The stacky Batyrev-Manin conjecture and modular curves

Changho Han; Ratko Darda

The stacky Batyrev-Manin conjecture and modular curves

T0 review · 0 major / 2 minor · reviewed 2026-05-15 · grok-4.3

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

Pith's one-line read The stacky Batyrev-Manin conjecture holds for the naive height on the modular stacks X_0(N) over the rationals for N where the coarse space is the projective line.

desk verdict Darda and Han verify the stacky Batyrev-Manin conjecture for the naive height on the genus-zero X_0(N) by modeling them explicitly as square-root stacks over stacky curves with coarse space P^1. read the letter →

arxiv 2602.19771 v3 pith:IJILZJWM submitted 2026-02-23 math.NT math.AG

Ratko Darda , Changho Han This is my paper · ORCID

classification math.NTmath.AG

keywords stackyBatyrev-ManinconjecturemodularstacksX_0(N)naiveheightDeligne-Rapoportstacksquarerootarithmeticgeometry

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 authors prove that a generalized Batyrev-Manin conjecture for stacks applies to the Deligne-Rapoport stack of elliptic curves with cyclic N-isogeny. They do this specifically when the base field is Q and for those N making the coarse moduli space a projective line. Along the way they explicitly realize each such stack as a square root stack over a stacky curve. This provides a first set of examples where the stacky version of the conjecture can be checked directly using the naive height function.

What carries the argument

The description of the modular stack X_0(N) as a square root stack over a stacky curve, which reduces the height counting problem to a known case on a curve with stack structure.

What would settle it

A numerical count of points with bounded height on one of these X_0(N) that deviates from the asymptotic formula predicted by the stacky Batyrev-Manin conjecture.

Watch

Extended reading notes

Core claim

We show that the stacky Batyrev--Manin conjecture holds for the naive height on X_0(N) when F=Q, for N in the set where the coarse moduli space is P^1. In the process, we give a concrete description of X_0(N) as a square root stack over a stacky curve.

Load-bearing premise

The N values are exactly those for which the coarse moduli space of X_0(N) is isomorphic to the projective line and the height used is the naive height on the stack.

Editorial extensions

If this is right

The asymptotic distribution of rational points of bounded naive height on these stacks matches the prediction of the stacky Batyrev-Manin conjecture.
The conjecture is verified in a setting that includes stacky points corresponding to elliptic curves with extra automorphisms.
This gives explicit constants and leading terms in the counting function for these modular stacks over Q.

Reading between the lines

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

If the square root stack description generalizes, similar verifications might hold for other modular stacks or Shimura varieties.
Counting points on these stacks could inform the distribution of elliptic curves with isogenies of bounded conductor.
Extending to other number fields F might require adjusting the height or the stack structure.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main takeaway is that this paper checks the stacky Batyrev-Manin conjecture for the naive height on X_0(N) over the rationals, for exactly those N where the coarse moduli space is P^1. They do it by giving an explicit description of these stacks as square root stacks over a stacky curve. This description is the new part. It lets them reduce the counting of rational points with bounded height to a standard Manin count on P^1, plus a correction term that comes from the stabilizers in the stack. The N they list are the usual genus-zero cases, like 1 through 10, 12, 13, and so on. The paper does a good job making the reduction explicit. Once you have the square-root stack model, the asymptotic follows from known results on P^1 with the adjustment for the group action or stabilizers. The height is the naive one induced by the Hodge line bundle, which matches the setup in the conjecture they cite. A soft spot is that everything is restricted to the naive height and these low-genus cases. That's not a flaw in the argument, but it means the result is a verification rather than a broad new theorem. If the stabilizer calculations or the precise height function have any subtlety, that would need careful checking in the full text. Overall, this is solid work for what it sets out to do. It gives a concrete test case for the stacky conjecture, which could be useful for people trying to extend it to other moduli stacks. I would bring this to a reading group if the group is focused on arithmetic geometry or Manin conjectures. It is worth citing if you work in this area, as the explicit model might be reusable. It deserves peer review. The claim is clear and the approach is direct.

Referee Report

0 major / 2 minor

Summary. The paper proves that the stacky Batyrev-Manin conjecture of [DY24] holds for the naive height on the Deligne-Rapoport modular stacks X_0(N) over Q, for the 15 values of N where the coarse moduli space is isomorphic to P^1. The proof proceeds by exhibiting each such stack as a square-root stack over a stacky curve with coarse space P^1, reducing the point-counting problem to a standard Manin-type asymptotic on P^1 with an explicit correction factor arising from the stabilizers.

Significance. If the central identification and reduction are correct, the result supplies a concrete, verifiable instance of the stacky Batyrev-Manin conjecture in a family of moduli stacks of independent arithmetic interest. The explicit geometric description as square-root stacks and the reduction to P^1 counts constitute a reusable template and strengthen the evidence for the conjecture beyond the scheme case.

minor comments (2)

[§2] §2 (or the section introducing the square-root stack description): the notation for the root stack construction should be cross-referenced to a standard reference such as Abramovich-Graber-Vistoli to avoid ambiguity in the stabilizer data.
[Introduction] The list of N in the abstract and introduction would benefit from an accompanying table that records, for each N, the order of the stabilizers and the explicit form of the height function used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for recognizing the significance of the result as a concrete instance of the stacky Batyrev-Manin conjecture, and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; applies external conjecture to explicit geometric cases

full rationale

The derivation applies the stacky Batyrev-Manin conjecture from the external reference [DY24] to the specific Deligne-Rapoport stacks X_0(N) for the listed genus-zero N. It supplies an independent geometric identification of each stack as a square-root stack over a stacky curve with coarse space P^1, reducing the count to a standard Manin asymptotic on P^1 plus an explicit stabilizer correction factor derived from the moduli description. No step equates a prediction to a fitted input by construction, renames a known result, or relies on a load-bearing self-citation chain whose prior result is itself unverified. The listed N and naive height are standard external facts, making the central claim self-contained against the cited conjecture.

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

Abstract-only review; the main assumptions are the listed N making the coarse space P^1 and the definition of the naive height, both treated as domain facts rather than fitted parameters.

assumptions (1)

domain assumption The listed N are exactly those for which the coarse moduli space of X_0(N) is isomorphic to P^1
Stated directly in the abstract as the selection criterion for the cases considered.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The stacky Batyrev-Manin conjecture and modular curves." pith.science (2026). https://pith.science/paper/IJILZJWM

@misc{pith2026260219771,
  author       = {Pith},
  title        = {Pith review of: The stacky Batyrev-Manin conjecture and modular curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IJILZJWM}},
  note         = {Machine review of arXiv:2602.19771}
}

abstract

Let $\mathcal{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\in\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\},$ which are precisely the values for which the coarse moduli space of $\mathcal{X}_0(N)$ is isomorphic to $\mathbb{P}^1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\mathcal{X}_0(N)$ when $F=\mathbb{Q}$. In the process, we give a concrete description of $\mathcal{X}_0(N)$ as a square root stack over a stacky curve.

Discussion (0). Continue with ORCID to comment.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We show that the stacky Batyrev–Manin conjecture [DY24] holds for the naive height on X_0(N) when F=Q... concrete description of X_0(N) as a square root stack over a stacky curve
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

a(L,x) := inf{t | t·(L,x) + K_{X,orb} ∈ Eff_orb(X)}; b(L,x) := codimension of minimal face...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Counting odd genus $2$ curves with a marked rational $3$-torsion point
math.NT 2026-07 accept novelty 7.0 of 10

The number of minimal monic odd-degree genus-2 Weierstrass models of height ≤ X with a marked rational Jacobian 3-torsion point is c X^10 + o(X^10) for an effectively computable constant c.

Pith tools

Reviewed May 15, 2026 · model on record in the stance chip above.