On algebraic integers all conjugates of which belong to a given compact subset of the complex plane

TL;DR

This paper investigates which compact subsets of the complex plane contain finitely or infinitely many algebraic integers with all conjugates within the set, using potential theory and capacity concepts.

Contribution

It applies capacity theory to characterize the finiteness or infiniteness of algebraic integers totally in a compact set, especially unions of real segments.

Findings

Capacity less than 1 implies finitely many algebraic integers

Capacity greater than 1 implies infinitely many algebraic integers

For real segments, the critical length is 4

Abstract

The study of Frobenius endomorphism provides numerous information about its corresponding Abelian variety. To understand the action of the Frobenius endomorphism, one may be interested in its eigenvalues. According to Weil's third conjecture ("Riemann hypothesis over finite fields"), they all have absolute value less than or equal to $2g\sqrt{p}$. Thus, the eigenvalues of the Frobenius endomorphism all belong to the same compact subset of the complex plane, and are roots of the same monic polynomial with integer coefficients (the characteristic polynomial of the Frobenius endomorphism). Such complex numbers are called algebraic integers "totally" in a compact subset, which means algebraic integers all conjugates of which belong to a same given compact subset of the complex plane. The study of such algebraic integers helps to understand the eigenvalues of the Frobenius endomorphism, especially their distribution. In this paper, we will study the following question : under which conditions a compact subset of the complex plane has a finite or infinite number of algebraic integers "totally" in it ? The problem can be studied in light of the notion of capacity of a compact subset, which comes from potential theory. In this paper, we will present the theory of capacity and some theorems (Fekete, Szeg\"o, Robinson) derived from it that partially answer the question: in the case of a union of real segments, when the capacity is smaller (resp. larger) than 1, it contains a finite (resp. infinite) number of algebraic integers totally in it. For instance, for real line segments, the limit length is 4. This paper is written as part of a collective project conducted in \'Ecole Polytechnique (France). It is aimed towards undergraduate audience in mathematics, with basic knowledge in algebra, topology, analysis, and dwells into a modern topic of research.