Algebraic integers with conjugates in a prescribed distribution

• 成果类型: Article
• 署名作者: Smith, Alexander
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.200.1.2
• 发表日期: 2024
• 页码: 71-122
• 关键词: Polynomials

摘要

Given a compact subset Sigma of R 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 & sdot;deg(alpha). We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts.