TITLE: Reference request: Diophantine equations
QUESTION [9 upvotes]: I am looking for a textbook, or preferably lectures, on the subject of Diophantine equations. I am familiar with the basic principles of modular arithmetic, conics and the Hasse Principle, and the basics of elliptic curves, Mordell's Theorem etc (though I'm not up to the point where I can understand the proof).
What I need is something that takes me beyond the basics. Something which will teach me the advanced theory, and also teach me about diophantine surfaces (not just curves).

REPLY [3 votes]: To the books mentioned above I would add one more:

Rational and Nearly Rational Varieties (Cambridge Studies in Advanced Mathematics) by J. Kollár, K. E. Smith, and A. Corti.

The authors present a more or less elementary approach to the rationality questions using a mix of classical and modern methods.