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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2510.22563 |
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| _version_ | 1866914190591000576 |
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| author | Bradley, Patrick Erik |
| author_facet | Bradley, Patrick Erik |
| contents | Kernel functions for Laplacian integral operators are constructed on $p$-adic analytic manifolds using charts and transition maps from an atlas with connected nerve complex. In the compact case, an operator of Vladimirov-Taibleson type parametrised by a real parameter $s$ is defined. Its kernel function uses a geodetic-like distance function on the nerve complex of its atlas. The $L^2$-spectrum of this operator is established, and it is shown that it gives rise to a Feller semigroup. In this way, the Cauchy problem for the corresponding heat equation is solved in the positive by a transition function of a Markov process. The existence of a heat kernel function and a Green function in the case $s > 1$ is proven. As an application, it is shown how to express the number of points on the reduction curve defined over the residue field of an elliptic curve with good reduction in terms of the eigenvalues of a Vladimirov-Taibleson-like operator. This provides for an alternative way of counting points on elliptic curves defined over finite fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22563 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffusion operators on $p$-adic analytic manifolds Bradley, Patrick Erik Analysis of PDEs 35P10, 11G07 Kernel functions for Laplacian integral operators are constructed on $p$-adic analytic manifolds using charts and transition maps from an atlas with connected nerve complex. In the compact case, an operator of Vladimirov-Taibleson type parametrised by a real parameter $s$ is defined. Its kernel function uses a geodetic-like distance function on the nerve complex of its atlas. The $L^2$-spectrum of this operator is established, and it is shown that it gives rise to a Feller semigroup. In this way, the Cauchy problem for the corresponding heat equation is solved in the positive by a transition function of a Markov process. The existence of a heat kernel function and a Green function in the case $s > 1$ is proven. As an application, it is shown how to express the number of points on the reduction curve defined over the residue field of an elliptic curve with good reduction in terms of the eigenvalues of a Vladimirov-Taibleson-like operator. This provides for an alternative way of counting points on elliptic curves defined over finite fields. |
| title | Diffusion operators on $p$-adic analytic manifolds |
| topic | Analysis of PDEs 35P10, 11G07 |
| url | https://arxiv.org/abs/2510.22563 |