Description
Book SynopsisL-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy-to-read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.
Trade Review'… a gentle introduction to this fascinating new subject. The presentation is very explicit and many examples are worked out with great detail … This book should be of great interest to students beginning with the theory of modular forms or for more advanced readers wanting to know about general L-functions.' Emmanuel P. Royer, Mathematical Reviews
'This book, whose clear and sometimes simplified proofs make the basic theory of automorphic forms on GL(n) accessible to a wide audience, will be valuable for students. It nicely complements D. Bump's book (Automorphic Forms and Representations, Cambridge, 1997), which offers a greater emphasis on representation theory and a different selection of topics.' Zentralblatt MATH
'Unfortunately, when n > 2 the GL(n) theory is not very accessible to the student of analytic number theory, yet it is increasing in importance. [This book] addresses this problem by developing a large part of the theory in a way that is carefully designed to make the field accessible … much of the literature is written in the adele language, and seeing how it translates into classical terms is both useful and enlightening … This is a unique and very welcome book, one that the student of automorphic forms will want to study, and also useful to experts.' Daniel Bump, SIAM Review
Table of ContentsIntroduction; 1. Discrete group actions; 2. Invariant differential operators; 3. Automorphic forms and L-functions for SL(2,Z); 4. Existence of Maass forms; 5. Maass forms and Whittaker functions for SL(n,Z); 6. Automorphic forms and L-functions for SL(3,Z); 7. The Gelbert–Jacquet lift; 8. Bounds for L-functions and Siegel zeros; 9. The Godement–Jacquet L-function; 10. Langlands Eisenstein series; 11. Poincaré series and Kloosterman sums; 12. Rankin–Selberg convolutions; 13. Langlands conjectures; Appendix. The GL(n)pack manual; References.
