I am soon to become a graduate student and so I started a personal project; I want to understand Faltings's proof of the Mordell conjecture.
I want to get into arithmetic geometry (since I always liked both algebraic geometry and number theory) and I thought that understanding this proof might be a good start (mostly because the first time a read about it I was quiet surprised and intrigued).
I am already trying to formalize my knowledge on modern algebraic geometry studying from Hartshorne's "Algebraic Geometry" and Liu's "Algebraic Geometry and Arithmetic Curves".
So, my question is where should I continue after I "finish" formalizing my knowledge on modern algebraic geometry?
I know there's a book called "Arithmetic Geometry" edited by Silverman and Cornell which actually contains Faltings's proof, but I am not sure if the material covered in Liu's and Hartshorne's books is enough to dive into this more specialized content.
I'm also aware of Milne's notes on Abelian Varieties which contains Faltings's proof, but I haven't look at these in detail yet.
Best Answer
I just wanted to make sure that you're aware that there is another proof of the Mordell conjecture that is in many ways more natural, and that has allowed great generalizations. This is the proof due to Vojta using ideas from Diophantine approximation. Vojta's proof was simplified by Bombieri, and that version does not require very much algebraic geometry, certainly far less than Faltings' original proof. Bombieri's article is quite readable, or you can find the proof with more exposition in my book with Hindry, Diophantine Geometry: An Introduction. Faltings subsequently generalized the methods in Vojta's article to prove strong results concerning rational and integral points on subvarieties of abelian varieties:
(Faltings) Let $A/K$ be an abelian variety defined over a number field. Theorem 1: Let $X\subset A$ be a subvariety. If $X$ contains no translates of abelian subvarieties of $A$, then $X(K)$ is finite. Theorem 2: Let $U$ be an affine open subset of $A$ and let $R\subset K$ be a ring of $S$-integers for some finite set of places $S$. Then $U(R)$ is finite.
This is not to take anything away from Faltings's first proof, which is a tour de force and well worth studying.