My goal for this project was to remedy some of my ignorance of the theory of automorphic forms. I hope these exercises will aid the reader in doing the same. If an exercise requires some sort of inspiration that isn't immediately obvious from the text, then I have tried to give at least a hint. I have attempted to write a detailed sketch or a full solution whenever an exercise was particularly difficult (for me). But it will be evident that I have violated both of these guiding principles at times with little rhyme or reason. An exercise marked with a * is particularly tricky (again, for me).

My thanks go to Dorian for the opportunity to be a part of this project, and to Joe for patiently answering loads of my questions. It's been a pleasure working with both of you. A National Science Foundation Postdoctoral Research Fellowship provided my funding during the completion of this project. Finally, I would like to thank my wife, Alana, for her unwavering support of my endeavors, especially those that detract from our time together.

The theory of L-functions is an old subject with a long history. In the 1940s Hecke and Maass rewrote the classical theory in the setting of automorphic forms, and it seemed as if the theory of L-functions had settled into a fairly final form. This view was effectively overturned with the publication of two major books: [Gelfand-Graev-Piatetski-Shapiro, 1969], [Jacquet-Langlands, 1970], where it was shown that the theory of L-functions could be recast in the language of infinite dimensional complex representations of reductive groups.

Another milestone in the recent theory of L-functions was the book by Roger Godement and Hervé Jacquet, [Godement-Jacquet, 1972], which defined for the first time the standard L-functions attached to automorphic representations of the general linear group, and proved their key properties by generalizing the seminal ideas of [Tate, 1950], [Iwasawa, 1952, 1992]. The proofs in [Godement-Jacquet, 1972] made fundamental use of matrix coefficients associated to automorphic representations. The standard L-functions of the general linear group are often called Godement-Jacquet L-functions. Although several other techniques have since been discovered to obtain the main analytic properties of such L-functions, none is more beautiful and elegant than the method of matrix coefficients, originally devised by Godement and Jacquet, which is a major theme of this book.

Modern research in the theory of automorphic representations and L-functions is largely focused in the direction of the Langlands program.