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arXiv · 2207.10822

An Inner Product on Adelic Measures: With Applications to the Arakelov-Zhang Pairing

Peter J. Oberly
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Abstract

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 $g$.

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Peter J. Oberly. 2022-07-22. An Inner Product on Adelic Measures: With Applications to the Arakelov-Zhang Pairing. https://arxiv.org/abs/2207.10822

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