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).



Reference request: Diophantine equations
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:


* 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:


* 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](https://arxiv.org/abs/math/9307231))


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


* Rational points on curves ([link](http://www.math.mcgill.ca/darmon/pub/Articles/Expository/12.Clay/paper.pdf))
* Rational points on modular elliptic curves ([link](http://www.math.mcgill.ca/darmon/pub/Articles/Research/36.NSF-CBMS/chapter.pdf))


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](https://math.rice.edu/%7Eav15/Files/LeidenLectures.pdf))
* Arithmetic of K3 surfaces ([link](https://math.rice.edu/%7Eav15/Files/AWS2015Notes.pdf))


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](http://wwwf.imperial.ac.uk/%7Eanskor/arith_geom_files/TCCnotes.pdf))


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


* Rational points on elliptic fibrations -- Course notes ([link](https://www.math.univ-paris13.fr/%7Eharpaz/IHP_course_notes.pdf))


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](https://www.math.brown.edu/%7Ebhassett/papers/budapest/budalect.pdf))


