Reference request: Diophantine equations 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).
 A: 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:

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*Marc Hindry, Joseph H. Silverman -- Diophantine Geometry: An Introduction, Graduate Texts in Mathematics 201, Springer (2000), https://doi.org/10.1007/978-1-4612-1210-2.

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

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*Mazur, Barry. Arithmetic on curves. Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 207--259. https://projecteuclid.org/euclid.bams/1183553167


*Mazur, Barry. On the passage from local to global in number theory (link)
Henri Darmon has a couple of nice articles on the topic of rational points on curves:

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*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:

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*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:

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*Arithmetic geometry: rational points (link)

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

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*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:

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*Potential density of rational points on
algebraic varieties (link)

A: E.g.

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*Number Theory: Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics 239, https://doi.org/10.1007/978-0-387-49923-9; and

*Number Theory: Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics 240, https://doi.org/10.1007/978-0-387-49894-2
by Henri Cohen.
A: 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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*Bjorn Poonen, Rational points on varieties, Graduate Studies in Mathematics 186 (2017), publisher page, Author pdf.

A: 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:

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*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.

A: To the books mentioned above I would add one more:

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*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.
