Required Background to Study L-Functions and Elliptic Curves

Question

I am a mathematics student about to enter graduate school. I have interests in many areas of mathematics, but two areas of study that sound interesting are $L$-functions and elliptic curves. What branches of mathematics would I need to understand and what types of courses might I need at the graduate level if I chose to study either of these subjects? I realize this question may not have a well-defined answer, but any suggestions are welcome.

Answer 1

It seems that your core interest is in number theory. For your purposes you will need to learn analytic number theory, algebraic number theory and algebraic geometry.

For analytic and algebraic number theory I recommend you start with Serre's "A Course in Arithmetic". The book is surprisingly short but has all the basic things you need to get started in number theory. It is by no mean easy, Serre writes concise proofs but you need to fill in the details. You will know what to do after understanding this book fully.

For the geometric side, if you already have a fairly good understanding of commutative algebra, you can read the renowned Hartshorne book. If not, I recommend Eisenbud's "Commutative Algebra -- with a View Toward Algebraic Geometry". Either of these books will take you a long time to finish, but you only need a fraction of them to get started on elliptic curves.

For elliptic curves, "Rational Points on Elliptic Curves" by Silverman and Tate require almost no prerequisite, you can start reading this one immediately. After having some basic knowledge of algebraic geometry, "The Arithmetic of Elliptic Curves" by Silverman is a good second step.