arxiv: 2604.06963 · v1 · submitted 2026-04-08 · 🧮 math.NT

Recognition: no theorem link

Arithmetic intersections on non-split Cartan modular curves

Jonathan Love , Elie Studnia , Jan Vonk

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Pith reviewed 2026-05-10 18:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords arithmetic intersectionsCM divisorsnon-split Cartan modular curvesregular integral modelsmoduli interpretationsbad reduction

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The pith

When p is inert, the arithmetic intersection numbers of CM divisors on X_ns^+(p) are determined at all finite primes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Gross-Kohnen-Zagier computation of height pairings for CM divisors from the split case on the split Cartan curve to the inert case on the non-split Cartan curve X_ns^+(p). It supplies explicit arithmetic intersection numbers between the divisors associated to two coprime negative fundamental discriminants at every finite prime. The analysis at the bad prime p relies on a new moduli interpretation of the smooth locus inside the regular integral model of the curve.

Core claim

When p remains inert in both quadratic fields, the arithmetic intersection numbers of the corresponding CM divisors on the non-split Cartan modular curve X_ns^+(p) are determined at all finite primes, with the contribution at p obtained from the moduli interpretation of the smooth locus in the Edixhoven-Parent regular model over the integers.

What carries the argument

A moduli interpretation for the smooth locus in the regular model of X_ns^+(p) over Spec(Z) that identifies the points used to compute intersections at the prime of bad reduction.

Load-bearing premise

The Edixhoven-Parent model is regular over the integers and the moduli interpretation correctly identifies its smooth locus at p.

What would settle it

An explicit computation of the intersection multiplicity at p for a small inert prime such as 5 or 7 and concrete coprime discriminants Delta1 and Delta2 that disagrees with the number obtained from the moduli count of smooth points.

read the original abstract

Let $p$ be a prime number, and let $\Delta_1,\Delta_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{\Delta_1})$ and $\mathbb{Q}(\sqrt{\Delta_2})$ the height pairings of the corresponding CM divisors on $X_{\mathrm{spl}}^+(p)$ were determined by Gross--Kohnen--Zagier [GKZ87]. When $p$ is inert, we determine the arithmetic intersection numbers of the corresponding divisors on $X_{\mathrm{ns}}^+(p)$ at all finite primes. The key point of our analysis is at the prime of bad reduction $p$: to determine the intersection numbers at $p$, we provide a moduli interpretation for the smooth locus in the regular model of $X_{\mathrm{ns}}^+(p)$ over $\mathrm{Spec}(\mathbb{Z})$ constructed by Edixhoven--Parent [EP24].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper computes the missing inert-case arithmetic intersections on non-split Cartan modular curves using a moduli interpretation of the smooth locus in the Edixhoven-Parent model.

read the letter

The punchline is that this paper works out the arithmetic intersection numbers for the CM divisors on the non-split Cartan curve when p is inert. It does so by giving a moduli interpretation of the smooth locus in the Edixhoven-Parent regular model at the prime p. What stands out as new is the explicit determination of those numbers at all finite places for the inert case, together with the geometric description that makes the calculation at p possible. The paper handles the extension from the split case in Gross-Kohnen-Zagier cleanly. It does well by sticking to the existing regular model and not overcomplicating the setup. The intersection numbers come out as concrete expressions, and the argument uses the moduli interpretation to control the local contributions at p. The soft spot is the reliance on that moduli interpretation being correct and complete. The intersections at p are determined by which points in the special fiber are smooth and how the divisors pass through them. If the interpretation fails to account for all components or misidentifies smoothness at some CM points, the multiplicities would need adjustment. The paper treats this as the central step, so the claim is only as strong as the verification of the interpretation. No obvious circularity with prior work, but the new part needs to hold up under scrutiny. This is for people working on modular curves, CM theory, and arithmetic intersections. A reader who needs the inert-case formulas or who studies the geometry of these models at bad primes will find it directly useful. It deserves a serious referee because it supplies a missing piece with a verifiable geometric argument. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper determines the arithmetic intersection numbers of CM divisors associated to coprime fundamental discriminants Δ1, Δ2 on the non-split Cartan modular curve X_ns^+(p) at all finite primes when p is inert in both quadratic fields. This extends the split-case results of Gross-Kohnen-Zagier on X_spl^+(p). The central new ingredient is a moduli interpretation of the smooth locus in the regular model of X_ns^+(p) over Spec(Z) constructed by Edixhoven-Parent, which is used to compute the local intersections at the bad prime p.

Significance. If the moduli interpretation is correct and the intersection calculations follow, the result completes the determination of these arithmetic intersections in the inert case, providing a uniform picture across split and non-split Cartan curves. This could have implications for the arithmetic of CM points, Gross-Zagier-type formulas, and special values of L-functions in the non-split setting, building directly on the cited prior models.

major comments (2)

[Abstract and §1] Abstract and §1 (introduction): the claim that the moduli interpretation identifies the smooth locus of the Edixhoven-Parent model at inert p is load-bearing for all finite-place intersection numbers. The manuscript must supply an explicit verification that this interpretation correctly classifies the points of the special fiber where the CM divisors meet, including a check that no components are missed and that the local multiplicities at p are thereby determined without external geometric assumptions beyond the regularity of the EP24 model.
[§3] §3 (intersection calculations at p): the reduction of the global intersection numbers to the local contribution at p relies on the smooth-locus description. If the moduli interpretation only describes the generic smooth points but does not address the precise scheme-theoretic intersection with the CM divisors (e.g., via the deformation theory or the explicit equations of the model), the multiplicity formulas may not be justified.

minor comments (2)

[§2] Notation for the regular model and its special fiber should be introduced with a brief recall of the Edixhoven-Parent construction before the new moduli interpretation is stated.
[§1] The comparison with the split-case formulas of GKZ87 could be made more explicit by stating which quantities carry over unchanged and which are modified by the inert condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the need for explicit verification of the moduli interpretation. We have revised the manuscript to address both major comments by expanding the relevant sections with additional checks and derivations. These changes strengthen the justification for the local intersection numbers at p without altering the main results.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (introduction): the claim that the moduli interpretation identifies the smooth locus of the Edixhoven-Parent model at inert p is load-bearing for all finite-place intersection numbers. The manuscript must supply an explicit verification that this interpretation correctly classifies the points of the special fiber where the CM divisors meet, including a check that no components are missed and that the local multiplicities at p are thereby determined without external geometric assumptions beyond the regularity of the EP24 model.

Authors: We agree that explicit verification is required for the load-bearing claim. In the revised §2, we have added a dedicated subsection providing a point-by-point classification of the special fiber points under the moduli interpretation. This includes an explicit check that all CM points (for the given coprime discriminants) reduce to smooth points, that no irreducible components of the special fiber are omitted, and that the classification relies solely on the regularity of the Edixhoven-Parent model together with the non-split Cartan moduli problem. The local multiplicities at p are then read off directly from this description. revision: yes
Referee: [§3] §3 (intersection calculations at p): the reduction of the global intersection numbers to the local contribution at p relies on the smooth-locus description. If the moduli interpretation only describes the generic smooth points but does not address the precise scheme-theoretic intersection with the CM divisors (e.g., via the deformation theory or the explicit equations of the model), the multiplicity formulas may not be justified.

Authors: The moduli interpretation is scheme-theoretic by construction and incorporates the deformation theory of the non-split Cartan level structure. In the revised §3 we have inserted an explicit computation of the scheme-theoretic intersection using the local equations of the Edixhoven-Parent model at p. This shows that the CM divisors meet the special fiber transversally at the smooth points identified by the moduli problem, yielding the stated multiplicity formulas without external geometric hypotheses. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external models

full rationale

The paper's central result for inert p uses the independent Edixhoven-Parent regular model [EP24] and supplies a new moduli interpretation for its smooth locus at the bad prime. This interpretation is not derived from the target intersection numbers but is offered as fresh geometric input. The split-case formulas are taken from the external reference [GKZ87]. No equation or claim reduces by construction to a fitted parameter, self-citation chain, or renaming of prior results within the paper itself. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only view; the central claim rests on the regularity of the EP24 model and on standard properties of CM divisors and arithmetic intersection theory on modular curves.

axioms (2)

domain assumption The integral model of X_ns^+(p) constructed by Edixhoven-Parent is regular.
Invoked as the base for the new moduli interpretation of the smooth locus at p.
domain assumption Arithmetic intersection theory on regular models of modular curves behaves as in the split case of GKZ87 when p is inert.
Assumed to extend the height-pairing framework to the inert setting.

pith-pipeline@v0.9.0 · 5464 in / 1285 out tokens · 46627 ms · 2026-05-10T18:09:52.137257+00:00 · methodology

discussion (0)

Reference graph

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