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

TL;DR

The work develops a general theory of $p$-adic diffusion by constructing pseudo-differential operators invariant under finite group actions and solving the associated Cauchy problems via twisted and invariant heat kernels. It then applies this framework to the Gerritzen–Herrlich Teichmüller spaces parametrising Schottky uniformisations of Mumford curves, producing a self-adjoint operator $H_{rak{G}}$ on each reduction type component $ormat{F}(rak{G})$ whose spectrum detects the corresponding reduction graph. In genus $2$ the paper gives explicit spectral criteria for the three possible reduction graphs, distinguishing graphs containing mouth-shaped subgraphs from those that do not, and extends the analysis to genus $g\, ext{ge}\,3$ via a mouth/corner degree condition. The main result ties the graph-theoretic structure of the reduction graph to the spectral properties of $H_{rak{G}}$, enabling a spectral read-off of Mumford curve reduction types. The approach broadens the toolbox for reading geometric information from spectra on $p$-adic spaces and paves the way for diffusion and stochastic constructions on more general $p$-adic manifolds.

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 Teichmüller 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.

Figures (2)

• Figure 1: The three possible stable reduction graphs for Mumford curves of genus $g=2$.
• Figure 2: A mouth-shaped graph. The broken lines depict paths of equal length.

Theorems & Definitions (40)

• Theorem 2.1: Mumford
• Theorem 2.2: Gerritzen
• proof
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Definition 3.4
• Remark 3.5
• Lemma 3.6