finite map of degree $>1$) is removed. Concretely, for every squarefree $d$ and the anticanonical height on $X_d = \mathrm{Res}_{\mathbb{Q}(\sqrt d)/\mathbb{Q}} \mathbb{P}^1$ induced from the natural height on $\mathbb{P}^1$, the set $U$ of points of $\mathrm{Sym}^2 X_d(\mathbb{Q})$ that neither come from two rational points nor from a conjugate pair over $L=\mathbb{Q}(\sqrt d)$ satisfies $N_{\mathbb{Q}}(U,B) = c_{\mathrm{Hilb}^2 X_d,\mathbb{Q}} B \log B + O_d(B(\log B)^{3/4})$, with the Peyre constant as leading coefficient. The removed set $Z$ itself satisfies $N_{\mathbb{Q}}(Z,B) \sim c_Z B \log B$, so the asymptotic for the full symmetric square would have the wrong leading constant without this removal. More broadly, the paper establishes that for any smooth weak Fano variety of dimension at least two the quadratic point count on $\mathrm{Sym}^2 X$ reduces to uniform estimates for $K$-points on $X_K$ over almost all quadratic fields $K$, and that for surfaces the sum of Tamagawa numbers $\tau(X_K)$ over fields of bounded discriminant grows like $\tau(\mathrm{Hilb}^2 X)\log Y$.

Load-bearing premise

The load-bearing premise is that the uniform counting error for each quadratic field, and the estimate used to add those errors up over all fields, are exactly as small as the paper claims; the proof of these bounds occupies the parts of Sections 9.3.5 and 9.4 not included in the material reviewed here, and the authors note in Remark 9.7 that the savings only just suffice.

Editorial extensions

If this is right

  • For the infinite family of non-split quadrics $X_d$, the quadratic Manin–Peyre prediction holds with the exact Peyre constant, extending the previously known cases $\mathbb{P}^2$ and $\mathbb{P}^1\times\mathbb{P}^1$.
  • The explicit thin set $Z$ contributes a full $B\log B$ term, so removing conjugate-pair loci is not an aesthetic choice but a numerical necessity for the predicted leading constant.
  • For any weak Fano variety of dimension at least three, the same field-by-field framework yields the quadratic point count whenever uniform estimates for $K$-points over almost all quadratic fields are available, because the sum of Tamagawa numbers over quadratic fields converges.
  • The averaged moment formula for $L(1,\rho\otimes\chi_K)$ gives asymptotics for all moments of class numbers of quadratic fields with error $Y^{r/2+2/3+\epsilon}$, improving previously known exponents.
  • Counting in $\mathcal{O}_F$-lattices reduces the effective dimension of lattice problems by $[F:\mathbb{Q}]$, producing stronger error terms than classical $\mathbb{Z}$-lattice counting in the same geometric setting.

Reading between the lines

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

  • Beyond the paper: the same field-by-field architecture should transfer to any surface whose uniform point counts over quadratic fields can be made explicit; the optimal cutoff, rather than a separate computation of $\alpha(\mathrm{Hilb}^2 X)$, would then be the main quantity to determine, and the identity $\alpha(\mathrm{Hilb}^2 X)=\tfrac12\gamma\,\alpha(X)$ found here is a plausible organizing re
  • Beyond the paper: the basis-free $\mathcal{O}_F$-lattice method is a general dimension-reduction principle for arithmetic statistics: whenever the region and lattice admit symmetry under the integers of a subfield, the relevant successive-minima information condenses into fewer factors, which should improve many uniform counting problems with varying fields.
  • Beyond the paper: the on-average constancy of $L(1,\rho\otimes\chi_K)$ suggests that other averages built from Euler products over quadratic fields, such as relative class numbers or field-counting statistics, will inherit the same $Y^{2/3+\epsilon}$ error structure; the paper does not develop those applications.
  • Beyond the paper: the thin set that must be removed is cut out by field-of-definition conditions rather than by a proper subvariety, so the accumulating sets for symmetric powers are likely governed by which quadratic fields contain pure points of small height; the same phenomenon should appear for higher symmetric powers $\mathrm{Sym}^r X$.
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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

2 major / 5 minor

Summary. The paper develops a systematic framework for counting rational points of bounded height on Sym^2 X for weak Fano varieties X/Q. The authors decompose Sym^2 X(Q) according to quadratic fields K, sum the corresponding Manin-Peyre counts uniformly over K, and relate the total to the Manin-Peyre prediction for Hilb^2 X when X is a surface. The main structural theorem (Theorem 6.10) evaluates the sum of Tamagawa numbers of X_K over quadratic fields K as τ(Hilb^2 X) log Y + O(1) in dimension 2 and shows absolute convergence in dimension at least 3. For the family of non-split quadrics X_d: x^2 - d y^2 = zw in P^3_Q, the paper uses the identification X_d = Res_{Q(√d)/Q} P^1 and an OF-lattice version of Davenport's lemma to prove (Theorem 9.1, equivalently Theorem 1.3) that, after removing an explicit thin set Z consisting of pairs of Q-points and conjugate L-points, the counting function for quadratic point pairs satisfies N_Q(U,B) = c_{Hilb^2 X_d, Q} B log B + O_d(B (log B)^{3/4}), with the Peyre constant as leading constant. The paper also proves an on-average truncation result for twisted Artin L-functions (Proposition 6.3) and derives consequences for moments of class numbers of quadratic fields (Theorem 1.9).

Significance. If the central claims hold, this is a substantial contribution to the quantitative arithmetic of higher-degree points. It is the first systematic treatment of the counting problem on Sym^2 X for general weak Fano varieties, and it produces the first infinite family of non-split surfaces for which the quadratic Manin-Peyre asymptotic is proved with the correct Peyre constant. The paper also introduces a genuinely basis-free theory of OF-lattice point counting, which reduces dimension and improves error terms, and a general on-average L-function truncation that yields new moment asymptotics for class numbers. A notable strength is that the visible core is derivation-heavy rather than heuristic: the Peyre constants are computed from geometric invariants, the cutoff γ = 1/2 is shown to be optimal, and no free parameters are fitted to match the asymptotic. The thin set that must be removed is of a genuinely new type-II flavour and the paper proves that its contribution has the same order of magnitude as the main term, so the removal is necessary for the leading constant.

major comments (2)
  1. [§9.4–§9.5, Theorem 9.1, Remark 9.7] The decisive summation over quadratic fields that turns the uniform point counts into the asymptotic of Theorem 9.1 is not present in the text available for review. Theorem 9.6 supplies a per-field error of order B^{7/8}(h_M R_M)^{3/4}(1 + res_{s=1} ζ_M(s))/|Δ_K|^{3/2} after the relevant substitution, and the summation of this error over |Δ_K| ≤ B^{1/2} is the load-bearing step for the main term B log B. Remark 9.7 explicitly states that the savings 'only just suffice' to prove Theorem 1.3. Since a lost power of |Δ_K| or a lost power of 1/λ_1(d,i) in Proposition 9.21 would make the accumulated error exceed o(B log B), the proof of Theorem 9.1 is incomplete until this summation is supplied and checked.
  2. [§6.2, Eq. (6.1), Proposition 6.3] The pointwise bound for the partial sums of the twisted Artin L-function coefficients, displayed as (6.1), is asserted via Brauer induction and a reference to [Gol70], but the authors explicitly note that the Brauer factorization may contain negative powers. For a general non-abelian representation ρ, the claimed uniformity in the conductor q(ρ⊗χ_Δ) and the cancellation in the coefficient sums are not immediate. This would be acceptable as a sketch for a peripheral tool, but Proposition 6.3 is the engine behind Theorem 6.10 and Corollary 6.12, so the argument should be either written out or backed by a precise statement with the required uniformity.
minor comments (5)
  1. [Title and Lemma 9.3(e)] The phrase 'non-squared' in Lemma 9.3(e) should read 'non-square d'.
  2. [Proposition 1.7] The condition 'Q(P)=K' should presumably be 'F(P)=K', matching the definition of a pure K-point.
  3. [§6.4, proof of Proposition 6.11] The phrase 'with cΔ equal to χ_K(b)g(Δ)' should read 'with c_Δ = χ_Δ(b)g(Δ)', since K is the summation variable.
  4. [§6.4, end of proof of Proposition 6.11] The sentence 'aA application of partial summation' contains a typo and should be 'An application of partial summation'.
  5. [Definition 4.11] The notation F_Ξ is used both for a set of quadratic fields and for the associated set of fundamental discriminants; this double use is potentially confusing and should be disambiguated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof chain is self-contained, with all constants computed from geometric invariants and proven analytic estimates rather than fitted inputs.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The asymptotic for Sym^2 X_d is obtained by summing uniform point counts over quadratic fields K with |Δ_K| ≤ B^{1/2}; each per-field count (Proposition 1.7 / Theorem 9.6) is proven from the O_L-lattice version of Davenport's lemma (Proposition 8.11) and from explicit unit-group and ideal-class estimates (Proposition 9.21), not by assuming the final Manin-Peyre asymptotic. The cutoff γ = 1/2 is not fitted: it is derived from height-discriminant inequalities and proven optimal via a conic lower bound (Lemma 9.3(e)). The Peyre constant c_{Hilb^2 X_d,Q} is computed from independent geometric invariants: α and β in Lemma 9.3 and the Tamagawa number via Theorem 5.4, which is a genuine computation of Peyre's recipe. The summation over quadratic fields is an independent analytic theorem (Theorem 6.10), proved using sums of multiplicative functions over fundamental discriminants (Proposition 6.2) and on-average truncation of twisted Artin L-functions (Proposition 6.3); the equality of the summed Tamagawa numbers with τ(Hilb^2 X)·log Y is a nontrivial result, not a definition. The thin set Z is identified geometrically (Lemma 4.1) and its contribution is computed by partial summation, not used to force the leading constant. The authors' Remark 9.7 that certain savings 'only just suffice' is a statement about the tightness of error terms and a correctness risk, not evidence of circular reasoning. No load-bearing self-citations occur; cited results (e.g., Loughran, Schmidt, Widmer, Le Rudulier) are external and independently established. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claims rest on classical analytic number theory, the standard geometry of numbers, and Peyre's local-to-global dictionary; the only paper-specific inputs are the choice of unit subgroup U and the choice of adelic heights in Section 9. No constants are fitted to data: the Peyre constants are computed from geometric invariants, and the cutoff gamma = 1/2 is derived and proved optimal. No new mathematical entities are introduced.

free parameters (1)
  • admissible anticanonical height family on X_d = chosen family (Definition 9.4 and Remark 9.32)
    The asymptotic is proved for specific anticanonical adelic heights induced from the restriction of scalars; the Peyre constant, cutoff, and counts depend on this normalization. The choice is an input, not a number fitted to data.
assumptions (6)
  • standard math Classical analytic number theory inputs: Selberg-Delange method, Heath-Brown quadratic large sieve, Polya-Vinogradov inequality, prime number theorem for Hecke L-functions via Brauer induction
    Used in Lemma 6.1, Lemma 6.5, Proposition 6.2 and Proposition 6.3; classical tools with the paper supplying precise variants.
  • domain assumption Peyre's local dictionary: at good primes the local Tamagawa measure equals #V(F_p)/p^dim, and convergence factors are L-function local factors (Peyre 1995)
    Invoked in Proposition 4.16 and Lemma 6.9(b) via [Pey95, Lem. 2.2.1]; standard for smooth proper varieties with good reduction.
  • domain assumption Weak Fano implies rationally connected, hence geometric Picard group torsion-free and Brauer group finite modulo constants
    Used in Section 4 (Lemma 4.9 and around) via [Zha06], and assumed throughout for the Manin-Peyre invariants to behave under base change.
  • domain assumption All but finitely many quadratic fields are linearly disjoint from the splitting field L of Pic X (the F^circ family)
    Proposition 4.4 and Definition 4.2; this is what makes rho, alpha, beta invariant under base change and makes the twist family non-principal in Proposition 6.3.
  • ad hoc to paper The unit subgroup U of M generated by fundamental units of the three quadratic subfields is used to parametrize the fundamental domain
    Definition 9.8 and Proposition 9.9; a paper-specific but proven device (index q_M divides 8) that makes the error terms in Section 9.3 explicit.
  • standard math Schanuel's constant for P^1 over a number field is the external main term
    c_{P^1,M} in Theorem 9.6 is Schanuel's constant [Sch79], used as the benchmark main term in the uniform count.

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Cite this review

Pith. "Pith review of Counting quadratic points on Fano varieties." pith.science (2026). https://pith.science/paper/GMFGLSQZ

@misc{pith2026250517940,
  author       = {Pith},
  title        = {Pith review of: Counting quadratic points on Fano varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMFGLSQZ}},
  note         = {Machine review of arXiv:2505.17940}
}
abstract

This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of surfaces, we relate this to the Manin--Peyre conjecture for $\text{Hilb}^2 X$, and prove the conjecture for an infinite family of non-split quadric surfaces. In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed. To establish our counting result for the specific family of surfaces, we generalise existing lattice point counting techniques to lattices defined over rings of integers. This reduces the dimension of the problem and yields improved error terms. Another key tool we develop is a collection of results for summing Euler products over quadratic extensions. We use this to show moments of $L$-functions at $s=1$ are constant on average in quadratic twist families.

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