##
**Elementary theory of Eisenstein series.**
*(English)*
Zbl 0268.10012

A Halsted Press Book. Tokyo: Kodansha Ltd.; New York etc.: John Wiley & Sons: xi, 110 p. £4.50 (1973).

This book is based on lectures that the author gave at the University of Maryland in 1970–1971 and covers his seminar-note with the same title published from University of Tokyo (No. 18 (1968), in Japanese).

According to the introduction, the aim of this book is to give an introduction to the theory of Selberg-Langlands an real analytic Eisenstein series [A. Selberg, J. Indian Math. Soc., n. Ser. 20, 47–87 (1956; Zbl 0072.08201); Proc. Int. Congr. Math., Stockholm 1962, 177–189 (1963; Zbl 0209.44001); R. P. Langlands, Proc. Symp. Pure Math. 9, 235–252 (1966; Zbl 0204.09603)] as the title of the book shows. To attain the aim quickly, only the group \(\mathrm{SL}(2,\mathbb R)\) and its discontinuous subgroups, automorphic functions with respect to these groups and the harmonic analysis in this situation are dealt with. General treatment an algebraic groups and harmonic analysis are omitted to make the streamline of the theory clear. In such a simplified situation the author keeps the reader well informed about where he is and where he is going. The author develops deep and elegant contents without introducing so many definitions and tools in such a small book, though the reviewer feels that the beginners need little more preliminary knowledge in harmonic analysis.

Roughly speaking the book contains the theory of analytic continuation of Eisenstein series, the spectral theory of Hilbert spaces \(L^2(\cdot\,\backslash\,\cdot)\) of square integrable automorphic functions including contribution of Eisenstein series, and discussion on the so called trace formula. It consists of seven chapters and an appendix:

Chapter I. Preliminaries.

This goes through definitions and facts in the fields of discontinuous subgroups and harmonic analysis.

Chapter II. Eisenstein series in the domain of convergence.

This begins with the definition of Eisenstein series \(E_i(z,s)\), \((z\in\mathcal H,\;s\in\mathbb C)\), where \(i\) ranges from \(1\) to \(h\), \(h\) is the number of elements in a representative system of inequivalent cusps with respect to a given discontinuous subgroup \(\Gamma\) of \(\mathrm{SL}(2,\mathbb R)\). (This definition is of course different from the classical Eisenstein series or the Poincaré series (cf. J. Lehner [Discontinuous groups and automorphic functions. Providence, R.I.: AMS (1964; Zbl 0178.42902)]).) After introducing the Hilbert space \(L^2(\Gamma\backslash \mathcal H)\), the inner product of the compact parts of \(E_i(z,s)\) and \(E_j(z,s')\) is given by making use of Green’s theorem.

Chapter III. Analytic continuation of Eisenstein series (First step).

Analytic continuation of \(E_i(z,s)\) to the domain \(\text{Re}\, s > 1/2\), \(s\notin (1/2,1]\) beyond the domain \(\text{Re}\, s > 1\) of absolute convergence is discussed. To the attempt the possibility of continuation of \(h^2\) functions \(\varphi_{ij}(s)\) is essential and proved where \(\varphi_{ij}(s)\) is the Fourier coefficient of the constant term of \(E_i(z,s)\) at the \(j\)-th cusp. Discussion on \(E_i(z,s)\) is transferred from one of \(\varphi_{ij}(s)\).

Chapter IV. Analytic continuation of Eisenstein series (second step).

Continuing discussion in Chapter III analytic continuation of \(\varphi_{ij}(s)\) and \(E_i(z,s)\) to the whole plane of \(s\) is completed. As the result an interesting number theoretic functional equation is given; \[ \mathcal E(z,s)=\Phi(s)\mathcal E(z,1-s), \] where \(\mathcal E(z,s)\) is a vector \((E_1(z,s), \ldots, E_n(z,s))\) and \(\Phi(s)\) is an \(h\times h\) matrix \((\varphi_{ij}(s))\). From this, another proof of the following famous equation is obtained without using the Poisson summation:

\[ \pi^{-s/2}\Gamma(s/2)\zeta(s)=\pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s) \] (cf. E. Hecke [Dirichlet series, modular forms and quadratic forms. Lecture notes, Institute for Advanced Study, Princeton (1938). Göttingen: Vandenhoeck & Ruprecht (1983; Zbl 0507.10015)]).

Chapter V. Spectral theory of \(L^2 (\Gamma\backslash \mathcal H)\).

An orthogonal decomposition of \(L^2 (\Gamma\backslash \mathcal H)\) is given from the view point of the spectrum of the Laplacian \(D\). A certain connection between the direct summands of \(L^2 (\Gamma\backslash \mathcal H)\) and the Eisenstein series or the cusp forms is discussed. The remainder part is devoted to an integral expression of the spectrum of \(D\) related to the Eisenstein series by the Selberg transformation.

Chapter VI. Generalization of Eisenstein series.

This and the next chapters contain similar arguments to ones in Chapters II–V, by treating \(G=\mathrm{SL}(2,\mathbb R)\) instead of \(\mathcal H\) (the complex upper-half plane) and a homogeneous space \(\Gamma\backslash G\). Two kinds of Eisenstein series \(E_i(g,s)_n\) (\(n\): even rational integer) and \(E_{i,\psi}(g,s)\) (\(\psi\): function on \(G\)) \((g\in G,\;s\in\mathbb C)\) are introduced as generalized Eisenstein series of \(E_i(z,s)\). Also analytic continuation and a functional equation are discussed for each kind of Eisenstein series.

Chapter VII. Spectral theory of \(L^2(\Gamma\backslash G)\).

Similar discussion to Chapter V is done. An orthogonal decomposition of \(L^2(\Gamma\backslash G)\) is given from the view point of decomposition of unitary representation of \(G\) in \(L^2(\Gamma\backslash G)\) into irreducible representations. Then a certain connection between the direct summands of \(L^2(\Gamma\backslash G)\) and the Eisenstein series \(E_{i,\psi}(g,s)\) or cusp forms is discussed. Furthermore the spectrum of the Hecke operator is explained related to a subspace spanned by cusp forms.

Appendix. Explicit trace formula.

Making use of Chapter V principally, the trace of an integral operator is expressed in a formula containing the Selberg transformation.

According to the introduction, the aim of this book is to give an introduction to the theory of Selberg-Langlands an real analytic Eisenstein series [A. Selberg, J. Indian Math. Soc., n. Ser. 20, 47–87 (1956; Zbl 0072.08201); Proc. Int. Congr. Math., Stockholm 1962, 177–189 (1963; Zbl 0209.44001); R. P. Langlands, Proc. Symp. Pure Math. 9, 235–252 (1966; Zbl 0204.09603)] as the title of the book shows. To attain the aim quickly, only the group \(\mathrm{SL}(2,\mathbb R)\) and its discontinuous subgroups, automorphic functions with respect to these groups and the harmonic analysis in this situation are dealt with. General treatment an algebraic groups and harmonic analysis are omitted to make the streamline of the theory clear. In such a simplified situation the author keeps the reader well informed about where he is and where he is going. The author develops deep and elegant contents without introducing so many definitions and tools in such a small book, though the reviewer feels that the beginners need little more preliminary knowledge in harmonic analysis.

Roughly speaking the book contains the theory of analytic continuation of Eisenstein series, the spectral theory of Hilbert spaces \(L^2(\cdot\,\backslash\,\cdot)\) of square integrable automorphic functions including contribution of Eisenstein series, and discussion on the so called trace formula. It consists of seven chapters and an appendix:

Chapter I. Preliminaries.

This goes through definitions and facts in the fields of discontinuous subgroups and harmonic analysis.

Chapter II. Eisenstein series in the domain of convergence.

This begins with the definition of Eisenstein series \(E_i(z,s)\), \((z\in\mathcal H,\;s\in\mathbb C)\), where \(i\) ranges from \(1\) to \(h\), \(h\) is the number of elements in a representative system of inequivalent cusps with respect to a given discontinuous subgroup \(\Gamma\) of \(\mathrm{SL}(2,\mathbb R)\). (This definition is of course different from the classical Eisenstein series or the Poincaré series (cf. J. Lehner [Discontinuous groups and automorphic functions. Providence, R.I.: AMS (1964; Zbl 0178.42902)]).) After introducing the Hilbert space \(L^2(\Gamma\backslash \mathcal H)\), the inner product of the compact parts of \(E_i(z,s)\) and \(E_j(z,s')\) is given by making use of Green’s theorem.

Chapter III. Analytic continuation of Eisenstein series (First step).

Analytic continuation of \(E_i(z,s)\) to the domain \(\text{Re}\, s > 1/2\), \(s\notin (1/2,1]\) beyond the domain \(\text{Re}\, s > 1\) of absolute convergence is discussed. To the attempt the possibility of continuation of \(h^2\) functions \(\varphi_{ij}(s)\) is essential and proved where \(\varphi_{ij}(s)\) is the Fourier coefficient of the constant term of \(E_i(z,s)\) at the \(j\)-th cusp. Discussion on \(E_i(z,s)\) is transferred from one of \(\varphi_{ij}(s)\).

Chapter IV. Analytic continuation of Eisenstein series (second step).

Continuing discussion in Chapter III analytic continuation of \(\varphi_{ij}(s)\) and \(E_i(z,s)\) to the whole plane of \(s\) is completed. As the result an interesting number theoretic functional equation is given; \[ \mathcal E(z,s)=\Phi(s)\mathcal E(z,1-s), \] where \(\mathcal E(z,s)\) is a vector \((E_1(z,s), \ldots, E_n(z,s))\) and \(\Phi(s)\) is an \(h\times h\) matrix \((\varphi_{ij}(s))\). From this, another proof of the following famous equation is obtained without using the Poisson summation:

\[ \pi^{-s/2}\Gamma(s/2)\zeta(s)=\pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s) \] (cf. E. Hecke [Dirichlet series, modular forms and quadratic forms. Lecture notes, Institute for Advanced Study, Princeton (1938). Göttingen: Vandenhoeck & Ruprecht (1983; Zbl 0507.10015)]).

Chapter V. Spectral theory of \(L^2 (\Gamma\backslash \mathcal H)\).

An orthogonal decomposition of \(L^2 (\Gamma\backslash \mathcal H)\) is given from the view point of the spectrum of the Laplacian \(D\). A certain connection between the direct summands of \(L^2 (\Gamma\backslash \mathcal H)\) and the Eisenstein series or the cusp forms is discussed. The remainder part is devoted to an integral expression of the spectrum of \(D\) related to the Eisenstein series by the Selberg transformation.

Chapter VI. Generalization of Eisenstein series.

This and the next chapters contain similar arguments to ones in Chapters II–V, by treating \(G=\mathrm{SL}(2,\mathbb R)\) instead of \(\mathcal H\) (the complex upper-half plane) and a homogeneous space \(\Gamma\backslash G\). Two kinds of Eisenstein series \(E_i(g,s)_n\) (\(n\): even rational integer) and \(E_{i,\psi}(g,s)\) (\(\psi\): function on \(G\)) \((g\in G,\;s\in\mathbb C)\) are introduced as generalized Eisenstein series of \(E_i(z,s)\). Also analytic continuation and a functional equation are discussed for each kind of Eisenstein series.

Chapter VII. Spectral theory of \(L^2(\Gamma\backslash G)\).

Similar discussion to Chapter V is done. An orthogonal decomposition of \(L^2(\Gamma\backslash G)\) is given from the view point of decomposition of unitary representation of \(G\) in \(L^2(\Gamma\backslash G)\) into irreducible representations. Then a certain connection between the direct summands of \(L^2(\Gamma\backslash G)\) and the Eisenstein series \(E_{i,\psi}(g,s)\) or cusp forms is discussed. Furthermore the spectrum of the Hecke operator is explained related to a subspace spanned by cusp forms.

Appendix. Explicit trace formula.

Making use of Chapter V principally, the trace of an integral operator is expressed in a formula containing the Selberg transformation.

Reviewer: Tetsuo Kodama (Fukuoka)

### MSC:

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

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

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

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

43A85 | Harmonic analysis on homogeneous spaces |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |