Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms

TL;DR

This paper constructs a heat operator on Mumford curves over non-archimedean fields, linking the spectrum of the heat equation to the genus of the curve through automorphic forms, enabling genus determination from spectral data.

Contribution

It introduces a novel self-adjoint operator on Mumford curves that connects spectral properties to geometric genus via automorphic forms.

Findings

Spectrum contains zero and finitely many limit points.

Zeros of a differential form correspond to non-eigenvalues.

Genus can be recovered from spectral data, especially in hyperelliptic cases.

Abstract

A self-adjoint operator is constructed on the $L_2$-functions on the $K$-rational points $X(K)$ of a Mumford curve $X$ defined over a non-archimedean local field $K$. It generates a Feller semi-group, and the corresponding heat equation describes a Markov process on $X(K)$. Its spectrum is non-positive, contains zero and has finitely many limit points which are the only non-eigenvalues, and correspond to the zeros of a given regular differential 1-form on $X(K)$. This allows to recover the genus of X from the spectrum. The hyperelliptic case allows in principle an explicit genus extraction.