A few questions on model theory, especially model theory of rings I have never really read anything proper about model theory, so I have a few questions:
Someone told me that a school of logicians managed to give a very short proof of Falting's Theorem using model theory (and apparently "elimination of quantifiers"); I have not been able to find any reference about this, any ideas?
I also remember being given some very interesting information about the model theory of some rings. I do not recall the specifics, but it did give precise insights about how something like the complex numbers is easy to work with, whereas for the integers it is much harder (with problems such as Matiyasevich's Theorem/Hilbert's Tenth Problem). I have not managed to find anything similar to what I remember about it.
Does anyone have any information about all this, or a good source to learn from (which would, hopefully, mention these examples, at least the second one as it seems like it must be fairly standard)?
 A: A reasonably good "beginner's" book on model theory is David Marker's Model Theory: An Introduction. I'm not sure if he talks about the specific examples you have in mind, but he definitely talks about quantifier elimination, and likely other tools that would be used in the proofs you seek.
A: This book starts on a quite basic level and is packed with small applications of model theory to ring theory. Nothing as deep as Mordell-Lang, but it gives a good impression of how it is possible at all to derive statements about rings using logic.
I like this example of the kind of things you have to deal with: It is ridiculously easy to prove (Thm 1.13 in the above source) that any two algebraically closed fields of the same characteristic statisfy exactly the same first order statements in the language of rings. At first one is impressed and tries to prove something interesting about all fields of char 0 by proving it for complex numbers. Then one notes: It is very hard to express an interesting statement in the 1st order language of rings, basically all one can talk about is polynomial equations. So the key step in these applications of logic is often to find an ingenious expression for the things one is interested, in in a simple enough language (e.g. you have to find axioms whose resulting theory then satisfies quantifier elimination).
A: Hrushovski's proof, as it is presented in Bouscaren's book, is very involving. It requires a lot of background, technical ingredients, and, let's say, a certain dose of faith. It is not a one-liner at all. I'd rather recommend reading a more straight forward approach to the ch.0 case (using jet spaces of differential fields) given by A. Pillay and M. Ziegler and written out very carefully by Paul Baginski in his undergrad thesis.
A: I don't know about "short", but I believe the model-theoretic proof for Mordell's Conjecture (Fatlings' Theorem) was given by Ehud Hrushovski. 
(EDIT: and just to clarify, I'm quite sure the proof is a good deal more complex than elimination of quantifiers per se. Also, Terry Tao is currently holding a reading seminar aimed at understanding another, more recent, paper of Hrushovski). 
A: You might find this book helpful, if you want to get into the details. Disclaimer: I found it while googling for something else and figured I'd mention it. I haven't read it myself.
A: "The complex numbers are easy to deal with, whereas for the integers it is much harder..."
What you might have heard about is that the complete first-order theory of the complex numbers with just the ring operations -- which is the same as the theory of algebraically closed fields of characteristic zero -- is decidable (i.e. "computable"), as was first proved by Tarski.  So in principle one could write a computer program where you could input any first-order sentence in the language of rings, and in a finite amount of time it would tell you whether or not this sentence is true in the complex numbers.
However, if you look at the ring of integers, no such thing is true; the first-order theory of the integers is undecidable.  This is a theorem of Alonzo Church, and is closely related to Goedel's famous incompleteness theorem.
The negative answer to Hilbert's Tenth Problem is a different issue -- this doesn't follow immediately from Church's Theorem, and was proved much later, by Davis, Putnam, Julia Robinson, and Matiyasevich.
I think of these things as more "logical folklore" than model theory, per se -- any reasonable introductory book on mathematical logic (e.g. Enderton's A Mathematical Introduction to Logic) will have a lot to say about them.
A: As far as I know, there is no model theoretic proof of Faltings' Theorem itself.  Hrushovski's proof applies only to algebraic varieties over function fields and fails for varieties over number fields. 
On the other hand, one of Abraham Robinson's last works, a joint paper with Roquette published posthumously, Robinson, A.; Roquette, P. On the finiteness theorem of Siegel and Mahler concerning Diophantine equations, J. Number Theory 7 (1975), 121--176, contains a proof of Siegel's theorem on the finiteness of the number of integral points on curves of positive genus via nonstandard analysis.      
