If you are interested in arithmetic geometry and the theory of elliptic curves (Diophantine equations with geometry involved) then I can recommend The Arithmetic of Elliptic Curves, written by Joseph H. Silverman. It may or may not be a little difficult to read, but it is a rewarding journey.
About the Book
The first two chapters is preliminaries from algebraic geometry. If you don't know this theory from before, then you may spend a little time on these two chapters and you may also want to learn from another source.
The next two chapters is about the understanding of the geometry of elliptic curves and the group structure one can give on the points of an elliptic curve, this gives rise to a Projective Group Scheme.
The next five chapters spends time to look at elliptic curves defined over different fields and arithmetic objects.
Some things you will see are (among other things)
- How elliptic curves over the complex numbers relates to elliptic integrals and elliptic functions.
- The Weil Conjectures for elliptic curves. You define a generating function which encodes the number of points of the elliptic curves defined over finite fields. This object is called a $\zeta$-function. The Weil Conjecture is then a list of statements regarding this newly defined object.
- See the Mordell-Weil Theorem for elliptic curves defined over the rational numbers. This means that you can find a finite list of rational points of the elliptic curve, from which you can generate all other points. This is an important result in the foundations of the Birch and Swinnerton-Dyer conjecture. This conjecture gives a fascinating relationship between arithmetic, geometry and analysis, I think.
- How Elliptic Curves relates to problems in Diophantine Approximations
I think the book is great, but it can be a little bit difficult to read - perhaps.
If you want something slightly easier, then you maybe want to have a look at Rational Points on Elliptic Curves, which is also by Silverman.
The Birch and Swinnerton-Dyer Conjecture is a great motivation for studying elliptic curves and their arithmetic properties. For that reason, I also want to add an extra book which I think is a really good introduction to that conjecture. The book is called Elliptic Tales: Curves, Counting, and Number Theory and is written by Avner Ash. It begins at a very elementary level and successively builds up to the conjecture. It is entertaining and describes most of the concepts in a non-technical language.