The marvelous documentary, The Proof, concerning Andrew Wiles’ victory over Fermat’s Last Theorem, contains a wonderful segment in which Peter Sarnak credits Martin Eichler with the aphorism that there are five, not four, fundamental operations of arithmetic: the usual addition, subtraction, multiplication, and division, supplemented by modular forms. It is the ubiquity of modular forms, or, more generally, automorphic functions, in number theory, which forms the background for Eichler’s joke, and the record is indeed spectacular. We need only consider Hecke’s work on the correspondence between modular forms and L-functions, Weil’s famous “converse theorem” to this Hecke correspondence, and Shimura’s gorgeous result tying in modular forms of weight n with modular forms of weight n/2, to get an idea of the influence of this quintessentially twentieth century subject. And then there is the Langlands Program, of course, one of the true centerpieces of current research at the frontier of number theory (and allied parts of mathematics).
By definition, an automorphic function is invariant under the action of a discrete group, so that we get multiply periodic functions. The spaces of such objects come in two flavors. On the one hand we have holomorphic (i.e. analytic) forms; on the other hand we encounter non-holomorphic forms, or Maaß wave forms, after their discoverer, Hans Maaß. Holomorphic forms go back to Poincaré and Klein, as we know from Bell’s florid account in Men of Mathematics, while doubtless the man primarily responsible for bringing modular forms into number theory was Erich Hecke. As we hinted above, Hecke’s primary motivation was to construct a bridge of sorts between modular forms and analytic number theory’s most important players, L-functions (i.e. Dirichlet series). This correspondence then formed the point of departure for Maaß. Says Baker on page 1 of the book under review: “Maaß sought to extend the work of Hecke on the L-functions associated to holomorphic automorphic forms. Working backwards from the Dirichlet series with Größencharakter attached to real quadratic fields he introduced non-holomorphic automorphic functions, eigenfunctions of the hyperbolic Laplacian … that would take on the role of holomorphic automorphic forms in Hecke’s work.” Thus, from their very inception, Maaß forms were intrinsically tied to the spectral theory of Laplacian operators and, to be sure, the last chapter of Kloosterman Sums and Maaß Forms bears the title, “The spectral theorem for ∆.”
It must be stressed, however, that to a number theorist the spectral theory of automorphic forms is ultimately still a tool, or, rather, a tool-kit, no matter how sophisticated or beautiful it is. Accordingly Baker spells out the raison d’être for his book in the Preface: “… we study the theory from as particular viewpoint. In the early 1980’s analytic number theorists began to apply results of Kuznetsov … to … the moments of the Riemann zeta function [and] classical problems in multiplicative number theory, especially prime numbers in arithmetic progressions to large moduli. Under [the second heading] … [w]ith several applications in mind (a divisor problem, a moment problem for Dirichlet polynomials, the Brun-Titchmarsh theorem …) [Deshouillers and Iwaniek] gave a wide variety of bounds for sums of Kloosterman sums with weight attached.” Kloosterman Sums and Maaß Forms is therefore intended to equip the reader with sufficient expertise for studying this seminal work by Jean-Marc Deshouillers and Henrik Iwaniek (still only a little over twenty years old) and a host of other recent analytic number theory besides, involving the spectral theory of automorphic forms.
The book is written in a terse style, but sacrifices nothing as far as completeness goes, and covers a huge amount of deep and important mathematics. The first five chapters set the stage for the last four, and they qualify as a superb first course in the theory of automorphic forms at the level of a first graduate seminar. Hyperbolic geometry, the modular group and congruence subgroups, special functions, Poincaré series, and Fourier expansions of Eisenstein series, are the major players here. But then, starting in the sixth chapter, spectral theory proper begins to stir: invariant integral operators, Fredholm theory, Hilbert-Schmidt theory, Green’s function, meromorphic continuation of Eisenstein series, and, finally, as we already mentioned, the spectral theorem for the Laplacian. There should be no misconceptions about the context into which this work should be fitted: this is very serious modern analytic number theory with a manifest orientation to harmonic analysis, and though the author is explicit in stressing that prerequisites consist in graduate real and complex analysis, and various references are given, a prospective student of this material had better possess considerable mathematical sophistication.
But the rewards of the indicated hard work will be great: Roger Baker’s Kloosterman Sums and Maaß Forms is a beautifully written introduction to an exceedingly important segment of modern number theory. I keenly look forward to Volume II.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles.