Lectures on the Theory of Automorphic Functions, First Volume

“Klein-Fricke” and “Fricke-Klein” — to number theorists these mantras invariably recall memories of early number theory seminars, be it in late undergraduate school or graduate school, or, more likely, both, when modular forms first made their appearance in the curriculum. I’m pretty sure that for me it happened in the late 1970s at UCLA in a memorable seminar run, in his usual inimitable style, by the late Basil Gordon. But doubtless they, Klein and Fricke, made appearances, too, in my graduate seminars at UCSD: my advisor, Audrey Terras, was always sure to refer back to classical sources whenever possible.

So it is that Felix Klein and Robert Fricke (his PhD student at Göttingen in its Golden Age), mentioned in that order, are the authors of the (two-volume) Lectures on the Theory of Elliptic Modular Forms, and the same pair, in reverse order, are responsible for the (also two-volume) Lectures on the Theory of Automorphic Forms. The original publication dates of these works are, respectively, 1880 and 1882 for the first set, and 1897 and 1912 for the second. Given that automorphic forms date their official discovery to Henri Poincaré in the 1880s, these encyclopedic tomes appeared as timely, if not avant garde, works. They made a huge impact.

As all of us who are familiar with E. T. Bell’s Men of Mathematics and Constance Reid’s Hilbert will immediately recall, this momentous discovery by Poincaré came in the setting of a competition with another young and brilliant mathematician who was hot on the trail, none other than Klein. Evidently the race nigh on brought the latter to nervous exhaustion, and, as I recall one of the aforementioned biographers saying, the result of the competition was essentially a tie. Here, by the way, for good measure, is Poincaré’s own account of his fabulous discovery (which he at first christened Fuchsian functions):

For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series; I had only to write out the results, which took but a few hours.

I am reminded of the aphorism usually ascribed to Paul Erdős, but evidently actually due to Alfréd Rényi, to the effect that a mathematician is a machine for turning caffeine into theorems.

In any event, Felix Klein emerged as one of the true early masters of this burgeoning field. While certainly at home in number theory, it required a lot of beautiful and new and living mathematics from other parts of the discipline, including, for example, hyperbolic geometry, the theory of transformation groups (and, before too long, representation theory), and all sorts of marvels from complex analysis, including the theory of Riemann surfaces. The number theoretic results properly so-called were, of course, nothing less than spectacular and the automorphic forms industry, so speak, continued to boom, especially after Klein brought David Hilbert to the Georg August Universität in Göttingen in 1895: one has merely to consider what just Carl Ludwig Siegel and Erich Hecke, both at Göttingen, brought about in this subject. And of course there is Robert Fricke, Klein’s student, and we now get to the books under review.

The titles making up Klein-Fricke plus Fricke-Klein are arranged as volumes 1–4 in the Classical Topics in Mathematics series published by Higher Education Press, Beijing, China, and distributed by the AMS. The books in question are offered as a paean to “Klein’s vision of the grand unity of mathematics,” and the books’ back covers accordingly allude to Klein’s vaunted Erlangen program. It is indeed hard to imagine anything more consonant with his philosophy than the theory of modular or automorphic forms, what with Klein’s Erlangen program centered on the fundamental role played by group theory in the structure and ensuing form of mathematics, particularly geometries. Just recall for a moment the now-familiar scenario: a discrete (e.g. Fuchsian) group acts on a symmetric space (e.g. the complex upper half plane), and one looks for a uniform way in which to describe what happens when the action is lifted to complex functions on such a space. A sort of functional equations yoga emerges, with the action by the group’s generators on such functions taking central stage. One defines such notions as weights, levels, etc., and makes certain natural identifications resulting in the delineation of fundamental domains for this data. Then comes some natural topology, the prototype being the emergence of the Riemann surfaces obtained from the (compactified) fundamental domains for the action of subgroups of the special linear group over the integers on the complex upper half plane: the theory of elliptic modular forms. Here the geometry is hyperbolic.

This is very beautiful mathematics in its own right, but more should be said regarding connections with perhaps more familiar objects in Zahlentheorie. Arguably the most effective illustration of this role played by modular forms is that of the Hecke correspondence. In rough terms what this powerful interplay is all about is the fact that via the services of the Mellin and inverse Mellin (integral) transforms, one obtains a correspondence between, on the one hand, certain modular forms, and, on the other hand, certain types of Dirichlet series. Modifying one set, e.g. by playing with weights, levels, and characters, on the modular forms end, brings about mirror effects (after a fashion) at the other end. Thus, results in one setting translate to results in the other setting, and we soon have at our disposal a set of very fecund tools to do analytic number theory.

It is worth noting that this material is heavily imbued with Fourier analysis (since we get Fourier expansions of modular forms at cusps of their fundamental domains), and the trailblazer along this magical path is none other than Riemann, and, yes, indeed, once again it’s in his magisterial paper, Über die Anzahl der Primzahlen unter einer gegebenen Größe. There is abundant reason for the famous quip by Martin Eichler that there are not four but five arithmetical operations, viz., \(+\), \(-\), \(\times\), \(\div\), and modular forms.

Very well, then, on to the four books under review. They all come equipped with the same (excellent) introduction by Lizhen Ji, titled, “Why should one open and read Klein-Fricke and Fricke-Klein?” This essay is in itself a gem, and addresses the following themes: why these books are (justly) classics; Klein vs. Poincaré; Klein’s “vision”; the historical role played by these books (and “unearthed treasure”); a rationale for modular and automorphic forms; how to prepare for reading these works; and finally a couple of rather piquant subsections: “Why does Fricke-Klein have a reputation for being difficult and not clear?,” and “A few facts about Robert Fricke.” These last two themes are interesting in themselves, and I won’t give the game away by elaborating too much: the interested reader should crack the books

To convey the special features of each of the four books would be an altogether herculean undertaking, so suffice it to say that the books’ tables of contents will give a good idea of what’s available. It should be noted, however, that Ji has very good reasons for including §6, “Preparatory reading for Klein-Fricke and Fricke-Klein,” in his introductory essay. Here, caveat lector, are a few of his remarks:

Both books … are not easy to read, and even the famous and more expository book [that Klein wrote on the icosahedron] is not easy to many people … [and] there have been several recent attempts to explain some ideas of [loc.cit.] in modern language … But these expositions do not seem to convey completely all the major ideas and results …, or its connections with other work of Klein, in particular [the present books]. Therefore, it is very valuable to read the original writing.

But even in English translation, the style is no longer what we are used to, and maybe it was even rough going for the readers of the late 19th and early 20th century, when the Satz-Beweis style that we are now so attuned to had not yet swept the field. Ji provides a related comment by none other than Jean-Pierre Serre (in connection with the aforementioned book on the icosahedron):

I am sorry to have been slow in answering your query about Klein’s icosahedron book. I have been looking at it, off and on, for the past weeks without being able to write anything.

Indeed, as regards the books under review, which are a horse of the same color as far as expository style goes, it is evident that while the authors’ scope and their corresponding attention to detail in each of the four books are meritorious indeed, there are no modern books written in the style of Klein-Fricke and Fricke-Klein. Here is a snippet from the first book in the quartet, just to give an illustration of what is going on. Book 1, p. 593:

… one can now define the Galoisian problem of 168th degree in form-theoretic form as follows: \(g_2,\,g_3,\,\Delta\) are given numerically, according to their [usual Weierstrassian] relations; from [earlier] equations, to which we also have to add the equation \(f(z_\alpha)=0\), one requires the associated solution system \(z_1,z_2,z_3\) to be calculated. We … designate this … as the form problem of the \(z_\alpha\). However, in order to state our problem in function-theoretic form, we set \(z_1=x,\, z_2=y,\, z_4=1\)  say, whereby [we get] the following equations: \(J:J-1:1 = \phi^3(x,y,1):\psi^2(x,y,1):-1728X^7(x,y,1),\, x^3+xy^3+y=0\). Our equation problem of 168th degree is now, to calculate, for given \(J\), from these equations, the associated solution system \(x,y\).

What a remarkable application of elliptic modular forms to Galois theory this is, and the reader unable to resist its appeal, but it is part of a long narrative, much along the lines of a verbatim transcript of a long lecture, and the same reader has to cultivate the requisite Sitzfleisch to get it all, from soup to nuts. And he had better have a sharp pencil ready. It would perhaps be a worthwhile — but highly non-trivial — enterprise to redo this theorem in modern form: Sätze und Beweise. But the scholar who does this has about 2000 more pages to play with: these four tomes are Teutonically beefy.

Even without the benefit of modern vernacular, however, any serious number theorist would do well to read these books. In fact, the added difficulty of dealing with an unfamiliar style, allowing for so much commentary beyond what today’s writing rubrics might dictate, can only add to the impact this beautiful mathematics would have on the patient and diligent reader. After all, Abel’s aphorism is ever so true: we should learn from the masters, not their pupils (allowing in this case for the fact that, in the beginning, Fricke was Klein’s student).

One final comment: these books each contain a copy (all isomorphic) of a set of commentaries by contemporary mathematicians. Richard Borcherds highlights the “bizarre properties” of Klein’s elliptic modular function, as well a lot more. Jeremy Gray addresses historical material. William Harvey writes on Fricke-Klein: automorphic functions as such. Barry Mazur’s essay is in itself an excellent review of much of the material in these books. There is one essay jointly by Caroline Series, David Mumford, and David Wright, and one essay by Domingo Toledo. Lastly, there is a collection of five very short commentaries, by Igor Dolgachev, Linda Keen, Robert Langlands, Yuri Manin, and Ken Ono. These address in broad terms why the enterprise of bringing Klein-Fricke and Fricke-Klein to a modern audience, in English, is so commendable. Ono is particularly encouraging:

It pays to read the seminal works by the greats, and with these translations these classic works will be reintroduced to generations of mathematicians.

Langlands, however, strikes a note of warning:

The indifference to the mathematical past and the distance from it will, I suppose, … become more pronounced if or when the Asian nations assume a major role in mathematics and even English ceases to be the principal, or even an adequate, medium of communication. The possibility of reducing mathematics to a trivial pursuit, a struggle for a large number of citations, for a prize, or just for a tenured position, is also there. It is difficult not to be pessimistic!

Against this threat, these four volumes stand as an obvious part of a necessary counteroffensive. It is noteworthy and encouraging that they are published in China: perhaps Langlands’ pessimism might be somewhat mitigated. Time will tell.

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