An Inner Product on Adelic Measures: With Applications to the Arakelov-Zhang Pairing
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866911879727677440 |
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| author | Oberly, Peter J. |
| author_facet | Oberly, Peter J. |
| contents | We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov-Zhang pairing from arithmetic dynamics. We prove a sharp lower bound on the norm of adelic measures with points of small adelic height. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with the Arakelov height on the space of rational functions with fixed degree. As a consequence, the Arakelov-Zhang pairing of two rational maps $f$ and $g$ can be bounded from below as a function of $g$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_10822 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An Inner Product on Adelic Measures: With Applications to the Arakelov-Zhang Pairing Oberly, Peter J. Number Theory Dynamical Systems 37P30, 37P05 We define an inner product on a vector space of adelic measures over a number field. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov-Zhang pairing from arithmetic dynamics. We prove a sharp lower bound on the norm of adelic measures with points of small adelic height. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with the Arakelov height on the space of rational functions with fixed degree. As a consequence, the Arakelov-Zhang pairing of two rational maps $f$ and $g$ can be bounded from below as a function of $g$. |
| title | An Inner Product on Adelic Measures: With Applications to the Arakelov-Zhang Pairing |
| topic | Number Theory Dynamical Systems 37P30, 37P05 |
| url | https://arxiv.org/abs/2207.10822 |