Roger Baker
Professor of Mathematics, Brigham Young University. Kendrick Press, Inc.
(2003) xiv+285pp. Paperback $75.00. ISBN 0-09740427-0-6.
This is a careful and detailed account of the spectral theory of automorphic
forms in the upper half plane. (The relevant differential operator is the
hyperbolic Laplacian r.) It has been designed
as a one-semester graduate course.
Several topics in analysis are developed for use later in the book (special
functions, Fredholm theory). Regarding prerequisites, the reader should
have taken standard graduate courses in real and complex analysis.
Chapter 1: Introduction. Chapter 2: Hyperbolic geometry. Chapter 3: The
modular group and its subgroups. Chapter 4: Special functions. Chapter 5:
Automorphic functions. Chapter 6: Interaction of r
with integral operators. Chapter 7: Integral equations and Green's function.
Chapter 8: Meromorphic continuation of Eisenstein series. Chapter 9: The
spectral theorem for r.
Readers can proceed to the study of more advanced textbooks and original
papers in this subject area. However, the author has a particular direction
in mind. In Volume 2, the Kuznetsov formulae will be derived and applied
to problems in analytic number theory, along the lines initiated by Deshouillers
and Iwaniec in their well-known Inventiones paper of 1982.
"…this is a highly welcome introduction to the spectral theory of automorphic
functions, starting from very modest prerequisites and leading up to deep
results. We may look forward with great excitement to the publication of
Volume 2." (J. Elstrodt, Zentralblatt Math.)