Make no mistake, this is number theory with a vengeance. It’s about pretty relatives of (elliptic) modular forms, some of the sexiest and most fecund things in mathematics — just recall that wonderful moment in the fantastic documentary “The Proof,” about Andrew Wiles’s victory over Fermat’s Last Theorem, when Wiles makes the comment that according to Martin Eichler there are five fundamental arithmetical operations, namely, addition, subtraction, multiplication, division and … modular forms.

Well, that said, what are modular forms and what, then, are Maass forms? First of all, a modular form is defined on the complex upper half plane and transforms in a special way relative to the action on this upper half plane by fractional linear transformations associated to certain distinguished subgroups of \(\mathrm{PSL}(2,\mathbb{Z}); there’s a multiplier involved in these transformation laws which determines the weight of the form (generally at worst a half-integer), and the nature of the subgroup determines other things, like the so-called level of the modular form. Additionally there are requirements in place concerning the analytic behavior of the putative modular form at the so-called cusps of the fundamental domain, the latter being a simply connected set of representatives for the equivalence relation coming from the action of the indicated group on the upper half plane. Hyperbolically the fundamental domain is polygonal, hence these cusps, and so it is that we have Fourier expansions for these modular forms at the cusps. A lot to reckon with, to be sure, but you certainly get what you pay for, as Wiles is the first to tell you — or any other number of number theorists, of course: this is really powerful stuff, and also very, very beautiful.

A Maass form shares properties with modular forms, such as weight and level, albeit the transformation laws are fancier, what with Jacobi (or extended Legendre) symbols entering in, but there are other requirements: a (harmonic) Maas form lives in the kernel of a so-called weight \(k\) hyperbolic Laplacian on the upper half-plane, and there are slick growth conditions in force at all cusps, rather stronger than holomorphy. But, as it says on p. 62 of the book under review, “weakly holomorphic modular forms are harmonic Maass forms.” So, you get the idea…

The book under review is split into three parts, and it makes a lot of sense to do this: the first part, “Background,” is a tour of the classical theory, with history given its just due. It all has to (and does) begin with elliptic functions, and, in addition to at least some of the usual suspects, Eisenstein and Weierstrass, we meet the redoubtable Martin Eichler, already referred to above. “All right,” some might say, “we have *some *of the usual suspects, but where is Jacobi?” Well, he’s next: the authors very properly devote a chapter to theta functions and holomorphic Jacobi forms, and we quickly find ourselves very much in the thick of it. After a tantalizing excursion into the area of Siegel modular forms, the authors hit “Classical Maass Forms,” and one is struck by the parallels with classical modular forms: Fourier expansions (of course), Eisenstein series (Aha!), L-functions (and Maass cusp forms: you’re a cusp form if there’s some vanishing going on at the cusps), and then real quadratic fields and Hecke theory.

A *caveat* is in order. Classical Maass forms are *not* the focus of the present book — say the authors on p. 49: “In 1949, Maass introduced nonholomorphic modular functions, his so-called *Maass forms *… This book is primarily about mock modular forms [and somewhere Ramanujan is smiling … ] and harmonic Maass forms, functions which are another type of nonholomorphic modular form.” Fair enough: see my hand-waving above to get an idea about these harmonic Maass forms.

The two classes of objects just cited are featured in the central part (two) of the book, titled, “Harmonic Maass Forms and Mock Modular Forms.” *A propos*, as also noted above, “…we can think of [this] theory of harmonic Maass forms as a natural extension of the classical theory of modular forms. And this heritage is certainly on display: in Part 2 we encounter, for example, “Zagier’s weight 3/2 Eisenstein series,” Weierstrass mock modular forms, and Hecke theory — *plus ça change, plus c’est la même chose*. But there’s more, including a lot of stuff on Ramanujan’s very famous mock theta functions (and this chapter starts very poignantly with a section titled, “Ramanujan’s last letter to Hardy”) and mock modular Eichler-Shimura theory. So, to be sure, in the thick of it and then some.

The book’s third and last part is devoted to “Applications,” as befits any discussion worth its salt, of modular forms and their offspring. Let’s just name six (*verbatim*): partitions and unimodal sequences, harmonic Maass forms as arithmetic and geometric generating functions, generalized Borcherds products, elliptic curves over \(\mathbb{Q}\), representation theory and mock modular forms (and it’s in the chapter so titled that the authors discuss Conway’s famous — or notorious — monstrous moonshine), and finally quantum modular forms (this chapter includes stuff by Kontsevich). So it’s quite a sweep, isn’t it?

It’s been many years since I had occasion to go at modular forms in any sort of nontrivial manner, seeing that my research has gone in another direction, despite there remaining certain deep implicit connections. This wonderful book is very exciting to me, as it presents some very exotic and beautiful stuff that I, for one, had no notion of, despite my past. I’m happy to be enlightened, if only in a preliminary way. But this book is tailor-made to lead wayward sons like me back home, and to light the right kind of fire under us. And this is in sync with what the authors say: “This book is intended to serve as a uniform and somewhat comprehensive introduction to the subject for graduate students and research mathematicians.” Excellent.

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.