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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.

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  • $\begingroup$ Not to discourage you or anything, but if you just go through Hartshorne or one of the other sources you mention carefully, that seems more than enough. $\endgroup$
    – Donu Arapura
    Commented Apr 12, 2015 at 6:42
  • $\begingroup$ (I mean for someone at your stage. Falting's proof requires more…) $\endgroup$
    – Donu Arapura
    Commented Apr 12, 2015 at 6:44
  • $\begingroup$ @DonuArapura: Sorry, I didn't understand that last comment. Did you mean that what is covered in Hartshorne's book is enough to start reading Silverman-Cornell? And what do you mean by "Falting's proof requires more…"? $\endgroup$
    – pjox
    Commented Apr 12, 2015 at 7:01
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    $\begingroup$ @user74230, I'm aware this is a long-term goal, and I know it is currently impossible for me to understand something as complex as this. That is why I want to read and educate myself, I mean, I've been doing that with algebraic geometry and I want to do it with arithmetic geometry. Also, you are telling me the proof is awesome, I think that's enough reason to want to understand it :-). Oh, and thank you so much for the Stanford notes! that was exactly what I was asking for. If you know other resources, they will also be appreciated. $\endgroup$
    – pjox
    Commented Apr 12, 2015 at 17:58
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    $\begingroup$ @user74230 Actually, there is now a proof of the isogeny theorem by Masser and Wustholz. $\endgroup$
    – naf
    Commented Apr 13, 2015 at 6:18

3 Answers 3

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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.

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    $\begingroup$ I'm well aware of Vojta's proof, and honestly I do want to study it as well, but I wanted to understand Faltings's proof first. I know that might be an odd decision, but the reason I want to do it that way is that I'm as interested in all the prerequisites as I'm interested in the theorem itself. Now, this approach might be mistaken; I'm not sure, that's why I'm asking for advice. But I'm sure I want to understand both proofs, that is my longer-term project. I don't know why, but I'm simply amazed and intrigued by this theorem! Thank you so much for you answer! $\endgroup$
    – pjox
    Commented Apr 13, 2015 at 17:44
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"Arithmetic Geometry" by Cornell-Silverman is a collection of more or less independently readable chapters of subjects playing a role in Faltings' proof. After some history in the first chapter, the second chapter is the translation of Faltings' original paper, thus you should skip this chapter on your first read. The remaining chapters should be well understandable with the knowledge of Hartshorne/Liu.

I think a very nice introduction to this topic is the paper "Arithmetic on Curves" by Mazur (https://projecteuclid.org/euclid.bams/1183553167). It requires only a very little background knowledge. Another paper, which helped me to get a good overview of Faltings' proof, is "Finiteness Problems in Diophantine Geometry" by Zarhin and Parshin (http://arxiv.org/abs/0912.4325). This paper is much more detailed than the one by Mazur, hence it would be helpful to have some knowledge from Cornell/Silverman in mind, especially about Abelian varieties.

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There is Bombieri-Gubler, Heights in Diophantine Geometry (Chapter 11), Faltings-Wüstholz, Rational Points and Milne's lecture notes http://www.jmilne.org/math/CourseNotes/av.html (Chapter IV; you already mentioned them).

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