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

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 enables the recovery of the genus of X from the spectrum. The hyperelliptic case permits an explicit determination.