I have read Hartshorne's *Algebraic Geometry* from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. 


* I want to know how to use scheme theory and its cohomology to solve arithmetic problems.
* I also want to learn something about moduli theory.


Would you please recommend some books or papers? Thank you very much!



Roadmap for studying arithmetic geometry
My suggestion, if you have really worked through most of Hartshorne, is to begin reading papers, referring to other books as you need them.


One place to start is Mazur's "Eisenstein Ideal" paper. The suggestion of Cornell--Silverman is also good. (This gives essentially the complete proof, due to Faltings, of the Tate conjecture for abelian varieties over number fields, and of the Mordell conjecture.) You might also want to look at Tate's original paper on the Tate conjecture for abelian varieties over finite fields,
which is a masterpiece. 


Another possibility is to learn etale cohomology (which you will have to learn in some form or other if you want to do research in arithemtic geometry). For this, my suggestion is to try to work through Deligne's first Weil conjectures paper (in which he proves the Riemann hypothesis), referring to textbooks on etale cohomology as you need them.


