I recently got a copy of a very interesting new textbook entitled A First Course in Modular Forms by Fred Diamond and Jerry Shurman. Fred was a student of Andrew Wiles at Princeton, and came here to Columbia as a junior faculty member at the same time I did. He now teaches at Brandeis.
The title of the book is a bit deceptive, what it is really about is what used to be called the Taniyama-Shimura-Weil (or some subset of those names) conjecture, but now is often known as the Modularity Theorem. Most of this theorem was proved by Andrew Wiles (with help from Richard Taylor), who famously used his result to prove Fermat’s last theorem. More recently, the proof of the full theorem was completed by Fred, together with collaborators Christophe Breuil, Brian Conrad and Richard Taylor. Stating the modularity theorem precisely requires some serious mathematical technology, an imprecise statement is the “All rational elliptic curves arise from modular forms”. This fits into the Langlands program of establishing a correspondence between arithmetic objects (in this case elliptic curves over the rational numbers), and analytic objects (in this case modular forms). If one can do this, typically the fact that the analytic objects are pretty well understood allows one to get a vast amount of very deep information about the more mysterious arithmetic objects (e.g. being able to count solutions to equations over the rationals or integers).
The book takes an interesting approach to the Modularity Theorem, not trying to actually prove it. The proof involves highly sophisticated mathematical technology, and really understanding it is still the province of experts. If one wants to try and learn this technology, two places to look are the volumes Modular Forms and Fermat’s Last Theorem and Arithmetic Algebraic Geometry, which are the proceedings of two different instructional conferences. Instead of trying to give a proof, Diamond and Shurman’s book explains exactly what the various related versions of the Modularity Theorem say. This covers a range of beautiful mathematical ideas, much of which hasn’t before had a particularly readable exposition. Until now, the main reference for some of this material has been Shimura’s Introduction to Arithmetic Theory of Automorphic Functions, a famously difficult text.
The book is advertised as “A First Course” and attempts to minimize the prerequisites necessary to read it, making it conceivable to even use the book with advanced undergraduates. This is a worthy goal, but may be a bit over-ambitious. I suspect most people will get more out of the book if they already have had exposure to some of this mathematics at a slightly more basic level. One place to get this is Neal Koblitz’s Introduction to Elliptic Curves and Modular Forms. But this really is a wonderful book, making accessible parts of the really beautiful mathematics which mathematicians have been making great progress in understanding over the last decade.