ETD

• diophantine equations
• families of norm form equations
• norm form equations
• thue equations

This thesis focuses on families of norm form equations. We introduce the more traditional methods used to study this kind of Diophantine equations, namely Skolem's p-adic method and the linear forms in logarithms. In particular, we apply Skolem’s method to solve a parametric norm form equation in three variables of degree five.
We then turn to a recent approach developed by Amoroso, Masser, and Zannier, which allows to study families of norm form equations depending on an integer parameter t. We apply this method in order to effectively solve some families of norm form equations. In particular, we study certain generalizations of the equation previously solved via Skolem's method. We provide two alternative strategies for detecting functional solutions. Additionally, we present an alternative resolution of a family of Thue equations of degree four previously studied by Pethö.
The main hypotheses required to apply the method of Amoroso, Masser, and Zannier are closely related to the structure of the group of units of the coordinate ring of a curve associated to the given family of norm form equations. In the final part of the thesis, we study these unit groups, proving that they are finitely generated modulo constants and deriving formulas for their ranks. We also provide explicit results and effective algorithms for computing some minimal sets of generators for such groups.