Automorphic forms and representations.

*(English)*Zbl 0868.11022
Cambridge Studies in Advanced Mathematics. 55. Cambridge: Cambridge University Press. xiv, 574 p. (1997).

This outstanding book covers, from a variety of points of view, the theory of automorphic forms on \(GL(2)\). Along the way the reader will encounter a host of information and techniques which go beyond \(GL(2)\). It comprises a solid foundation for the serious student or the researcher interested in automorphic forms, and contains a wealth of information which is not readily accessible in textbook form elsewhere. The book also features numerous and well-thought-out exercises, at a variety of levels.

The book is actually four courses in one: each of its four chapters is a complete course, which may be studied on its own; however, the four chapters are complementary. The first chapter is written in a classical language (functions but, except in 1.8, neither representations nor the adeles), but it is far more comprehensive and sophisticated then most introductions to the theory. It begins with a treatment of Dirichlet \(L\)-functions. This is followed by a discussion of the modular group \(SL(2,{\mathbb{Z}})\), and of holomorphic modular forms. Hecke operators are defined and used to obtain the Euler product associated to such a form. Weil’s converse theorem is stated and proved. Next, the non-holomorphic Eisenstein series are introduced, and the Rankin-Selberg method for \(GL(2)\times GL(2)\) is explained. After a brief review of relevant algebraic number theory, Hecke characters are defined, and the continuation of their \(L\)-functions is proved following Hecke. Hilbert modular forms are defined, and the base change theorem of Doi and Naganuma is stated. This is the result that given a Hecke eigenform \(f\) for \(SL(2,{\mathbb{Z}})\) and a real quadratic field \(K\), there is a Hilbert modular form \(F\) whose \(L\)-series is the product of \(L(s,f)\) and \(L(s,f,\chi_K)\), where \(\chi_K\) is the quadratic character associated to \(K/{\mathbb{Q}}\). Artin \(L\)-functions are defined, and the Langlands conjectures stated in rough form. In particular, it is shown how the quadratic base change result is a special case of the functoriality conjecture. The strategy exploited by Doi and Naganuma is explained heuristically. Maass forms are introduced, and the Maass forms associated to a real quadratic extension are produced. Finally, the quadratic base change map is produced following Doi and Naganuma: by taking the Rankin-Selberg convolution of the given modular form \(f\) with the Maass forms associated to \(K\), one obtains the continuation of the desired product and its twists, and by Weil’s converse theorem, the result is thus the \(L\)-function of a Hilbert modular form.

The second chapter systematically develops the connection between the representation theory of \(GL(2,{\mathbb{R}})\) and automorphic forms on the Poincaré upper half plane. The focus is on the compact quotient case. The chapter begins by posing the question as to why there are precisely two types of automorphic forms on the Poincaré upper half plane, classical holomorphic modular forms and Maass forms, and no others. The theory of the Laplacian is developed, and the spectral problem is formulated. Basic Lie theory is introduced, and the Laplacian recast in terms of the center of the universal enveloping algebra. Gelfand’s result that the spherical Hecke algebra for \(GL(n)\) is commutative is proved. Hilbert-Schmidt theory is introduced, and the discreteness of the spectrum is proved. Then the basic vocabulary of the representation theory of \(GL(n,{\mathbb{R}})\) is given: smooth vector, \(K\)-finite vector, admissible representation, \(({\mathfrak g},K)\)-module, infinitesimal equivalence. The irreducible \(({\mathfrak g},K)\)-modules for \(GL(2,{\mathbb{R}})\) are classified, and their unitarizability is studied. Finally, the representation theory developed is used to refine the spectral problem and to answer the question posed at the beginning of the chapter. The chapter concludes with brief sections on Whittaker models and on a result of Harish-Chandra concerning convolutions.

Chapter three provides an introduction to the modern theory of automorphic forms. The chapter begins with the adeles and Tate’s thesis. The spectral theory is revisited, this time for the non-compact quotient case, and the fundamental result of Gelfand, Graev, and Piatetski-Shapiro is used to decompose the space of cuspidal square integrable functions into discrete irreducible subspaces of finite multiplicity. Automorphic forms on \(GL(n)\) are defined, and the tensor product theorem of Flath is proved in full detail. Whittaker models are introduced, and the existence and uniqueness of global Whittaker models for automorphic representations of \(GL(2)\) is proved (with some local results stated here which are then proved in Chapter 4). The functional equation for the standard \(L\)-function on \(GL(2)\) is also proved (once again with some local results quoted from Chapter 4). The adelization of classical automorphic forms is covered. Next the author describes the \(GL(2)\) Eisenstein series, and proves their analytic continuation and functional equation from an examination of their Fourier expansions. The constant term in the Fourier expansion leads to the intertwining integrals, which are developed further in Chapter 4. The Rankin-Selberg \(L\)-function for \(GL(2)\times GL(2)\) is defined and studied. The Langlands functoriality conjecture and the \(L\)-group are described. The chapter concludes with a brief discussion of the triple-product \(L\)-function.

The last chapter concerns the representation theory of \(GL(2)\) over a local field. The emphasis is on techniques which generalize to \(GL(n)\) or beyond, and as in the previous chapters, some results are stated for \(GL(n)\) but proved for \(GL(2)\). The chapter begins with a brief discussion of \(GL(2)\) over a finite field, including the Weil representation; exercises include such important topics as the Stone-Von Neumann Theorem and Harish-Chandra’s philosophy of cusp forms. Turning to a local field, smooth and admissible representations are introduced and their basic properties are studied. The tools of distributions and sheaves are introduced, following Bernstein and Zelevinsky, in order to ultimately extend Mackey theory to locally compact groups. The uniqueness of the local Whittaker model is proved, and conversely it is shown that an irreducible admissible representation of \(GL(2)\) which has no Whittaker functional must be one dimensional. In proving this, Jacquet functors are introduced and proved to be exact. Next the principal series representations are constructed by induction from a Borel subgroup. The spherical representations are studied, and the Macdonald formula for the spherical functions and the explicit formula for the spherical Whittaker function are proved by the method of Casselman-Shalika. The unitarizability of the local principal series is discussed. The local functional equation is proved; this was needed in Chapter 3 above. Supercuspidal representations are constructed from the Weil representation, following Jacquet-Langlands. The compatibility of the local gamma factors with parabolic induction (up to \(GL(2)\)) is also proved, using the Weil representation. The last section describes the local Langlands correspondence, and how the construction of supercuspidals achieved above fits in to this.

In summary, this book covers a large amount of material concerning automorphic forms on \(GL(2)\), and the material is presented from a vantage far broader. The careful reader will have taken a significant step on the path to contemporary research in this important and beautiful area.

Reviewer’s Remark: The author is maintaining a web page on the book, with links to a virtual study group and to a list of errata, at http://math.stanford.edu/~bump/book.html.

The book is actually four courses in one: each of its four chapters is a complete course, which may be studied on its own; however, the four chapters are complementary. The first chapter is written in a classical language (functions but, except in 1.8, neither representations nor the adeles), but it is far more comprehensive and sophisticated then most introductions to the theory. It begins with a treatment of Dirichlet \(L\)-functions. This is followed by a discussion of the modular group \(SL(2,{\mathbb{Z}})\), and of holomorphic modular forms. Hecke operators are defined and used to obtain the Euler product associated to such a form. Weil’s converse theorem is stated and proved. Next, the non-holomorphic Eisenstein series are introduced, and the Rankin-Selberg method for \(GL(2)\times GL(2)\) is explained. After a brief review of relevant algebraic number theory, Hecke characters are defined, and the continuation of their \(L\)-functions is proved following Hecke. Hilbert modular forms are defined, and the base change theorem of Doi and Naganuma is stated. This is the result that given a Hecke eigenform \(f\) for \(SL(2,{\mathbb{Z}})\) and a real quadratic field \(K\), there is a Hilbert modular form \(F\) whose \(L\)-series is the product of \(L(s,f)\) and \(L(s,f,\chi_K)\), where \(\chi_K\) is the quadratic character associated to \(K/{\mathbb{Q}}\). Artin \(L\)-functions are defined, and the Langlands conjectures stated in rough form. In particular, it is shown how the quadratic base change result is a special case of the functoriality conjecture. The strategy exploited by Doi and Naganuma is explained heuristically. Maass forms are introduced, and the Maass forms associated to a real quadratic extension are produced. Finally, the quadratic base change map is produced following Doi and Naganuma: by taking the Rankin-Selberg convolution of the given modular form \(f\) with the Maass forms associated to \(K\), one obtains the continuation of the desired product and its twists, and by Weil’s converse theorem, the result is thus the \(L\)-function of a Hilbert modular form.

The second chapter systematically develops the connection between the representation theory of \(GL(2,{\mathbb{R}})\) and automorphic forms on the Poincaré upper half plane. The focus is on the compact quotient case. The chapter begins by posing the question as to why there are precisely two types of automorphic forms on the Poincaré upper half plane, classical holomorphic modular forms and Maass forms, and no others. The theory of the Laplacian is developed, and the spectral problem is formulated. Basic Lie theory is introduced, and the Laplacian recast in terms of the center of the universal enveloping algebra. Gelfand’s result that the spherical Hecke algebra for \(GL(n)\) is commutative is proved. Hilbert-Schmidt theory is introduced, and the discreteness of the spectrum is proved. Then the basic vocabulary of the representation theory of \(GL(n,{\mathbb{R}})\) is given: smooth vector, \(K\)-finite vector, admissible representation, \(({\mathfrak g},K)\)-module, infinitesimal equivalence. The irreducible \(({\mathfrak g},K)\)-modules for \(GL(2,{\mathbb{R}})\) are classified, and their unitarizability is studied. Finally, the representation theory developed is used to refine the spectral problem and to answer the question posed at the beginning of the chapter. The chapter concludes with brief sections on Whittaker models and on a result of Harish-Chandra concerning convolutions.

Chapter three provides an introduction to the modern theory of automorphic forms. The chapter begins with the adeles and Tate’s thesis. The spectral theory is revisited, this time for the non-compact quotient case, and the fundamental result of Gelfand, Graev, and Piatetski-Shapiro is used to decompose the space of cuspidal square integrable functions into discrete irreducible subspaces of finite multiplicity. Automorphic forms on \(GL(n)\) are defined, and the tensor product theorem of Flath is proved in full detail. Whittaker models are introduced, and the existence and uniqueness of global Whittaker models for automorphic representations of \(GL(2)\) is proved (with some local results stated here which are then proved in Chapter 4). The functional equation for the standard \(L\)-function on \(GL(2)\) is also proved (once again with some local results quoted from Chapter 4). The adelization of classical automorphic forms is covered. Next the author describes the \(GL(2)\) Eisenstein series, and proves their analytic continuation and functional equation from an examination of their Fourier expansions. The constant term in the Fourier expansion leads to the intertwining integrals, which are developed further in Chapter 4. The Rankin-Selberg \(L\)-function for \(GL(2)\times GL(2)\) is defined and studied. The Langlands functoriality conjecture and the \(L\)-group are described. The chapter concludes with a brief discussion of the triple-product \(L\)-function.

The last chapter concerns the representation theory of \(GL(2)\) over a local field. The emphasis is on techniques which generalize to \(GL(n)\) or beyond, and as in the previous chapters, some results are stated for \(GL(n)\) but proved for \(GL(2)\). The chapter begins with a brief discussion of \(GL(2)\) over a finite field, including the Weil representation; exercises include such important topics as the Stone-Von Neumann Theorem and Harish-Chandra’s philosophy of cusp forms. Turning to a local field, smooth and admissible representations are introduced and their basic properties are studied. The tools of distributions and sheaves are introduced, following Bernstein and Zelevinsky, in order to ultimately extend Mackey theory to locally compact groups. The uniqueness of the local Whittaker model is proved, and conversely it is shown that an irreducible admissible representation of \(GL(2)\) which has no Whittaker functional must be one dimensional. In proving this, Jacquet functors are introduced and proved to be exact. Next the principal series representations are constructed by induction from a Borel subgroup. The spherical representations are studied, and the Macdonald formula for the spherical functions and the explicit formula for the spherical Whittaker function are proved by the method of Casselman-Shalika. The unitarizability of the local principal series is discussed. The local functional equation is proved; this was needed in Chapter 3 above. Supercuspidal representations are constructed from the Weil representation, following Jacquet-Langlands. The compatibility of the local gamma factors with parabolic induction (up to \(GL(2)\)) is also proved, using the Weil representation. The last section describes the local Langlands correspondence, and how the construction of supercuspidals achieved above fits in to this.

In summary, this book covers a large amount of material concerning automorphic forms on \(GL(2)\), and the material is presented from a vantage far broader. The careful reader will have taken a significant step on the path to contemporary research in this important and beautiful area.

Reviewer’s Remark: The author is maintaining a web page on the book, with links to a virtual study group and to a list of errata, at http://math.stanford.edu/~bump/book.html.

Reviewer: S.Friedberg (Chestnut Hill)

##### MSC:

11Fxx | Discontinuous groups and automorphic forms |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

22E50 | Representations of Lie and linear algebraic groups over local fields |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

11S37 | Langlands-Weil conjectures, nonabelian class field theory |

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

11F11 | Holomorphic modular forms of integral weight |

11F12 | Automorphic forms, one variable |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

11R37 | Class field theory |

11R42 | Zeta functions and \(L\)-functions of number fields |

11F25 | Hecke-Petersson operators, differential operators (one variable) |

11F27 | Theta series; Weil representation; theta correspondences |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |