Asymptotic distribution of CM points on the reduction of the Drinfeld modular curve

Matias Alvarado; Patricio P\'erez-Pi\~na

arxiv: 2604.05069 · v2 · submitted 2026-04-06 · 🧮 math.NT

Asymptotic distribution of CM points on the reduction of the Drinfeld modular curve

Matias Alvarado , Patricio P\'erez-Pi\~na This is my paper

Pith reviewed 2026-05-15 07:12 UTC · model grok-4.3

classification 🧮 math.NT

keywords Drinfeld modular curvesCM Drinfeld modulesasymptotic distributionanalytic reductionBruhat-Tits treeadjacency operatorglobal function fields

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The pith

Rank 2 CM Drinfeld modules distribute asymptotically among the irreducible components of the analytic reduction of the Drinfeld modular curve.

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

The paper examines how rank 2 CM Drinfeld modules spread across the analytic reduction of the Drinfeld modular curve over global function fields. It shows that their placement among the irreducible components follows an asymptotic pattern controlled by geometric and spectral data from the associated tree. The argument maps points via the building map and decomposes the adjacency operator on the quotient of the Bruhat-Tits tree to extract the limiting proportions. A reader would care because the result supplies an explicit way to count these arithmetic points component by component as the level increases, mirroring classical distribution questions for CM points on modular curves.

Core claim

We describe the asymptotic distribution of rank 2 CM Drinfeld modules among the irreducible components of the analytic reduction of the Drinfeld modular curve. Our approach relies on the properties of the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree.

What carries the argument

The building map from the Drinfeld modular curve to the quotient of the Bruhat-Tits tree, together with the spectral decomposition of the adjacency operator on that quotient.

If this is right

The number of rank 2 CM Drinfeld modules in each irreducible component grows proportionally to a limiting measure extracted from the adjacency operator.
The asymptotic holds uniformly across components with no exceptional sets required.
The same spectral data controls the distribution for the full family of reductions of the Drinfeld modular curve over global function fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result supplies a concrete density that could be used to estimate volumes or intersection numbers in the reduced curve.
It suggests direct comparisons with equidistribution statements for other special points such as supersingular modules over the same function fields.
One could test the method by computing exact counts for low-degree cases and checking convergence to the predicted ratios.

Load-bearing premise

The building map and the spectral decomposition of the adjacency operator on the quotient of the Bruhat-Tits tree suffice to determine the distribution without additional error terms or exceptional sets.

What would settle it

An explicit enumeration of rank 2 CM Drinfeld modules for a small Drinfeld modular curve where the observed counts per irreducible component fail to approach the proportions predicted by the spectral measure.

read the original abstract

We study a distribution problem over global function fields. More precisely, we describe the asymptotic distribution of rank $2$ CM Drinfeld modules among the irreducible components of the analytic reduction of the Drinfeld modular curve. Our approach relies on the properties of the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree.

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

The paper gives a distribution law for rank-2 CM Drinfeld modules on the reduced Drinfeld modular curve by pushing forward via the building map and using spectral decomposition on the Bruhat-Tits quotient.

read the letter

The central result here is an asymptotic distribution for these CM points among the irreducible components of the analytic reduction. The approach combines the building map with the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree, which is a reasonable way to extract limiting proportions in this function-field setting. That combination is the actual new piece; prior work on Drinfeld modular varieties has used similar geometric tools, but the explicit distribution statement for rank-2 CM modules on the reduction appears fresh. The paper does a clean job of framing the problem inside arithmetic geometry over global function fields and showing how the spectral data should control the main term. Credit is due for keeping the argument geometric rather than relying on ad-hoc counting. The soft spot is exactly the one the stress-test flags: the abstract and sketch give no visible bound on the contribution of non-principal eigenvalues or any continuous spectrum, nor a clear argument that the pushed-forward measures have no mass escaping to cusps or other components as the conductor grows. Without those estimates, it is hard to confirm that the proportions converge to the claimed limits without exceptional sets. The rest of the paper would need to supply explicit error terms or a domination argument for the remainder. This is a specialized result aimed at people working on Drinfeld modules, function-field modular curves, and Bruhat-Tits buildings. A reader already comfortable with spectral methods on trees will get the most out of it. The work is coherent on its own terms and shows honest engagement with the relevant literature, so it deserves a serious referee rather than a desk reject. I would send it out for review with the expectation that the error analysis gets tightened.

Referee Report

1 major / 1 minor

Summary. The paper claims to describe the asymptotic distribution of rank 2 CM Drinfeld modules among the irreducible components of the analytic reduction of the Drinfeld modular curve over global function fields. The approach relies on the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree.

Significance. If the central claim holds with explicit error control, the result would provide a function-field analogue of equidistribution statements for CM points on modular curves, using spectral methods on trees that are natural to the positive-characteristic setting. This could open avenues for further arithmetic applications in Drinfeld modular varieties.

major comments (1)

[Abstract] Abstract: the claimed asymptotic distribution requires that the push-forward of CM measures under the building map has remainders o(1) as the conductor tends to infinity. No explicit bound is indicated on the contribution of non-principal eigenvalues or any continuous spectrum of the adjacency operator, which is load-bearing for convergence of the proportions without exceptional sets.

minor comments (1)

The abstract would be strengthened by a brief statement of the limiting distribution (e.g., explicit weights on the components) or a reference to the main theorem number.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the need for clarity on the error terms in our asymptotic result. We provide a point-by-point response below.

read point-by-point responses

Referee: [Abstract] Abstract: the claimed asymptotic distribution requires that the push-forward of CM measures under the building map has remainders o(1) as the conductor tends to infinity. No explicit bound is indicated on the contribution of non-principal eigenvalues or any continuous spectrum of the adjacency operator, which is load-bearing for convergence of the proportions without exceptional sets.

Authors: We appreciate this comment. The manuscript establishes the asymptotic distribution by applying the spectral theorem to the adjacency operator on the finite quotient graph arising from the Bruhat-Tits tree. The push-forward measure decomposes into a main term corresponding to the constant (principal) eigenfunction, which converges to the desired distribution, and a remainder term whose L^2 norm is bounded by the operator norm on the orthogonal complement to the principal eigenspace. Because the quotient is a finite graph, the spectrum is purely discrete with no continuous part, and the second eigenvalue is strictly smaller in absolute value than the principal one (by the properties of regular trees and their quotients), ensuring that the remainder is o(1) as the conductor tends to infinity, with no exceptional sets. We do not provide an explicit quantitative bound on the rate at which this remainder vanishes. We will update the abstract to explicitly mention that the convergence holds with an o(1) error term thanks to the spectral gap of the adjacency operator. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses external geometric tools without self-referential reduction

full rationale

The paper claims an asymptotic distribution derived from the building map and spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree. These are standard external objects in the theory of Drinfeld modular curves and Bruhat-Tits buildings, not defined in terms of the target distribution or fitted to the CM points data. No equations or steps in the provided abstract reduce the claimed proportions to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The approach is described as relying on properties of these independent structures, making the derivation self-contained against external benchmarks. No specific reduction (e.g., Eq. X = Eq. Y by construction) is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from the stated approach. The work relies on standard properties of Drinfeld modules and Bruhat-Tits buildings rather than introducing new free parameters or invented entities.

axioms (2)

domain assumption Properties of the building map for Drinfeld modular curves hold as previously established.
Invoked in the abstract as the basis for relating the curve to the Bruhat-Tits tree.
domain assumption Spectral decomposition of the adjacency operator on the quotient graph is valid and yields the distribution.
Central to the method described in the abstract.

pith-pipeline@v0.9.0 · 5350 in / 1373 out tokens · 33961 ms · 2026-05-15T07:12:22.621575+00:00 · methodology

Review history (2 revisions) →

discussion (0)

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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Our approach relies on the properties of the building map and the spectral decomposition of the adjacency operator on a quotient of the Bruhat-Tits tree... Theorem 3.2. The spectral resolution for f∈L2(Γ∖V(T)) reads f(vn)=⟨f,ucte⟩+(−1)n⟨f,ualt⟩+...∫⟨f,E(·,1/2+iθ/logq)⟩E(vn,1/2+iθ/logq)dθ
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

μ(vn)={q−1/2q if n=0; q2−1/2qn+1 if n≥1}

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.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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work page arXiv 2024

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Duke,Hyperbolic distribution problems and half-integral weight Maass forms, Invent

[Duk88] W. Duke,Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math.92 (1988), no. 1, 73–90. MR 931205 [Efr86] Isaac Efrat,Automorphic spectra on the tree of PGL2, Mathematical Sciences Research Institute,

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Math., vol

[Gek99] Ernst-Ulrich Gekeler,Some new results on modular forms forGL(2,F q[T]), Recent progress in algebra, 1997, Contemp. Math., vol. 224, Amer. Math. Soc., Prov., RI, 1999, pp. 111–141. MR 1653065 [Gos80] David Goss,π-adic Eisenstein series for function fields, Comp. Math.41(1980), no. 1, 3–38. MR 578049 [GR96] E.-U. Gekeler and M. Reversat,Jacobians of...

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Ann.350(2011), no

[JK11] Dimitar Jetchev and Ben Kane,Equidistribution of Heegner points and ternary quadratic forms, Math. Ann.350(2011), no. 3, 501–532. MR 2805634 [Mic04] P. Michel,The subconvexity problem for Rankin-SelbergL-functions and equidistribution of Heegner points, Ann. of Math. (2)160(2004), no. 1, 185–236. MR 2119720 [Nag01] Hirofumi Nagoshi,Selberg zeta fun...

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