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Duke for Drinfeld

T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read CM points equidistribute on Drinfeld-Stuhler modular curves as discriminants grow.

desk verdict The paper adapts Duke equidistribution to Drinfeld-Stuhler curves over function fields via Weyl, Waldspurger, and RH, but uniformity of the formula at arbitrary levels is the unverified piece. read the letter →

arxiv 2606.21163 v1 pith:EI4FE53C submitted 2026-06-19 math.NT

Francesco Maria Saettone This is my paper · ORCID
classification math.NT
keywords equidistributionCMpointsDrinfeld-StuhlermodularcurvesfunctionfieldsDuketheoremtoricperiodsWaldspurgerformulaRiemannHypothesis
topics The Riemann Hypothesis
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 paper proves a function field analogue of Duke's equidistribution theorem for CM points on Drinfeld-Stuhler modular curves. Equidistribution follows from reducing the problem via a Weyl criterion to the decay of toric periods, which Waldspurger's formula converts into central values of automorphic L-functions. These central values are bounded in Lindelöf form by the Riemann Hypothesis over function fields. The argument works at arbitrary level structures and in every positive characteristic, extending both Duke's theorem on the modular curve and Zhang's equidistribution result for Shimura curves.

What carries the argument

Weyl criterion reduction of equidistribution to toric period decay, expressed via Waldspurger's formula as central values of automorphic L-functions and bounded using the Riemann Hypothesis over function fields.

What would settle it

A sequence of CM points on some Drinfeld-Stuhler curve whose empirical distribution measure fails to converge weakly to the uniform hyperbolic measure would falsify the equidistribution claim.

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Extended reading notes

Core claim

We prove a function field analogue of Duke's equidistribution theorem for CM points, in the setting of Drinfeld--Stuhler modular curves. Our results thus extend, to the Drinfeld setting, both Duke's theorem on the modular curve and S.-W. Zhang's equidistribution in the case of Shimura curves. Equidistribution is reduced via a Weyl criterion to the decay of toric periods, which Waldspurger's formula expresses through central values of automorphic L-functions, bounded in Lindelöf-strength form by the Riemann Hypothesis over function fields. We work at arbitrary level structures and in every positive characteristic.

Load-bearing premise

Waldspurger's formula expresses the toric periods through central values of automorphic L-functions, which are then bounded in Lindelöf form by the Riemann Hypothesis over function fields.

Editorial extensions

If this is right

  • Equidistribution of CM points holds at arbitrary level structures.
  • The result applies in every positive characteristic.
  • The theorem extends Duke's original equidistribution on the modular curve to the Drinfeld setting.
  • The theorem extends Zhang's equidistribution on Shimura curves to the Drinfeld setting.

Reading between the lines

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

  • The same reduction and bound strategy could be tested on other families of modular curves defined over function fields.
  • Equidistribution statements of this type may supply effective versions of the Chebotarev density theorem for Galois representations attached to Drinfeld modules.
  • Removing the dependence on the function-field Riemann Hypothesis would require new unconditional estimates for the relevant toric periods.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper proves a function field analogue of Duke's equidistribution theorem for CM points on Drinfeld-Stuhler modular curves. Equidistribution is reduced via the Weyl criterion to decay of toric periods; Waldspurger's formula then relates these periods to central values of automorphic L-functions, which are bounded in Lindelöf form using the Riemann Hypothesis over function fields. The results are claimed to hold at arbitrary level structures and in every positive characteristic, extending both Duke's theorem on the modular curve and S.-W. Zhang's equidistribution for Shimura curves.

Significance. If the central reduction and uniformity statements hold, the result supplies a function-field counterpart to Duke's theorem with the added flexibility of arbitrary levels and all characteristics, thereby strengthening the dictionary between number-field and function-field arithmetic geometry.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: the reduction relies on Waldspurger's formula expressing toric periods through central L-values with uniformity in arbitrary level structures and every positive characteristic for the quaternion-algebra setting over global function fields; the manuscript must explicitly cite the precise theorem (or derive the required uniformity) because this identity is load-bearing for passing from period decay to the L-value bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need to make the citation of the Waldspurger formula explicit. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the reduction relies on Waldspurger's formula expressing toric periods through central L-values with uniformity in arbitrary level structures and every positive characteristic for the quaternion-algebra setting over global function fields; the manuscript must explicitly cite the precise theorem (or derive the required uniformity) because this identity is load-bearing for passing from period decay to the L-value bound.

    Authors: We agree that the load-bearing appeal to Waldspurger's formula requires an explicit reference, particularly for the claimed uniformity across arbitrary level structures and all positive characteristics. In the revised manuscript we will insert a precise citation to the relevant statement of the Waldspurger formula in the quaternion-algebra setting over global function fields (specifically, the version that supplies the required uniformity in level and characteristic). The body of the paper already relies on this result; the revision will make the dependence visible already in the abstract and introduction. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation uses external theorems

full rationale

The abstract describes a standard reduction: equidistribution via Weyl criterion to toric period decay, then Waldspurger's formula relating periods to central L-values, then Lindelöf bounds from the known Riemann Hypothesis over function fields. These steps invoke established external results (Waldspurger, function field RH) rather than any self-definitional relation, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or claims in the provided text reduce the target result to its own inputs by construction. The paper works at arbitrary levels but presents this as an extension of prior external theorems (Duke, Zhang), with no visible internal circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The proof strategy rests on the applicability of Waldspurger's formula and the Riemann hypothesis for function fields to the Drinfeld setting; no free parameters or new entities are mentioned in the abstract.

assumptions (2)
  • domain assumption Waldspurger's formula holds for the relevant toric periods on Drinfeld-Stuhler curves
    Used to express periods via L-function central values.
  • domain assumption Riemann Hypothesis over function fields supplies Lindelöf-strength bounds on the L-values
    Provides the decay needed for equidistribution.

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Cite this review

Pith. "Pith review of Duke for Drinfeld." pith.science (2026). https://pith.science/paper/EI4FE53C

@misc{pith2026260621163,
  author       = {Pith},
  title        = {Pith review of: Duke for Drinfeld},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EI4FE53C}},
  note         = {Machine review of arXiv:2606.21163}
}
abstract

We prove a function field analogue of Duke's equidistribution theorem for CM points, in the setting of Drinfeld--Stuhler modular curves. Our results thus extend, to the Drinfeld setting, both Duke's theorem on the modular curve and S.-W. Zhang's equidistribution in the case of Shimura curves. Equidistribution is reduced via a Weyl criterion to the decay of toric periods, which Waldspurger's formula expresses through central values of automorphic $L$-functions, bounded in Lindel\"of-strength form by the Riemann Hypothesis over function fields. We work at arbitrary level structures and in every positive characteristic.

Discussion (0). Continue with ORCID to comment.

Reference graph

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