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.)