The Mean field equation on the Tate curve

Yaojia Sun

arxiv: 2603.15706 · v2 · submitted 2026-03-16 · 🧮 math.NT · math.AP· math.RT

The Mean field equation on the Tate curve

Yaojia Sun This is my paper

Pith reviewed 2026-05-15 10:25 UTC · model grok-4.3

classification 🧮 math.NT math.APmath.RT

keywords mean field equationTate curveGreen's functionLaplacian spectrumnon-Archimedean geometryelliptic curveexistence and uniqueness

0 comments

The pith

Solutions to the mean field equation exist and are unique on the Tate curve for suitable parameters.

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

This paper shows that the mean field equation on the Tate curve has solutions constructed as limits of solutions on finite quotients of the curve. It builds the Green's function of the Laplacian explicitly as a finite sum, giving a direct non-Archimedean counterpart to the Green's function on the flat torus. The construction yields uniqueness of solutions when the equation parameter lies in certain ranges. A reader would care because the result transfers well-posedness from the classical Archimedean setting to this non-Archimedean elliptic curve, opening a path to study the same equation uniformly across different places.

Core claim

On the Tate curve the Laplacian spectrum is determined and its Green's function is constructed as a finite sum. Solutions to the mean field equation are then obtained by proving that solutions on finite quotients converge to a solution on the full curve, with uniqueness holding in a parameter region. The resulting well-posedness mirrors the corresponding statements for the mean field equation on the flat torus.

What carries the argument

The Green's function on the Tate curve, constructed as an explicit finite sum that serves as the integral kernel for the mean field equation.

If this is right

The mean field equation on the Tate curve can be solved by taking limits of solutions defined on its finite quotients.
Uniqueness of solutions holds for the mean field equation when the parameter belongs to a suitable open set.
The spectrum of the Laplacian is explicitly available, permitting the finite-sum Green's function.
Well-posedness properties of the mean field equation transfer from the Archimedean torus to the non-Archimedean Tate curve.

Where Pith is reading between the lines

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

The finite-quotient approximation may supply a practical numerical scheme for computing solutions on the Tate curve.
Similar quotient-and-limit arguments could apply to mean field equations on other p-adic elliptic curves.
The explicit Green's function may allow direct comparison of solution sets across Archimedean and non-Archimedean places.

Load-bearing premise

Solutions constructed on finite quotients converge to a solution on the full Tate curve.

What would settle it

A specific parameter value for which the limit of solutions on finite quotients fails to satisfy the mean field equation on the Tate curve.

read the original abstract

In this paper, we study the spectrum of the Laplacian on the Tate curve and construct the associated Green's function as a finite sum, which can be viewed as the non-Archimedean counterpart of the Green's function on the flat torus in the Archimedean case. Moreover, we establish existence and uniqueness results of the mean field equation on this space. To address the problem, we first prove the structure of solutions on finite quotients, and prove the existence on the Tate curve by the convergence of such solutions. We also prove the uniqueness of the solutions for some parameter region. Notably, the well-posedness of the solution resembles that in the Archimedean case.

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 an explicit finite-sum Green's function on the Tate curve and proves mean-field existence and uniqueness via finite-quotient approximations, with the convergence step as the main point to verify.

read the letter

The main takeaway is that Sun constructs the Green's function for the Laplacian on the Tate curve explicitly as a finite sum and then establishes existence and uniqueness for the mean field equation by solving on finite quotients and passing to the limit. This supplies a direct non-Archimedean counterpart to the classical results on flat tori. The finite-sum expression for the Green's function is the clearest new piece; it makes the spectrum of the Laplacian concrete in this p-adic setting and gives a usable tool rather than an abstract existence statement. The existence proof follows the natural route of building solutions level by level and taking limits, while uniqueness holds in a stated parameter region that matches the Archimedean behavior. That part of the argument is straightforward and worth having on record. The potential soft spot is the convergence step itself. The abstract claims the finite-quotient solutions converge to a solution on the full Tate curve, but this requires uniform bounds or compactness that do not deteriorate with the level of the quotient. If the estimates depend on the approximation or fail to control the non-Archimedean Green's function sum properly in the limit, the limiting object may not satisfy the equation or may lose uniqueness. The stress-test note flags exactly this issue, and it is the part that needs the most careful reading in the full proof. The work is aimed at specialists in p-adic analysis and arithmetic geometry who already care about mean-field equations or Green's functions on curves over local fields. A reader in that subfield will get a usable construction and a clear analogy to the real case. It is narrow enough that it will not reshape broader mathematics, but the explicit construction is solid enough to merit referee time rather than a desk rejection. I would send it out for review, with the expectation that the convergence details and any error estimates get tightened or clarified.

Referee Report

2 major / 2 minor

Summary. The paper studies the spectrum of the Laplacian on the Tate curve, constructs the associated Green's function explicitly as a finite sum (the non-Archimedean analog of the flat-torus Green's function), and proves existence of solutions to the mean-field equation by solving the equation on finite quotients and passing to the limit on the Tate curve; uniqueness is established in a restricted parameter region, with the well-posedness claimed to parallel the Archimedean case.

Significance. If the convergence step is made rigorous, the work supplies a concrete non-Archimedean model for mean-field equations on curves, with the finite-sum Green's function offering an explicit, computable object that has no direct Archimedean counterpart. This could serve as a test case for p-adic geometric PDEs and arithmetic applications.

major comments (2)

[Existence via convergence (abstract and main existence argument)] The existence proof relies on convergence of solutions constructed on finite quotients to a solution on the full Tate curve. No uniform a-priori bounds (independent of the level of the quotient) or compactness argument in a suitable function space controlling the non-Archimedean Green's function sum are supplied; without these, the limiting object may fail to satisfy the mean-field equation.
[Uniqueness result] The uniqueness statement is restricted to 'some parameter region,' but the precise interval of the parameter (and the proof that uniqueness holds exactly there) is not compared quantitatively with the corresponding Archimedean threshold; this leaves the claimed resemblance unverified.

minor comments (2)

[Green's function construction] The abstract states that the Green's function 'can be viewed as' the non-Archimedean counterpart; an explicit comparison of the finite-sum formula with the classical torus formula would clarify the analogy.
[Notation and setup] Notation for the finite quotients and the passage to the p-adic completion should be introduced once and used consistently; several symbols appear without prior definition in the convergence step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed review and valuable suggestions. We address each major comment below and plan to incorporate the necessary revisions to strengthen the manuscript.

read point-by-point responses

Referee: The existence proof relies on convergence of solutions constructed on finite quotients to a solution on the full Tate curve. No uniform a-priori bounds (independent of the level of the quotient) or compactness argument in a suitable function space controlling the non-Archimedean Green's function sum are supplied; without these, the limiting object may fail to satisfy the mean-field equation.

Authors: We thank the referee for pointing this out. The explicit finite-sum form of the Green's function on the Tate curve does permit uniform a-priori bounds independent of the quotient level, owing to the valuation structure controlling the sum. To make the argument fully rigorous, we will add explicit estimates and a compactness argument (via a non-Archimedean analogue of Arzelà–Ascoli) in the revised manuscript, ensuring the limit satisfies the mean-field equation. revision: yes
Referee: The uniqueness statement is restricted to 'some parameter region,' but the precise interval of the parameter (and the proof that uniqueness holds exactly there) is not compared quantitatively with the corresponding Archimedean threshold; this leaves the claimed resemblance unverified.

Authors: We agree that greater precision and a direct comparison would strengthen the parallel with the Archimedean case. In the revision we will state the exact parameter interval for uniqueness and include a quantitative comparison with the corresponding classical threshold. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses explicit construction and independent limit passage

full rationale

The paper constructs the Green's function directly as a finite sum on the Tate curve and obtains existence via convergence of solutions built on finite quotients. No equation or claim reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing step rests solely on self-citation. The convergence argument is presented as an independent approximation procedure, and uniqueness is handled separately in a restricted parameter region. The derivation therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the paper assumes standard background results on the Tate curve, rigid analytic spaces, and the Laplacian operator in non-Archimedean geometry.

axioms (1)

domain assumption The Tate curve admits a well-defined Laplacian operator with discrete spectrum
Invoked when studying the spectrum and constructing the Green's function.

pith-pipeline@v0.9.0 · 5401 in / 1135 out tokens · 58268 ms · 2026-05-15T10:25:04.515057+00:00 · methodology

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

construct the associated Green's function as a finite sum... eigenvalues of D ... λ_{n,l} = -p^{n-1}-p^{n-2} + (p^{-l}+p^{l+1-m})/p ... G(x,y) expressed as finite sum over characters
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

existence on the Tate curve by the convergence of such solutions... uniqueness for ρ ∈ (0, ρ*)

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

30 extracted references · 30 canonical work pages

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Uniqueness of least energy solutions to a semilinear elliptic equation inR 2

Chang Shou Lin. Uniqueness of least energy solutions to a semilinear elliptic equation inR 2. Manuscripta Math., 84(1):13–19, 1994. Yaojia Sun, 24210180117@m.fudan.edu.cn School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China 35

work page 1994

[1] [1]

Kazdan and F

Jerry L. Kazdan and F. W. Warner. Curvature functions for open 2-manifolds.Ann. of Math. (2), 99:203–219, 1974

work page 1974

[2] [2]

Concentration phenomena of two- vortex solutions in a chern–simons model.Annali della Scuola Normale Superiore di Pisa

Chiun-Chuan Chen, Chang-Shou Lin, and Guofang Wang. Concentration phenomena of two- vortex solutions in a chern–simons model.Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 3(2):367–397, 2004

work page 2004

[3] [3]

Topological degree for a mean field equation on riemann surfaces.Communications on Pure and Applied Mathematics, 56(12):1667–1727, 2003

Chiun-Chuan Chen and Chang-Shou Lin. Topological degree for a mean field equation on riemann surfaces.Communications on Pure and Applied Mathematics, 56(12):1667–1727, 2003

work page 2003

[4] [4]

Existence of bubbling solutions for chern–simons model on a torus.Archive for Rational Mechanics and Analysis, 207(2):353–392, 2013

Chang-Shou Lin and Shusen Yan. Existence of bubbling solutions for chern–simons model on a torus.Archive for Rational Mechanics and Analysis, 207(2):353–392, 2013

work page 2013

[5] [5]

Uniqueness of solutions to mean field equations of liouville type in two- dimension

Chang-Shou Lin. Uniqueness of solutions to mean field equations of liouville type in two- dimension. InGeometry and analysis. No. 1, volume 17 ofAdvanced Lectures in Mathematics (ALM), pages 419–446. 2011

work page 2011

[6] [6]

Mean field equation of liouville type with singular data: topological degree.Communications on Pure and Applied Mathematics, 68(6):887–947, 2015

Chiun-Chuan Chen and Chang-Shou Lin. Mean field equation of liouville type with singular data: topological degree.Communications on Pure and Applied Mathematics, 68(6):887–947, 2015

work page 2015

[7] [7]

Existence and non-existence of solutions of the mean field equations on flat tori.Proceedings of the American Mathematical Society, 145(9):3989–3996, 2017

Zhijie Chen, Ting-Jung Kuo, and Chang-Shou Lin. Existence and non-existence of solutions of the mean field equations on flat tori.Proceedings of the American Mathematical Society, 145(9):3989–3996, 2017

work page 2017

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Even solutions of some mean field equations at non- critical parameters on a flat torus.Proceedings of the American Mathematical Society, 150(4):1577–1590, 2022

Ting-Jung Kuo and Chang-Shou Lin. Even solutions of some mean field equations at non- critical parameters on a flat torus.Proceedings of the American Mathematical Society, 150(4):1577–1590, 2022

work page 2022

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Kazdan-warner equation on graph.Cal- culus of Variations and Partial Differential Equations, 55(4):Art

Alexander Grigor’yan, Yong Lin, and Yunyan Yang. Kazdan-warner equation on graph.Cal- culus of Variations and Partial Differential Equations, 55(4):Art. 92, 13, 2016

work page 2016

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Kazdan-warner equation on infinite graphs.Journal of the Korean Mathematical Society, 55(5):1091–1101, 2018

Huabin Ge and Wenfeng Jiang. Kazdan-warner equation on infinite graphs.Journal of the Korean Mathematical Society, 55(5):1091–1101, 2018

work page 2018

[11] [11]

Existence of solutions to mean field equations on graphs.Communications in Mathematical Physics, 377(1):613–621, 2020

An Huang, Yong Lin, and Shing-Tung Yau. Existence of solutions to mean field equations on graphs.Communications in Mathematical Physics, 377(1):613–621, 2020

work page 2020

[12] [12]

A heat flow for the mean field equation on a finite graph.Calc

Yong Lin and Yunyan Yang. A heat flow for the mean field equation on a finite graph.Calc. Var. Partial Differential Equations, 60(6):Paper No. 206, 15, 2021

work page 2021

[13] [13]

Brouwer degree for mean field equation on graph.Bull

Yang Liu. Brouwer degree for mean field equation on graph.Bull. Korean Math. Soc., 59(5):1305–1315, 2022

work page 2022

[14] [14]

Brouwer degree for Kazdan-Warner equations on a connected finite graph.Adv

Linlin Sun and Liuquan Wang. Brouwer degree for Kazdan-Warner equations on a connected finite graph.Adv. Math., 404:Paper No. 108422, 29, 2022

work page 2022

[15] [15]

A heat flow with sign-changing prescribed function on finite graphs.J

Yang Liu and Mengjie Zhang. A heat flow with sign-changing prescribed function on finite graphs.J. Math. Anal. Appl., 528(2):Paper No. 127529, 17, 2023. 34

work page 2023

[16] [16]

Variational approach to the mean field equation on finite graphs

Yi Li and Qianwei Zhang. Variational approach to the mean field equation on finite graphs. J. Math. Anal. Appl., 551(1):Paper No. 129614, 21, 2025

work page 2025

[17] [17]

Elliptic functions, green functions and the mean field equations on tori.Annals of Mathematics, 172(2):911–954, 2010

Chang-Shou Lin and Chin-Lung Wang. Elliptic functions, green functions and the mean field equations on tori.Annals of Mathematics, 172(2):911–954, 2010

work page 2010

[18] [18]

A function theoretic view of the mean field equations on tori

Chang-Shou Lin and Chin-Lung Wang. A function theoretic view of the mean field equations on tori. InRecent advances in geometric analysis, volume 11 ofAdvanced Lectures in Mathematics (ALM), pages 173–193. 2010

work page 2010

[19] [19]

Zamolodchikov and Al

A. Zamolodchikov and Al. Zamolodchikov. Conformal bootstrap in liouville field theory.Nu- clear Physics B, 477(2):577–605, 1996

work page 1996

[20] [20]

Analytic continuation of liouville theory

Daniel Harlow, Jonathan Maltz, and Edward Witten. Analytic continuation of liouville theory. Journal of High Energy Physics, (12):071, i, 104, 2011

work page 2011

[21] [21]

Resurgence in liou- ville theory.Journal of High Energy Physics, (1):Paper No

Nathan Benjamin, Scott Collier, Alexander Maloney, and Viraj Meruliya. Resurgence in liou- ville theory.Journal of High Energy Physics, (1):Paper No. 38, 44, 2025

work page 2025

[22] [22]

W. A. Z´ u˜ niga Galindo.Pseudodifferential equations over non-Archimedean spaces, volume 2174 ofLecture Notes in Mathematics. Springer, Cham, 2016

work page 2016

[23] [23]

Green’s functions for Vladimirov derivatives and Tate’s thesis.Communications in Number Theory and Physics, 15(2):315–361, 2021

An Huang, Bogdan Stoica, Shing-Tung Yau, and Xiao Zhong. Green’s functions for Vladimirov derivatives and Tate’s thesis.Communications in Number Theory and Physics, 15(2):315–361, 2021

work page 2021

[24] [24]

Quadratic reciprocity from a family of adelic conformal field theories

An Huang, Bogdan Stoica, and Xiao Zhong. Quadratic reciprocity from a family of adelic conformal field theories. arXiv preprint arXiv:2202.01217, 2022

work page arXiv 2022

[25] [25]

arXiv preprint arXiv:2509.09446, 2025

Hazem Hassan.p-adic higher green’s functions for stark-heegner cycles. arXiv preprint arXiv:2509.09446, 2025

work page arXiv 2025

[26] [26]

Green’s function on the tate curve

An Huang, Rebecca Rohrlich, Yaojia Sun, and Eric Whyman. Green’s function on the tate curve. arXiv preprint arXiv:2512.24935, 2025

work page arXiv 2025

[27] [27]

Boundary value problems for p-adic elliptic parisi-z´ u˜ niga diffusion

Patrick Erik Bradley. Boundary value problems for p-adic elliptic parisi-z´ u˜ niga diffusion. arXiv preprint arXiv:2504.06288, 2025

work page arXiv 2025

[28] [28]

Diffusion operators onp-adic analytic manifolds

Patrick Erik Bradley. Diffusion operators onp-adic analytic manifolds. arXiv preprint arXiv:2510.22563, 2025

work page arXiv 2025

[29] [29]

An Huang and Christian B. Jepsen. A glimpse into the ultrametric spectrum. arXiv preprint arXiv:2601.03738, 2026

work page arXiv 2026

[30] [30]

Uniqueness of least energy solutions to a semilinear elliptic equation inR 2

Chang Shou Lin. Uniqueness of least energy solutions to a semilinear elliptic equation inR 2. Manuscripta Math., 84(1):13–19, 1994. Yaojia Sun, 24210180117@m.fudan.edu.cn School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China 35

work page 1994