The Mean field equation on the Tate curve

TL;DR

This paper analyzes the Laplacian spectrum and Green's function on the Tate curve, establishing existence and uniqueness of solutions to the mean field equation in a non-Archimedean setting.

Contribution

It constructs the Green's function as a finite sum and proves existence and uniqueness of solutions to the mean field equation on the Tate curve, a non-Archimedean analogue.

Findings

Constructed Green's function as a finite sum on the Tate curve.

Proved existence of solutions to the mean field equation via convergence of finite quotient solutions.

Established uniqueness of solutions in certain parameter regions.

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.