Adelically summable normalized weights and adelic equidistribution of effective divisors having small diagonals and small heights on the Berkovich projective lines

TL;DR

This paper introduces adelically summable normalized weights on Berkovich projective lines and proves an adelic equidistribution theorem for effective divisors with small heights and diagonals, generalizing previous results.

Contribution

It defines adelically summable normalized weights and establishes a new adelic equidistribution theorem for divisors with small heights and diagonals on Berkovich lines.

Findings

Introduces the concept of adelically summable normalized weights.

Proves adelic equidistribution of divisors with small heights and diagonals.

Generalizes Ye's equidistribution results for algebraic numbers.

Abstract

We introduce the notion of an adelically summable normalized weight $g$, which is a family of normalized weights on the Berkovich projective lines satisfying a summability condition. We then establish an adelic equidistribution of effective $k$-divisors on the projective line over the separable closure $k_s$ in $\overline{k}$ of a product formula field $k$ having small $g$-heights and small diagonals. This equidistribution result generalizes Ye's for the Galois conjugacy classes of algebraic numbers with respect to quasi-adelic measures.