Ebook: Holomorphic Hilbert Modular Forms

 Garrett's introduction presents a diagram of the tortuous sequence of implications leading to the Arithmetic Structure Theorem, and one realizes with some astonishment that his derivation of this result uses little more than elementary complex analysis, measure theory, and elementary algebraic number theory. One also realizes that these simple ingredients have been combined to present exemplary applications of most of the standard techniques of the analytic theory of automorphic forms. Better still, starting from scratch, Garrett has succeeded in developing his subject matter to the point of presenting results which, if not exactly the cutting edge of the field, certainly come close. Such results include not only the Arithmetic Structure Theorem but also simplified versions of a series of theorems of Shimura, dating from the '70s, which relate special values of L-functions to periods of integrals. Apart from their intrinsic interest, Shimura's theorems are the starting point for the kind of arithmetic applications discussed above, and Garrett's exposition makes Shimura's difficult theorems seem completely natural.
 Anyone interested in the arithmetic of number fields will eventually have to learn something about Hilbert modular forms. As an introduction to the analytic and arithmetic aspects of the subject, Garrett's book may be the best place to start.