Math7280 : References

As well as the main text, Shimura's book on modular forms, you might find it useful to refer to some of the following. I will be adding more references, and more comments later, including suggestions for possible prereading, and also more advanced reading. The following are all at about the level of the course.

&bull G. Shimura, Introduction to the arithmetic theory of modular forms

The course text book. This was chosen because it is a classic text, written by a master in the field, and is packed with a great deal of information, so is also useful as a reference. It's also not too expensive, as it's available as a paperback. We will cover about the first four chapters.

&bull J.-P. Serre, A Course in Arithmetic

This covers modular forms for the level 1 case. What we cover over the first few weeks will be similar to chapter VII of this book. You do not need to read the previous chapters in detail to follow chapter VII. Chapter II gives an introduction to p-adic numbers, which we will cover later in the course

&bull Cassels, Lectures on elliptic curves

A good, fairly consise introduction to elliptic curves, from a concrete point of veiw. We will not cover elliptic curves in this much detail, but we will probably at least mention most of the material in the first 12 chapters, especially the group law (chapter 7) and p-adics (chapter 2).

&bull Silverman, The arithmetic of elliptic curves
- Silverman, Advanced topics in the arithmetic of elliptic curves

These are great books on elliptic curves. We will not cover anything like as much as what's in these, but they are good reference material. You need the second (advanced) book for the relationship between modular forms and elliptic curves. These are good books to fill in some of the gaps in the presentation I will give on elliptic curves.

&bull T. Miyake, Modular forms
- S. Lang, Introduction to modular forms
- N. Koblitz, Introduction to elliptic curves and modular forms
- Knapp, Elliptic curves
- Huesemoeller, Elliptic curves
- Diamond and Im, Modular forms and modular curves

These are all good books on modular forms and/or elliptic curves, which you can refer to for further information.

&bull Gouvea, p-adic numbers. An introduction

We will need p-adic numbers in this course; we just need basic definitions, but if you want to explore further, this book is a good place to start.

&bull Jones and Singerman, Complex functions. An algebraic and geometric viewpoint.

This is a great book on the complex analytic aspects of what we will cover. Although it is not obvious from the title, some of the chapters are perfect for an introduction to modular forms, and especially fundamental domains. This would be good as prereading before the course.