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Mean field equation and Green's function on the Tate curve

Yaojia Sun

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. And we establish the existence and uniqueness results of the mean field equation on this space. To address this problem, we begin by presenting the solution structure and proving its uniqueness under specific conditions on the finite quotient. We then establish convergence of the solutions to obtain existence results. Notably, the well-posedness of the solution resembles that in the Archimedean case.

Mean field equation and Green's function on the Tate curve

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. And we establish the existence and uniqueness results of the mean field equation on this space. To address this problem, we begin by presenting the solution structure and proving its uniqueness under specific conditions on the finite quotient. We then establish convergence of the solutions to obtain existence results. Notably, the well-posedness of the solution resembles that in the Archimedean case.
Paper Structure (13 sections, 21 theorems, 120 equations)

This paper contains 13 sections, 21 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Spectrum of D and the Green's function
  4. Multiplicative Characters of Zp*
  5. Eigenvalues and eigenfunctions for D when m=1
  6. Construction of the Green's function when m=1
  7. General m case
  8. Mean field equation on the finite quotient
  9. Structure of the solution
  10. Uniqueness of the solution
  11. Mean field equation on the Tate curve
  12. Convergence to the Tate curve
  13. Uniqueness of the solution

Key Result

Theorem 1.3

The eigenvalues of $D$ for $\mathbb{Q}_p^{\times}/p^{m\mathbb{Z}}$ are $\lambda_{0,l}\in\sigma(\boldsymbol{R})$ with $\lambda_{0,0}=0$, and $\lambda_{n,l}=-p^{n-1}-p^{n-2}+\frac{p^{-l}+p^{l+1-m}}{p}$ for $n\geq 1$, with the corresponding eigenfunctions $\phi_0^l(x)=\sum_{k=0}^{m-1}q_{k,l}\boldsymbol when $v_p(x)=v_p(y)=l$ and $v_p(d(x,y))=M$.

Theorems & Definitions (42)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3: The spectrum of $D$ and the corresponding Green's function
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 32 more