Diophantine Approximation and Integral Points on Algebraic Curves

The thesis lies within the field of Diophantine approximation, which aims to study how well elements of a number field can be approximated by rational num- bers or by elements of simpler subfields. This approach, originally developed in the study of Diophantine equations, has made it possible to deepen the understanding of the interactions between the arithmetic properties of numbers and the structures of number fields. To address such problems it is essential to analyze the properties of absolute values on number fields, through which one can define the notion of distance between two elements, as well as the “height” of an element, understood informally as a measure of its arithmetic size and complexity. This perspective natu- rally leads to the study of algebraic varieties and curves, with particular attention to elliptic curves, which provide a privileged setting in which Diophantine techniques can be applied effectively. We will see how elliptic curves are endowed with a group structure, which will be essential in determining properties related to height and distances defined on the curve. Finally, the analysis culminates in Siegel’s Theo- rem, which provides a fundamental result concerning the finite number of integral points on elliptic curves defined over number fields. The thesis aims to illustrate how the ideas of Diophantine approximation, combined with geometric and arithmetic tools, lead to concrete results in the understanding of integral solutions of algebraic equations.