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Algebraic integers with conjugates in a prescribed distribution

Alexander Smith

Abstract

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.

Algebraic integers with conjugates in a prescribed distribution

Abstract

Given a compact subset of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in . The distribution of conjugates of such an integer defines a probability measure on ; our main result gives a necessary and sufficient condition for a given probability measure on to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers with . We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts.
Paper Structure (10 sections, 33 theorems, 283 equations)

This paper contains 10 sections, 33 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on measures and polynomials
  3. Approximating polynomials
  4. Squarefree polynomials and the geometry of numbers
  5. Finding real polynomials to adjust
  6. Proof of Theorem \ref{['thm:main']}
  7. Towards the proof of Proposition \ref{['prop:early_ints_better']}
  8. Limits of measures and Serre's example
  9. The limit of Smyth's method
  10. Serre's example

Key Result

Theorem 1.1

We have $\lambda_{\textup{SSS}} < 1.89831$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 68 more