I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of scheme is of utmost importance in number theory but what about the cohomology of sheaves ? Why is étale cohomology useful in number theory for instance ?
Note : About the étale cohomoloy groups, I already know that they are very important examples of representations of absolute Galois groups.
This is slightly too log to be a comment, but certainly too short to be a defintive answer. Apologees in advance.
I would suggest reading the volume 2 book ("Cohomology of algebraic varieties", by Danilov) of the EMS series "Algebraic Geometry". You will learn a lot about étale cohomology, in a non-technical way in this book. The basic motivation for Grothendieck to develop étale cohomology was to micmick an argument of Serre, who proved some analogues of the Weil conjectures over the field of complex numbers (see Analogues K"ahleriennes de certaines conjecture de Weil).
Grothendieck noticed early on that if we had a cohomology theory for varieties over finite fields having nice properties (like 6 operations, Lefschetz trace formula and so on) then the Weil conjectures "would open like a nutshell when the time is ripe : hand pressure is enough, the shell opens like a perfectly ripened avocado!" (the quote is from Recoltes et Semailles, page 552-3)
It was realized, only a few years later, that an analogue of the Hodge conjecture in étale cohomology could be formulated in terms of Galois actions on étale cohomology groups of a smooth projective variety over a finite field. This is the famous Tate cojecture. This conjecture is really remarkable as it would, for instance, imply most of the Grothendieck's standard conjectures. In the aftermath of this conjecture, some area of representation theory merged with arithmetic in order to study in more details the etale cohomology groups as representation spaces.
