Algebraic integers with conjugates in a prescribed distribution
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866929279624806400 |
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| author | Smith, Alexander |
| author_facet | Smith, Alexander |
| contents | Given a compact subset $Σ$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $Σ$. The distribution of conjugates of such an integer defines a probability measure on $Σ$; our main result gives a necessary and sufficient condition for a given probability measure on $Σ$ to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers $α$ with $tr(α) < 1.89831\cdot deg(α)$. We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_12660 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Algebraic integers with conjugates in a prescribed distribution Smith, Alexander Number Theory 11R06 (Primary) 11G10, 11G25 (Secondary) Given a compact subset $Σ$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $Σ$. The distribution of conjugates of such an integer defines a probability measure on $Σ$; our main result gives a necessary and sufficient condition for a given probability measure on $Σ$ to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers $α$ with $tr(α) < 1.89831\cdot deg(α)$. We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts. |
| title | Algebraic integers with conjugates in a prescribed distribution |
| topic | Number Theory 11R06 (Primary) 11G10, 11G25 (Secondary) |
| url | https://arxiv.org/abs/2111.12660 |