Areal Weil Heights

TL;DR

This paper introduces areal Weil heights, extending the areal Mahler measure to an adelic setting, and establishes their properties including analogs of Kronecker's theorem and equidistribution results.

Contribution

It generalizes areal Mahler measures to adelic heights, constructs a p-adic analog, and proves fundamental theorems on small points and distribution behavior.

Findings

Proves an analog of Kronecker's theorem for areal Weil heights.

Establishes equidistribution theorems with unique or multiple limiting distributions.

Provides examples of algebraic integers distributing to disks with computed limiting heights.

Abstract

In 2008, Pritsker introduced the areal Mahler measure, which is defined using an integral over the unit disk, as opposed to the classical Mahler measure which is defined using an integral over the unit circle. In this paper we introduce areal Weil heights, which generalize the areal Mahler measure to the adelic setting. We use the framework of adelic heights established by Favre and Rivera-Letelier and we construct a $p$-adic analog for the area measure on a disk in $\mathbb{C}$. For areal Weil heights we prove an analog of Kronecker's theorem, which characterizes their small points and essential minima. Furthermore, we determine equidistribution theorems for areal Weil heights. In some cases, they have a unique limiting distribution for small points, while in others there are infinitely many limiting distributions. We conclude with examples. In one of our examples, we determine for which radii $r$ there exist sequences of conjugate sets of algebraic integers which uniformly distribute to the disk $D(0,r) \subset \mathbb{C}$, and we compute the limiting height for such sequences.