Equidistribution and the torsor method

Christian Bernert; Florian Wilsch; Ulrich Derenthal

REVIEW 2 major objections 3 minor 2 references

Equidistribution and the torsor method

T0 review · 2 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Rational points outside the ten lines on a smooth split quintic del Pezzo surface are equidistributed in the adelic space with the Tamagawa measure, for any anticanonical height.

desk verdict Strong and novel equidistribution result for quintic del Pezzo surfaces with a fixable but real gap in the congruence-counting step. read the letter →

arxiv 2607.14866 v1 pith:XYSCXJLH submitted 2026-07-16 math.NT math.AG

Christian Bernert , Ulrich Derenthal , Florian Wilsch This is my paper · ORCID

classification math.NTmath.AG MSC 11G3511D4514G05

keywords Manin'sconjectureequidistributiondelPezzosurfaceuniversaltorsorTamagawameasureanticanonicalheightadelicpointsmethod

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

This paper proves that the rational points outside the ten lines on a smooth split quintic del Pezzo surface over any number field are equidistributed in the space of adelic points, with limit measure equal to the Tamagawa measure, when the points are ordered by any anticanonical height. This is the first adelic equidistribution theorem for a del Pezzo surface obtained through the torsor method. As a direct consequence, Manin's conjecture—the predicted power-law count of rational points of bounded height—holds for arbitrary anticanonical height functions, not only the special heights treated in earlier counting results. The proof is built on a general abstract equidistribution theorem that converts torsor-based counting asymptotics for many equivalent height functions into equidistribution statements, and on a congruence-counting result that handles the finite places.

What carries the argument

The load-bearing mechanism is the abstract equidistribution theorem (Theorem 1.1). It says that for a projective variety equipped with a big semiample line bundle, if one has asymptotic formulas for the height functions induced by all finite sets of global sections that are $S$-integer linear combinations of a fixed generating set—and the associated measures satisfy the compatibility $d\tau_{P'} = \left(\frac{H_P}{H_{P'}}\right)^a d\tau_P$—then the rational points are equidistributed in the $S$-adelic points with respect to the induced measure. The proof passes to the affine cone over projective space, constructs a topological basis of the adelic cone from regions defined by inequalities involving $S$-integer linear forms, and

What would settle it

For a specific surface over $\mathbb{Q}$, take a small prime $p$ and a residue class $r$ modulo $p$, and compute the number of rational points of anticanonical height $\le B$ that reduce to $r$. If the relative error to the predicted asymptotic $\frac{\tau_H(X(A_K))}{|X(\mathcal{O}_K/p)|} B (\log B)^4$ does not tend to 0 as $B$ grows, Proposition 3.1 and Theorem 1.7 would be false.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.7: for a smooth split quintic del Pezzo surface $X$ over a number field $K$, the set $U(K)$ of rational points outside the ten lines is equidistributed in $X(A_K)$ with limit measure $\tau_H$ for any anticanonical height $H$. In particular, Manin's conjecture holds for every anticanonical height function. The supporting framework is a general theorem (Theorem 1.1) stating that if a family of height functions coming from finite sets of global sections satisfies asymptotic counting formulas with leading constant proportional to a compatible measure, then the rational points are equidistributed with respect to that measure. The paper also records a converse equidistribution theo

Load-bearing premise

The per-class asymptotic of Proposition 3.1 relies on the assertion that adding congruence conditions only shrinks the error terms inherited from the prior counting theorem; if this monotonicity fails for some residue classes modulo $q$, the finite-place equidistribution in Theorem 1.7 would not follow.

Editorial extensions

If this is right

Rational points outside the lines on a smooth split quintic del Pezzo surface are equidistributed in the full adelic space with the Tamagawa measure for any anticanonical height, not just the heights treated in earlier counting theorems.
Manin's conjecture holds for arbitrary anticanonical height functions on these surfaces, with the predicted leading constant α(X) τ_H(X(A_K)).
The abstract equidistribution theorem upgrades any torsor-based counting proof satisfying its hypotheses to an equidistribution statement; this yields, for instance, archimedean equidistribution for integral points on the complement of a line.
The per-residue counting shows the local distribution is uniform: each residue class modulo an ideal q receives proportion 1/|X(o_K/q)| of the points, once the Tamagawa measure is trivialized.
The converse theorem in the appendix shows that equidistribution is equivalent to counting asymptotics for a dense family of reweighted height functions, so the two perspectives on Manin's problem are interchangeable in this generality.

Reading between the lines

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

A natural next step would be to apply the congruence-counting method to integral points on the log-anticanonical model of X\D; the authors note this should be possible, and an explicit per-residue asymptotic for that model would give the first adelic equidistribution theorem for integral points on a surface.
The abstract theorem suggests a general recipe for other Fano varieties: any existing torsor-based proof of Manin's conjecture that uses height conditions closed under S-integer linear combinations can be upgraded to equidistribution without new harmonic analysis, just by verifying the measure-compatibility condition.
One could stress-test the uniformity prediction numerically: on a quintic del Pezzo surface over Q, the proportion of points of height ≤ B in a fixed residue disk mod 2 should tend to the product of the real density of the disk and the 2-adic factor; a large-scale computation would provide an independent check of Theorem 1.7.
The paper's approach to finite places via congruence conditions isolates the place-dependence: if a future counting theorem for another variety does not satisfy the error-term monotonicity under congruence restrictions, the finite-place part of equidistribution would need a different argument, such as a direct sieve on the torsor.

Share X Bluesky LinkedIn Reddit HN

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

Know this: Theorem 1.7 is the first adelic equidistribution result for a del Pezzo surface outside the toric/equivariant settings, and it implies Manin's conjecture for every anticanonical height on smooth split quintic del Pezzo surfaces. The abstract theorem (1.1) is a genuinely useful tool for the torsor method. But there is a real gap in the congruence step (Section 3.4/Proposition 3.4): the claim that adding congruence conditions 'only leads to smaller error terms' is not justified, and without it Corollary 2.8's per-class hypothesis is not met.

The strengths first. Theorem 1.1 is a clean generalization of Peyre's and Chambert-Loir–Tschinkel's equidistribution principles, tailored so that torsor-counting results yield equidistribution. The proof via cones and geometry of numbers (Lemmas 2.2, 2.4) is careful; the measure-theoretic basis argument is sound. Corollary 2.8's reduction to congruence counting is also a nice formulation. Section 3 then adapts the authors' earlier counting theorem [BD25b] to residue classes. The local Euler factor computations look plausible, and the final constant matches Peyre's Tamagawa measure. If the per-class asymptotic holds, Theorem 1.7 follows.

The soft spot is exactly where the stress-test points. In Proposition 3.4 the error term is inherited from [BD25b] with the explanation that the sets are subsets, so errors are smaller. That is not a valid inference for a fixed residue class: signed errors in the total can cancel, while an individual class can have large error that cancels a negative error elsewhere. The paper never tracks the dependence on q through the lattice-point count or Möbius inversions. The same assertion appears in Proposition 3.6. This is a gap as written, not a known counterexample. It might be fixable by running [BD25b] with q in the constants and allowing implied constants to depend on q—but then uniformity in q is precisely what Corollary 2.8 needs, and that isn't discussed. The final reduction from admissible to arbitrary anticanonical heights is also compressed; the appendix converse theorem helps, but the proof of Theorem 1.7 is brief.

The heavy reliance on [BD25b] is not itself a problem: that theorem is published and independent. The remark about locally accumulating lines (Figure 1) is a nice observation, not central.

Verdict: worth a serious referee. The main theorem is important if correct, and the gap is identifiable and likely repairable. I would send this to peer review and ask the authors to either prove the per-class error bound or weaken Theorem 1.7 to admissible heights. For me it is a revise-and-resubmit, not an accept as-is.

Referee Report

2 major / 3 minor

Summary. The paper proves a general abstract equidistribution theorem for the torsor method (Theorem 1.1, generalizing results of Peyre and Chambert-Loir–Tschinkel), which converts asymptotic counting formulas for many height functions into equidistribution of rational points in adelic spaces. The authors then apply this theorem to smooth split quintic del Pezzo surfaces over number fields, using their earlier counting result [BD25b] and a new congruence-counting extension (Proposition 3.1), to prove Theorem 1.7: the rational points outside the ten lines are equidistributed in the adelic space with respect to any anticanonical height, with limit measure the corresponding Tamagawa measure. This implies Manin's conjecture for arbitrary anticanonical height functions on such surfaces. The paper also contains a converse abstract equidistribution theorem in an appendix.

Significance. If the result is fully established, it is a significant contribution: it provides a widely applicable framework for deriving equidistribution from torsor-method counting theorems, and it gives the first adelic equidistribution result for quintic del Pezzo surfaces with arbitrary anticanonical heights, not just the standard model heights. The abstract theorem is carefully developed with measure-theoretic lemmas, and the paper is commendably explicit about the hypotheses and the continuity-set argument. The application, however, rests on a congruence refinement of [BD25b] whose proof is only sketched; this is the main technical risk. The self-citation to [BD25b] and [BD25a] is appropriate because those theorems are published and are used as inputs, not as consequences of the equidistribution claim.

major comments (2)

[§3.4 (Prop. 3.4) and §3.6 (Prop. 3.6)] The proof of Proposition 3.4 asserts that because the congruence conditions (3.7) define subsets of the sets counted in [BD25b], the error terms 'only lead to smaller error terms'. The same assertion is repeated in the proof of Proposition 3.6. This monotonicity is not valid for signed errors: the total error bound in [BD25b] controls |N_total(B) − main_total(B)|, but the per-class difference |N_r(B) − main_r(B)| is not automatically bounded by that total error, since cancellations between residue classes can occur. Since Corollary 2.8 requires the per-class asymptotic for every ideal q and every residue r, and Proposition 3.1 is the only finite-place counting input for Theorem 1.7, this is a load-bearing gap. The manuscript does not track the dependence on q through the lattice-point count and Möbius inversions. A complete proof, or a precise reference to a congruence-compatible version
[Proof of Theorem 1.7] Theorem 1.7 claims equidistribution with respect to any anticanonical height, but Proposition 3.1 is stated only for 'admissible' heights defined by a set P ⊂ o_K[Y1,Y2,Y3]. The proof of Theorem 1.7 cites only Corollary 2.8, which concerns the restricted family P' in Theorem 1.1. The passage from admissible heights to arbitrary anticanonical heights requires either the counting hypotheses of Corollary 1.2 for all P' with K-linear combinations, or an explicit approximation argument such as Theorem A.1. This step is not spelled out in the proof and should be clarified, since the full generality of the statement depends on it.

minor comments (3)

[Lemma 2.2] The inequality 'max{|u_{i,j}|_v, |w_{i,j}|_v} > κκ_v' appears to mix the global constant κ and the local factor κ_v; the intended normalization should be clarified.
[Proposition 3.1] The exponent d in the error term O(B(log B)^4 (log log B)^{-1/(3d+1)}) is not defined in the statement; it should be identified (presumably as [K:Q] or as in [BD25b]).
[Figure 1] The caption refers to 'brown rational points', but the figure appears in grayscale in the arXiv version; consider describing the shading or adding a color version.

Circularity Check

0 steps flagged · score 2.0 of 10

No derivation step uses its own conclusion; reliance on [BD25b] is self-citation but not circularity.

full rationale

The paper's derivation chain is: prove an abstract equidistribution criterion (Theorem 1.1, Corollary 2.8), prove per-residue counting asymptotics (Proposition 3.1) by rerunning the torsor arguments of [BD25b] with congruence conditions, then apply Corollary 2.8 to obtain Theorem 1.7. The finite-place hypothesis of Corollary 2.8 (N(P',r,B) ~ c c_r tau_{P'}(X(A_K)) B^a (log B)^{b-1} for every q, r, and P') is materially stronger than the conclusion and is discharged by Proposition 3.1; it is not assumed. No parameter is fitted to data, and the limit measure tau_H is not defined in terms of the counting functions. The main counting input [BD25b] is a published, peer-reviewed theorem by two of the three authors; under the stated rules this is real independent evidence and does not make the argument circular. The asserted monotonicity in Sections 3.4 and 3.6 ('subsets ... only lead to smaller error terms') is the least secure step: it is not proved with explicit q-dependence, and Corollary 2.8 requires per-class o(1) errors. This is a correctness/rigor gap, not a circularity; even if the gap were filled by a different method, the logic of the paper would be unchanged. Likewise, the equality tau({x = r mod q}) = c_r tau(X(A_K)) in Remark 2.9 is a property of Tamagawa measures, not an input identical to the counting conclusion. Score 2 reflects only the heavy self-citation context and the unproved monotonicity assertion; no circular reduction was identified.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted to data; the constants in the asymptotic are computed from the geometry of the surface and the Tamagawa measure, not chosen to match a sample. No new geometric or physical entities are introduced. All load-bearing inputs are the external counting theorem [BD25b] and the Tamagawa measure formalism.

assumptions (3)

domain assumption The counting theorem of [BD25b] for points of bounded height on smooth split quintic del Pezzo surfaces over number fields, with the stated error term, is valid for admissible heights.
Proposition 3.1 and Section 3 import [BD25b]'s main result and notation wholesale; the paper does not re-derive the base count.
domain assumption Tamagawa measures satisfy the scaling relation (1.1) and the congruence-measure identity (2.6) for globally generated models, per Peyre [Pey95] and Salberger [Sal98].
Used in Corollary 2.8 and Remark 2.9 to identify local congruence measures with 1/|X(o_K/q)| and to transfer asymptotics between height functions.
standard math The lattice-point lemma of [Hau21, Lem. 3.2]: if the volume of a family of sets grows faster than the volume of its F-boundary, then it contains an S-integer lattice point for all sufficiently large parameters.
This is the key existence step in Lemma 2.2 for constructing the topological basis of the adelic affine cone; it is cited, not proven in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Equidistribution and the torsor method." pith.science (2026). https://pith.science/paper/XYSCXJLH

@misc{pith2026260714866,
  author       = {Pith},
  title        = {Pith review of: Equidistribution and the torsor method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XYSCXJLH}},
  note         = {Machine review of arXiv:2607.14866}
}

read the original abstract

We prove equidistribution and Manin's conjecture for rational points outside the lines on smooth split quintic del Pezzo surfaces over number fields with respect to any anticanonical height. The proof is based on a general theorem that is broadly usable to deduce equidistribution when using the torsor method with several equivalent height functions to treat variants of Manin's problem.

Figures

Figures reproduced from arXiv: 2607.14866 by the authors.

**Figure 1.** Rational points of height at most 500 on a del Pezzo surface X of degree 5 with a subset of X(R) in the shape of a triangle and the density function for the limit probability measure. of lines through two centers of the blow-up are shown in gray, while the remaining four lines are exceptional divisors and contracted along this map. The points of bounded height are color-coded according to the 15 possible residue dis… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

2 extracted references · 2 linked inside Pith

[193]

Séminaire Bourbaki. Vol. 2003/2004, Exp. No. 931. [Pey95] E. Peyre,Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79(1995), no. 1, 101–218. [PSTVA21] M. Pieropan, A. Smeets, S. Tanimoto, and A. Várilly-Alvarado,Campana points of bounded height on vector group compactifications, Proc. Lond. Math. Soc. (3)123 (2021), no. 1, 57–101. ...

arXiv 2003
[2022]

[LRS22] D

arXiv:2111.01509v2. [LRS22] D. Loughran, N. Rome, and E. Sofos,The leading constant for rational points in families, 2022. Preprint, arXiv:2210.13559. [LS16] D. Loughran and A. Smeets,Fibrations with few rational points, Geom. Funct. Anal. 26(2016), no. 5, 1449–1482. [McK07] D. McKinnon,A conjecture on rational approximations to rational points, J. Algebr...

arXiv 2022

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [193]

Séminaire Bourbaki. Vol. 2003/2004, Exp. No. 931. [Pey95] E. Peyre,Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79(1995), no. 1, 101–218. [PSTVA21] M. Pieropan, A. Smeets, S. Tanimoto, and A. Várilly-Alvarado,Campana points of bounded height on vector group compactifications, Proc. Lond. Math. Soc. (3)123 (2021), no. 1, 57–101. ...

arXiv 2003

[2] [2022]

[LRS22] D

arXiv:2111.01509v2. [LRS22] D. Loughran, N. Rome, and E. Sofos,The leading constant for rational points in families, 2022. Preprint, arXiv:2210.13559. [LS16] D. Loughran and A. Smeets,Fibrations with few rational points, Geom. Funct. Anal. 26(2016), no. 5, 1449–1482. [McK07] D. McKinnon,A conjecture on rational approximations to rational points, J. Algebr...

arXiv 2022