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arXiv · 2402.02869

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

Patrick Erik Bradley
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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.

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Patrick Erik Bradley. 2024-05-28. Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms. https://arxiv.org/abs/2402.02869

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