Some applications of modular forms.

*(English)*Zbl 0721.11015
Cambridge Tracts in Mathematics, 99. Cambridge: Cambridge University Press. x, 111 p. £17.50; $ 29.95 (1990).

The book under review deals with the following three long outstanding difficult problems (A), (B), (C) which are seemingly unrelated with modular forms but which have recently been solved by an application of deep results of the theory of modular forms.

(A) Ruziewicz’s problem. Ruziewicz posed the problem of the uniqueness of rotationally invariant finitely additive (finite) measures defined on the \(\sigma\)-algebra of Lebesgue measurable subsets of the n-sphere \(S^ n\) [see S. Banach, Fundamenta Math. 4, 7-33 (1923; JFM 49.0145.03)]. Banach already proved that uniqueness fails to hold for \(n=1\), whereas the problem for \(n\geq 2\) remained open until recently. For \(n\geq 2\) the problem of uniqueness is equivalent to the question whether the Lebesgue measure on \(S^ n\) is the unique rotationally invariant mean on \(L^{\infty}(S^ n)\). The affirmative answer to this question is for \(n\geq 4\) due to Margulis and Sullivan and for \(n=2,3\) to Drinfel’d who used the Jacquet-Langlands theory of automorphic forms on GL(2). The proof of the uniqueness for \(n\geq 2\) in Chapter 2 of the present book is new. For \(n=2\) it is based on the theory of Hecke operators on \(L^ 2(S^ 2)\) as developed by Lubotzky, Phillips and Sarnak; the case \(n\geq 3\) is dealt with by an induction argument. The crucial notion is that of an \(\epsilon\)-good set of rotations. The solution presented to Ruziewicz’s problem has the advantage of being effective and explicit in its construction of the relevant \(\epsilon\)-good sets.

(B) Ramanujan graphs. A basic problem in graph theory with highly important practical applications is that of explicitly constructing highly connected sparse graphs. Of the various notions of high connectivity the most important one seems to be the so-called expansion property. The problem then is to construct a family of expanders on n vertices (n\(\to \infty)\) with expansion coefficient c as large as possible. The existence of expanders can rather easily be shown by counting arguments. The explicit construction of such graphs is much more difficult and is carried out here. In some respect these graphs are even optimal and they enjoy the additional property of being the first explicit examples of graphs with large girth and large chromatic number. The construction of these so-called Ramanujan graphs is due to Lubotzky, Phillips and Sarnak. The expansion property is shown to be connected with the spectrum of the adjacency matrix. The Ramanujan graphs are constructed as Cayley graphs of PGL(2,\({\mathbb{Z}}/q{\mathbb{Z}})\) relative to a suitably selected set of generators.

(C) Linnik’s Problem. If \(n\in {\mathbb{N}}\) is large and is a sum \(n=m^ 2_ 1+m^ 2_ 2+m^ 2_ 3\) of three squares of integers then the points \(m/\| m\|\) \((m=(m_ 1,m_ 2,m_ 3)\), \(\| m\|^ 2=n)\) become equidistributed on \(S^ 2\) as \(n\to \infty\). Linnik proved this under certain hypotheses. Using Iwaniec’s estimation of Fourier coefficients of modular forms of half-integral weight the author gives an unconditional proof of the equidistribution property.

What links problems (A),(B),(C) is that they are all reduced to the Ramanujan conjectures and their generalizations. The necessary theory of modular forms and in particular the necessary theory of theta series and the estimations of Fourier coefficients of modular forms are developed in Chapters 1 and 4. An important role is played by the estimates of various exponential sums (Kloosterman sums, Salié sums). The book under review effectively bridges the gap between teaching and current research. It is strongly recommended to anybody interested in the field of modular forms or in the fields of problems (A), (B), (C).

(A) Ruziewicz’s problem. Ruziewicz posed the problem of the uniqueness of rotationally invariant finitely additive (finite) measures defined on the \(\sigma\)-algebra of Lebesgue measurable subsets of the n-sphere \(S^ n\) [see S. Banach, Fundamenta Math. 4, 7-33 (1923; JFM 49.0145.03)]. Banach already proved that uniqueness fails to hold for \(n=1\), whereas the problem for \(n\geq 2\) remained open until recently. For \(n\geq 2\) the problem of uniqueness is equivalent to the question whether the Lebesgue measure on \(S^ n\) is the unique rotationally invariant mean on \(L^{\infty}(S^ n)\). The affirmative answer to this question is for \(n\geq 4\) due to Margulis and Sullivan and for \(n=2,3\) to Drinfel’d who used the Jacquet-Langlands theory of automorphic forms on GL(2). The proof of the uniqueness for \(n\geq 2\) in Chapter 2 of the present book is new. For \(n=2\) it is based on the theory of Hecke operators on \(L^ 2(S^ 2)\) as developed by Lubotzky, Phillips and Sarnak; the case \(n\geq 3\) is dealt with by an induction argument. The crucial notion is that of an \(\epsilon\)-good set of rotations. The solution presented to Ruziewicz’s problem has the advantage of being effective and explicit in its construction of the relevant \(\epsilon\)-good sets.

(B) Ramanujan graphs. A basic problem in graph theory with highly important practical applications is that of explicitly constructing highly connected sparse graphs. Of the various notions of high connectivity the most important one seems to be the so-called expansion property. The problem then is to construct a family of expanders on n vertices (n\(\to \infty)\) with expansion coefficient c as large as possible. The existence of expanders can rather easily be shown by counting arguments. The explicit construction of such graphs is much more difficult and is carried out here. In some respect these graphs are even optimal and they enjoy the additional property of being the first explicit examples of graphs with large girth and large chromatic number. The construction of these so-called Ramanujan graphs is due to Lubotzky, Phillips and Sarnak. The expansion property is shown to be connected with the spectrum of the adjacency matrix. The Ramanujan graphs are constructed as Cayley graphs of PGL(2,\({\mathbb{Z}}/q{\mathbb{Z}})\) relative to a suitably selected set of generators.

(C) Linnik’s Problem. If \(n\in {\mathbb{N}}\) is large and is a sum \(n=m^ 2_ 1+m^ 2_ 2+m^ 2_ 3\) of three squares of integers then the points \(m/\| m\|\) \((m=(m_ 1,m_ 2,m_ 3)\), \(\| m\|^ 2=n)\) become equidistributed on \(S^ 2\) as \(n\to \infty\). Linnik proved this under certain hypotheses. Using Iwaniec’s estimation of Fourier coefficients of modular forms of half-integral weight the author gives an unconditional proof of the equidistribution property.

What links problems (A),(B),(C) is that they are all reduced to the Ramanujan conjectures and their generalizations. The necessary theory of modular forms and in particular the necessary theory of theta series and the estimations of Fourier coefficients of modular forms are developed in Chapters 1 and 4. An important role is played by the estimates of various exponential sums (Kloosterman sums, Salié sums). The book under review effectively bridges the gap between teaching and current research. It is strongly recommended to anybody interested in the field of modular forms or in the fields of problems (A), (B), (C).

Reviewer: J.Elstrodt (Münster)

##### MSC:

11F11 | Holomorphic modular forms of integral weight |

11F30 | Fourier coefficients of automorphic forms |

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

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

28A12 | Contents, measures, outer measures, capacities |

05C35 | Extremal problems in graph theory |

11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |

11K36 | Well-distributed sequences and other variations |

11L07 | Estimates on exponential sums |

11L05 | Gauss and Kloosterman sums; generalizations |