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arXiv · 2312.03570

Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields

Patrick Erik Bradley
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Abstract

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.

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Patrick Erik Bradley. 2023-12-06. Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields. https://doi.org/10.2140/pjm.2025.334.13

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