First, I apologize for my poor English.
I like number theory such as "when can prime $p$ be written as $x^2 + y^2$?" and "find the integer solutions of this equation." Because I've heard that these problems can be solved by arithmetic geometry, I want to study it.
So I've read Hartshorne's Algebraic Geometry and Neukirch's Algebraic Number Theory and so on. (I've heard that these are fundamental for arithmetic geometry.) And next I’m about to study Abelian varieties and etale cohomology for the same reason. But after reading list of contents of these books, I feel like that these are very abstract, and very distant from my first purpose.
So the question: what are applications of abelian variety and etale cohomology? I know the theory of etale cohomology solved the Weil's conjecture. But only this? I know this is the great achievement, but I want to know more applications. (and I do not know applications of the theory of abelian varieties at all.)
And the second question is: what is arithmetic geometry? What books should I read to study it's mainstream? I want, for example, books which tells me not only the definitions and fundamental properties of etale cohomology, but also interesting and elementary (like "find the integer solutions of this equation") applications. Or books which tells me many theories and techniques which are used in frontier researches.