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arXiv · 2111.12660

Algebraic integers with conjugates in a prescribed distribution

Alexander Smith
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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.

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Alexander Smith. 2021-11-24. Algebraic integers with conjugates in a prescribed distribution. https://arxiv.org/abs/2111.12660

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