Moscow Mathematical Journal

Volume 25, Issue 4, October–December 2025 pp. 447–478.

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

Authors: Patrick Erik Bradley (1)
Author institution:(1) Institute of Geodesy, Karlsruhe Institute of Technology, Englerstr. 7, 76131 Karlsruhe, Germany

Summary:

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.

2020 Math. Subj. Class. 14H42, 58J35.

Keywords: Mumford curves, automorphic forms, $p$-adic analysis, heat equation.

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