Elementary Effective Theory of Rational and Integer Points on Rational Curves
Prerequisiti
The prerequisites are modest: basic algebra and basic algebraic geometry.
The seminar series can be followed by students from III year onwards, and may be of interest also to PhD or post-doc students.
Programma
We shall discuss the problem of effectivity for the search of integer and rational points on algebraic curves. This problem is still open in genus >1 and in genus 1 is solved only for integral points.
In this course we shall completely discuss the case of genus 0, that is, the issue for rational curves.
For simplicity we shall limit ourselves to the case of the field \Q and the ring \Z, but of course we shall occasionally mention more general results.
We shall give complete proofs for rational and integral points on lines and conics.
We shall also illustrate the Theorem of Runge, which sometimes applies to curves of higher genus, and we shall present some applications of this.
Obiettivi formativi
This seminar series may be of interest for people interested in Number Theory, but it contains basic material on several topics which may be of interest to any mathematician.
Also, we stress that effectivity issues are very important from a conceptual viewpoint, as well as for problems of computational nature.
Riferimenti bibliografici
My book "Lecture Notes on Diophantine Analysis" Edizioni della Normale, contains some of the topics that I will present.
generally speaking, every good book in diophantine theory is relevant (e.g. Serre's, Lectures on the Mordell Weil Theorem, Hindry-Silverman, Bombieri-Gubler, Lang's diophantine geometry).
However the course will not follow the pattern of any book.
Other references will be given during the lectures.