Spectral methods of automorphic forms. 2nd ed. (English) Zbl 1006.11024

Graduate Studies in Mathematics. 53. Providence, RI: American Mathematical Society (AMS). Madrid: Revista Matemática Iberoamericana, xi, 220 p. (2002).
The book under review is the second edition of the monograph “Introduction to the spectral theory of automorphic forms” (Madrid: Biblioteca de la Revista Matemática Iberoamericana 1995) by the same author, which was reviewed in Zbl 0847.11028. The text of this new edition agrees with that of the first edition up to a few minor revisions, additions, rearrangements and updates. So the review of the first edition applies unchanged to the new one.
The book gives a superb introduction to real analytic automorphic forms and their applications in number theory starting at a modest level and leading up to topics of current research. Clearly this book belongs in any institutional library. If you did not succeed in buying your copy of the first edition hurry up and order your copy of this new one.


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F12 Automorphic forms, one variable
11F30 Fourier coefficients of automorphic forms
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
11L05 Gauss and Kloosterman sums; generalizations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations


Zbl 0847.11028