The Mean field equation on the Tate curve

Summary · qwen2.5:32b

The paper explores the spectrum of the Laplacian on the Tate curve and constructs a Green's function as a finite sum, analogous to its Archimedean counterpart on the flat torus. It establishes existence and uniqueness for solutions of the mean field equation on this space, with proofs involving convergence of solutions from finite quotients. Notably, the well-posedness properties mirror those in the Archimedean setting.

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Exploring the Laplacian spectrum on Tate curve in this paper, leading to insights about Green's function and mean field equation, with connections to both rendering (for understanding the structure of solutions) and mathematics (for the core topic of study).

Excerpt

arXiv:2603.15706v3 Announce Type: replace 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.