Algebraic integers with conjugates in a prescribed distribution

TL;DR

This paper characterizes which probability measures on a real set can be limits of conjugate distributions of algebraic integers, and applies this to find algebraic integers with specific trace properties and to construct abelian varieties with extreme point counts.

Contribution

It provides a necessary and sufficient condition for a probability measure to be a limit of conjugate distributions of algebraic integers, linking distribution theory with algebraic number properties.

Findings

Characterization of limit distributions of conjugates of algebraic integers.

Existence of infinitely many totally positive algebraic integers with bounded trace-to-degree ratio.

Application to constructing abelian varieties over finite fields with extremal point counts.

Abstract

Given a compact subset $\Sigma$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability measure on $\Sigma$; our main result gives a necessary and sufficient condition for a given probability measure on $\Sigma$ to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers $\alpha$ with $tr(\alpha) < 1.89831\cdot deg(\alpha)$. We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts.