A First Course in Modular Forms

Description

this book only assumes complex analysis and simple group theory, yet manages to cover surprisingly many modern results in the theory of modular forms. the first 4 chapters present the basics, as covered in any intro to modular forms. but chapter 5 is a readable account of atkin-lehner theory from the '70s. although it's not discussed in this book, this theory leads directly into p-adic modular forms, an active topic in current number theory research. this is the first place i've seen all these results set out in one place. the proofs are similar in style to those in joe silverman's books on elliptic curves, although they are perhaps little more terse in this book (which i prefer). some of the material is presented in (actually!) straightforward exercises, for which there are ample hints at the back. i haven't read the the second half of the book yet, but apparently it aims to "explain" the modularity theorem. i don't know what they put in and what they leave out, but at the very least it seems like it would be a good starting place if you want to find out about L-functions, another important topic in current number theory research.there's the odd bamboozling typo, but that's pretty standard for the springer GTM's, but other than that it's very solid, and i would say the perfect companion to silverman's books, as the starting point in a number theory/arithmetic geometry library,