Generalised diffusion on moduli spaces of $p$-adic Mumford curves

TL;DR

This paper constructs a pseudo-differential operator on non-archimedean fields, applies it to Mumford curves, and uses its spectrum to analyze properties of their reduction graphs.

Contribution

It introduces a new pseudo-differential operator invariant under group actions and applies it to study the spectral properties of Mumford curves.

Findings

Constructed a pseudo-differential operator on non-archimedean fields.

Solved the associated Cauchy problem for this operator.

Used the spectrum to infer properties of Mumford curves' reduction graphs.

Abstract

A construction of a pseudo-differential operator on non-archimedean local fields invariant under a finite group action is given together with the solution of the corresponding Cauchy problem. This construction is applied to parts of the Gerritzen-Herrlich Teichm\"uller space in order to obtain a self-adjoint operator whose spectrum can decide about certain properties of the reduction graph of the corresponding Mumford curves.