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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).

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    $\begingroup$ I don’t think a single textbook or lecture series will provide you with a robust picture of the “advanced theory” of half of the topics you mentioned in the post. For example, Hasse principle is still a subject of intense research. $\endgroup$
    – Stanley Yao Xiao
    Commented Aug 18, 2020 at 1:56
  • $\begingroup$ Fair enough. I'm just looking for anything which can push forward my knowledge in this area. $\endgroup$
    – Thomas
    Commented Aug 18, 2020 at 3:29

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This may be a good choice for someone who (like yourself) is already superficially acquainted with some of the definitions and methods of Diophantine geometry:

The following two are great expository articles (especially the first), which provided me with plenty of inspiration back in the day:

Henri Darmon has a couple of nice articles on the topic of rational points on curves:

  • Rational points on curves (link)

  • Rational points on modular elliptic curves (link)

Anthony Varilly-Alvarado has a number of very good introductions to the topic of rational points on different types of surfaces:

  • Lectures on the Arithmetic of del Pezzo surfaces (link)

  • Arithmetic of K3 surfaces (link)

Alexei Skorobogatov taught a course in 2013 on the topic of rational points on surfaces and higher-dimensional varieties. The notes strike a great balance between accessibility and generality:

  • Arithmetic geometry: rational points (link)

Then there are these notes by Yonatan Harpaz on rational points on elliptic surfaces:

  • Rational points on elliptic fibrations -- Course notes (link)

Finally (for now), Brendan Hassett has a nice article on the topic of potential density of rational points on varieties, which is very interesting as well:

  • Potential density of rational points on algebraic varieties (link)
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E.g.

by Henri Cohen.

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It is difficult to get far in the modern theory without some algebraic geometry.

This is the approach taken in the book:

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If you are interested in applications of Baker's method, Schmidt's subspace theorem etc., then you might like the following recent books by Evertse and Győry:

  • Discriminant equations in Diophantine number theory, New Mathematical Monographs, 32, Cambridge University Press, Cambridge, 2017.
  • Unit equations in Diophantine number theory, Cambridge Studies in Advanced Mathematics, 146, Cambridge University Press, Cambridge, 2015.
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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.

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