# Course 18.784: Seminar in Number Theory

## Basic Info

Instructor:
Scott Carnahan
Class time:
MWF, 11am-12pm
Location:
Building 2, room 102
Office Hours:
Monday 2-3pm
Tuesday 1-2pm
Wednesday 8-?pm
You are encouraged to request appointments if these times are inconvenient.
Prerequisites:
You should know what a group action is, and you should be reasonably familiar with complex numbers (e.g., periodicity of the function e2πix).
Format:
Most of the class time will be given to student presentations. Before a talk is given, the speaker will come up with simple exercises for the class to do in groups after the talks. I will help the speakers prepare for this. I will also assign some short homework, but it will mostly involve individually writing up the short exercises. Finally, the institute requires a coherent mathematics paper of about 10 pages on a subject related to the course. I have a list of topics to suggest.

## Recommended Texts:

Serre, A Course in Arithmetic
This will be the basic reference for the class. We will follow chapter VII until we run out of material. It is somewhat dense, so you may want to draw on other sources for additional examples and motivation. This is reasonably close to my favorite possible treatment, although there are some anomalies, such as the unorthodox Bernoulli number convention.
Stein, Modular Forms, A Computational Approach
This book was published very recently. You can download a pdf, but the author would like it if you bought a copy. It has substantially more material than Serre, and gives you guidance for realizing explicit examples on a computer.
Apostol, Modular functions and Dirichlet series in number theory
I like the exposition a lot. The beginning of this class has a lot of overlap with the material starting with chapter 2.
Dolgachev, Lecture Notes
This may assume a little more geometry than you have seen, but it has interesting exercises.
Milne, Modular Functions and Modular Forms
The beginning seems to be roughly a permuted expansion of Serre.
Koblitz, Introduction to elliptic curves and modular forms
This is an interesting treatment of the subject, where all of the structure is motivated by the congruent number problem. If you haven't seen concrete arithmetic applications of elliptic curves, this is a good place to look.
Silverman, Advanced topics in the Arithmetic of Elliptic Curves
Only the first chapter is relevant to the class, but it's pretty good.
Ribet, Lecture notes
Shimura, Introduction to the arithmetic theory of automorphic functions
This is not an easy read, but can be a useful reference.
Lang, Introduction to modular forms
I haven't actually looked at this book, but some people like it.