Università Roma Tre

Introduction to algebraic number theory:
Rings of integers in number fields and unique factorisation of ideals.
Absolute values in number fields.

The Weil Height and the Mahler measure:
Definitions and properties.
The product Formula
Northcott’s Theorem.
Kroneker’s Theorem.

Thue equations:
Thue’s Theorem on diophantine approximation.
Siegel’s Lemma.
Thue equations have a finite number of integer solutions.

Arithmetic dynamics:
(Pre)periodic points.
The canonical height.
Rational functions.

Diophantine equations in roots of unity:
Revision about roots of unity and cyclotomic polynomials.
The Theorem of Ihare-Serre-Tate.

Equidistribution:
Definitions and examples.
Bilu’s Theorem.
Bogomolov’s Conjecture.

Lecture notes