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 is a very nicetwo are great expository articlearticles (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. 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:
Anthony Varilly-Alvarado has a number of very good introductions to the topic of rational points on different types of surfaces:
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)
