The Heat Kernel and Theta Inversion on \(\mathrm{SL}_2(\mathbb{C})\)

The late Serge Lang was, by all accounts, a very colorful character, extremely fierce in both his opinions and the methods he employed to defend them, and possessed of a correspondingly idiosyncratic style in his pursuit of mathematics. Lang’s prevailing foci were number theory and the algebraic geometry that touches on number theory, although his prolific output also includes books on, say, graduate level analysis and algebra. Of course, given that modern number theory is unthinkable without a huge palette of background mathematics at one’s disposal, this makes eminent sense.

I belong to the generation of university students for whom Lang’s Algebra (in its old red binding) figured ever so prominently when the time for qualifying examinations was upon us and we had to prepare to pay the piper. Additionally, as I went on to study number theory itself, Lang’s Algebraic Number Theory entered the game. Over the years, I have had occasion to look at a number of other books by Lang. Again, given his level of productivity and the quality of his work, it could hardly be otherwise.

By and by, regarding my own work, starting with modular forms in undergraduate and graduate school and continuing over the years to include work on theta functions, abstract Fourier analysis, and allied arcana, Lang’s well-known book SL₂(R) proved to be the most relevant: it can always be relied on for effective and accessible discussions of the various topics of interest. The book under review, The Heat Kernel and Theta Inversion on SL₂(C), co-written with Jay Jorgenson, is something of a sequel to SL₂(R) in that “[l]ike… SL₂(R)… [it] provides an introduction to the general theory of semisimple or reductive groups G, with symmetric space G/K (K maximal compact).” However, it is a lot more than that.

Indeed, the Introduction very quickly and effectively lays out an ambitious plan of attack, the book’s raison d’être, which centers on the subject of theta inversion, connections with zeta functions, and with both analytic number theory and algebraic (and analytic) geometry making an appearance in relatively short order. Much of what the authors have to say in this opening oration, however, is along the lines of ideology (to use Manin’s term), in that they propose that what they do in these pages sets the stage for the indicated further investigations: The Heat Kernel and Theta Inversion on SL₂(C) is only the beginning.

It is clear that the greater objective is to start a movement of sorts; in fact, Jorgenson, in his Preface, says as much: “Beginning in the 1990s Lang became fascinated with the prospect of using heat kernels and heat kernel analysis in analytic number theory. Specifically, we developed a program of study where one would define Selberg-type zeta functions associated to finite volume quotients of symmetric spaces, and we speculated that each such zeta function would admit a functional equation where lower level Selberg-type zeta functions would appear. In the case of the Riemann surface associated to PSL₂(Z), the functional equation has a functional equation which involves the Riemann zeta function. Lang and I began the work necessary to carry out our proposed analysis for quotients of SL_n(C) by SL_n(Z[i]). As with other… works undertaken by Lang, he wanted to develop the foundations himself. The present book is the result of establishing, as only Lang could, the case n = 2.”

One immediately gleans from this passage that the authors’ approach to their larger theme is very much in keeping with what has evolved in the twentieth century vis à vis the main parts of analytic number theory, with Selberg taking center stage, having taken the baton from Hecke and Siegel, who took it from Hilbert (or Poincaré, or Klein, or (I suspect) Landau), who took it from Dedekind and Dirichlet, who took it from Riemann himself, if I may be forgiven a shameful amount of historical license.

It certainly was Riemann who, in what is arguably the most dramatic paper in mathematical history (yes: I’m willing to defend this position!), Über die Anzahl der Primzahlen unter einer gegebenen Grösse, started the ball rolling: [{zeta functions + Mellin transforms →  theta functions (and back again)} + Fourier analysis] → theta- & zeta-functional equations + the inner life of primes. Fast-forward to Hecke: L-functions and theta functions (half-integral weight modular forms for a certain discrete, or Fuchsian, group) and functional equations; fast-forward to Selberg: the trace formula(s). Voilá: the emergence of modern analytic number theory.

But what of the heat kernel in all this? Well, at possibly its deepest level, it’s all Fourier analysis, of course: suppressing scaling or normalizing constants, the heat equation, u_xx = u_t, as a boundary value problem (meaning that one adds suitable (sometimes asymptotic) boundary requirements (and before too long distributions or, at least, Dirac δ-functions come knocking on the door) can be persuaded to yield Doppelgängers of exp(-x²t), and even (theta-) series of such (think Sturm-Lioiuville), coming about as a consequence of the eigenvalue problem for this PDE obtained via separation of variables. Thus, from the PDE standpoint, the natural habitat of theta functions is precisely the theory of the heat equation, and one can only agree with Lang: exploring the heat kernel (a.k.a. Gaussian kernel, Gaussian density, etc.) for such analytic number theoretic purposes is an irresistible prospect, and one that more than justifies the book under review.

The Heat Kernel and Theta Inversion on SL₂(C) has five main parts, respectively, “Gaussians, Spherical Inversion, and the Heat Kernel,” “The General Trace Formula,” “The Heat Kernel on ΓG/K,” “Fourier-Eisenstein Eigenfunction Expansions,” and “The Eisenstein-Cuspidal Affair.” As already mentioned, this is a fair representation of many of the main thrusts in modern analytic number theory as focused on automorphic forms, and the heat kernel perspective delivers very interesting results.

However, the book is not written for the general mathematical reader. As a Springer Monograph it is not meant to be a book for beginners. It is obviously meant for experienced insiders, even as the presentation of the material is excellent and accessible. A well-prepared graduate student would do well with this book. More experienced analytic number theorists will find it enjoyable and spellbinding. (Providentially the material Jorgenson and Lang deal with is of great interest to me in connection with my own research in which theta functions play a huge role: I expect to have many good uses for The Heat Kernel and Theta Inversion on SL₂(C)!)

When I was a graduate student at UCSD, Lang came to visit my advisor, Terras. I recall I was missing in action, probably because I had again been derelict in some of my academic duties, no doubt, and was keeping a very low profile. So I missed the opportunity of meeting Lang in person. In studying The Heat Kernel and Theta Inversion on SL₂(C) I will have occasion to meet him again in his writing: Jorgenson closes his Preface with the phrase, “I have refrained from making any changes in order to preserve Lang’s style of exposition.” Given that the draft of The Heat Kernel and Theta Inversion on SL₂(C) was completed only one month before Lang’s passing, and, that, as such, this book is Lang’s last, this is the right touch. I very much look forward to dissecting the fascinating material presented in this book which I heartily recommend to other analytic number theorists of a similar disposition.

Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.