Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910460800925696 |
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| author | Bradley, Patrick Erik |
| author_facet | Bradley, Patrick Erik |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms Bradley, Patrick Erik Number Theory Algebraic Geometry 14H42, 58J35 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. |
| title | Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms |
| topic | Number Theory Algebraic Geometry 14H42, 58J35 |
| url | https://arxiv.org/abs/2402.02869 |