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

TL;DR

This paper introduces an inner product on adelic measures over a number field, linking it to the Arakelov-Zhang pairing and providing bounds related to heights and rational maps in arithmetic dynamics.

Contribution

It defines a new inner product on adelic measures, connects it to the Arakelov-Zhang pairing, and establishes bounds involving heights and rational maps.

Findings

The norm governs weak convergence at each place of the number field.

The Arakelov-Zhang pairing equals the square of the norm difference of adelic measures.

A sharp lower bound on the norm of adelic measures with small adelic height is proved.

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 $g$.