Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910767207415808 |
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| author | Bradley, Patrick Erik |
| author_facet | Bradley, Patrick Erik |
| contents | A diffusion operator on the $K$-rational points of a Tate elliptic curve $E_q$ is constructed, where $K$ is a non-archimedean local field, as well as an operator on the Berkovich-analytification $E_q^{an}$ of $E_q$. These are integral operators for measures coming from a regular $1$-form, and kernel functions constructed via theta functions. The second operator can be described via certain non-archimedan curvature forms on $E_q^{an}$. The spectra of these self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the $K$-rational points of $E_q$ is retrieved from the spectrum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_03570 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields Bradley, Patrick Erik Number Theory Algebraic Geometry 14H52 (Primary), 58J35 (Secondary) A diffusion operator on the $K$-rational points of a Tate elliptic curve $E_q$ is constructed, where $K$ is a non-archimedean local field, as well as an operator on the Berkovich-analytification $E_q^{an}$ of $E_q$. These are integral operators for measures coming from a regular $1$-form, and kernel functions constructed via theta functions. The second operator can be described via certain non-archimedan curvature forms on $E_q^{an}$. The spectra of these self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the $K$-rational points of $E_q$ is retrieved from the spectrum. |
| title | Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields |
| topic | Number Theory Algebraic Geometry 14H52 (Primary), 58J35 (Secondary) |
| url | https://arxiv.org/abs/2312.03570 |