Spectral Methods of Automorphic Forms
The book under review is the second edition of the monograph “Introduction to the spectral theory of automorphic forms” (the first edition was published at Biblioteca de la Revista Matemática Iberoamericana, Madrid, 1995) by the same author. It reflects the fact that the book has grown out of lectures given by the author in Spain. Automorphic forms play a central role on the border between analytic number theory, algebra, analysis and geometry; hence it is not an easy task to write a comprehensive and readable introduction to this classical subject. The book under review demonstrates the author’s mastery in both a conceptual and a pedagogical direction; it gives a very nice introduction to real analytic automorphic forms and their applications in number theory. Two introductory chapters deal with the hyperbolic metric and eigenfunctions of the Laplacian on the upper half plane, and with Fuchsian groups. The next five chapters form the core of the book. The reader is made familiar with the basic facts concerning cusp forms, Eisenstein series, and their meromorphic continuation based on the Selberg method, using the Fredholm theory of integral equations. Then the author devotes his attention to spectral theory, Kloosterman sums and trace formulas. The Selberg trace formula is applied to the problem of distribution of eigenvalues. The next chapter considers a geometric application, the lattice point problem in the hyperbolic upper half plane of complex numbers. The book provides a very readable textbook on the spectral theory of automorphic forms, not only for those willing to enter this fascinating subject but also for those who need some help to orient themselves in the theory. For the latter group of readers, two appendices provide some necessary background material from classical analysis and special functions.