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**Introduction to the spectral theory of automorphic forms.**
*(English)*
Zbl 0847.11028

Madrid: Biblioteca de la Revista Matemática Iberoamericana, xiii, 247 p. (1995).

Widely different books might be written under this title. This book does not concern itself with analysis on general Lie groups, but is content with functions on the upper half plane that are invariant under a discrete group of hyperbolic motions. If such functions are also eigenfunctions of the hyperbolic Laplace operator, they are called (real analytic) automorphic forms of weight 0, or Maass forms. Even with these restrictions, the subject is large, and the author has made further choices, as becomes clear from a description of the contents.

Two introductory chapters discuss the hyperbolic metric on the upper half plane \(\mathcal H\), eigenfunctions of the Laplacian on \(\mathcal H\), and Fuchsian groups \(\Gamma\) for which the volume of \(\Gamma\backslash {\mathcal H}\) is finite.

Chapters 3-7 are the core of the book. Here one finds cusp forms and Eisenstein series, and the meromorphic continuation of the latter. These automorphic forms build the spectral decomposition of the Laplace operator in the square integrable functions on \(\Gamma\backslash {\mathcal H}\). The continuation of the Eisenstein series is obtained by a method due to Selberg. This method is based on the meromorphy of the resolvent kernel of certain integral operators.

Central theorems in the spectral theory of automorphic forms are the trace formula of Selberg and the sum formula of Kuznetsov. After deriving these results, the author shows how the latter leads to an estimate of sums of (generalized) Kloosterman sums. The former formula is used to prove a distribution result on the length of closed geodesics.

Another geometric application discussed in the book is the hyperbolic lattice point problem. On the spectral side, there are results on the distribution of the eigenvalues of the Laplace operator, and of the size of cusp forms.

Two appendices provide the reader with background from classical analysis and special functions.

The book has grown out of lectures the author has given in Spain. It is directed at advanced graduate students. I think that it gives a superb introduction to real analytic automorphic forms and their application in number theory. It gives also new insights to mathematicians to which the subject is not new. I quote from the introduction to Chapter 8: “The main goal of the chapter is to establish auxiliary estimates,…. Crude results would often do. However, since it is effortless to be general, sharp and explicit in some cases, we go beyond the primary objective.” This last sentence characterizes the whole book. At many places I enjoyed reading sharper estimates than I knew. I warmly recommend this introduction to real analytic modular forms.

Two introductory chapters discuss the hyperbolic metric on the upper half plane \(\mathcal H\), eigenfunctions of the Laplacian on \(\mathcal H\), and Fuchsian groups \(\Gamma\) for which the volume of \(\Gamma\backslash {\mathcal H}\) is finite.

Chapters 3-7 are the core of the book. Here one finds cusp forms and Eisenstein series, and the meromorphic continuation of the latter. These automorphic forms build the spectral decomposition of the Laplace operator in the square integrable functions on \(\Gamma\backslash {\mathcal H}\). The continuation of the Eisenstein series is obtained by a method due to Selberg. This method is based on the meromorphy of the resolvent kernel of certain integral operators.

Central theorems in the spectral theory of automorphic forms are the trace formula of Selberg and the sum formula of Kuznetsov. After deriving these results, the author shows how the latter leads to an estimate of sums of (generalized) Kloosterman sums. The former formula is used to prove a distribution result on the length of closed geodesics.

Another geometric application discussed in the book is the hyperbolic lattice point problem. On the spectral side, there are results on the distribution of the eigenvalues of the Laplace operator, and of the size of cusp forms.

Two appendices provide the reader with background from classical analysis and special functions.

The book has grown out of lectures the author has given in Spain. It is directed at advanced graduate students. I think that it gives a superb introduction to real analytic automorphic forms and their application in number theory. It gives also new insights to mathematicians to which the subject is not new. I quote from the introduction to Chapter 8: “The main goal of the chapter is to establish auxiliary estimates,…. Crude results would often do. However, since it is effortless to be general, sharp and explicit in some cases, we go beyond the primary objective.” This last sentence characterizes the whole book. At many places I enjoyed reading sharper estimates than I knew. I warmly recommend this introduction to real analytic modular forms.

Reviewer: R.W.Bruggeman (Utrecht)

### MSC:

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11F12 | Automorphic forms, one variable |

11F30 | Fourier coefficients of automorphic forms |

11F37 | Forms of half-integer weight; nonholomorphic modular forms |

11L05 | Gauss and Kloosterman sums; generalizations |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

53C22 | Geodesics in global differential geometry |