I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the seminal work of Selberg who developped a a very broad theory which applies to any weakly symmetric Riemannian spaces and not just to symmetric spaces coming from $GL_2$.

A very good reference that I found on the subject is the book of Tomio Kubota entitled "Elementary theory of Eisenstein series". I found that book extremely well written and very accessible for somebody who wants to learn the subject. I have noticed that this book is not so often cited (as I think it should be) as compared for example to the more recent book of Iwaniec entitled "Spectral methods of Automorphic forms". In fact, I find it very curious that Kubota's book is not even cited once in Iwaniec's book, since after reading the two books, I found a lot of overlaps, especially on the topic surrounding $GL_2$ real analytic Eisenstein series.

For a paper that I'm currently writting, I'm planning to cite Kubota's work and give him the proper credits, but before doing so I would like to make sure that I'm not missing an earlier reference on $GL_2$-real analytic Eisenstein series (besides of course the paramount work of Selberg, Langlands, Harish-Chandra, Gelfand, Pyatetskii-Shapiro who worked in much greater generality).


1 Answer 1


R. Rankin's 1939 paper giving a non-trivial estimate on Ramanujan's $\tau$ function used the "real-analytic Eisenstein series" for $SL_2(\mathbb Z)$, at least. Selberg's related paper just-slightly later seemed to express the same awareness for such cases, as opposed to the more general situations treated in the 1950's. In some informal remarks I saw elsewhere, Rankin credited his advisor, Ingham, with knowing how to meromorphically continue such Eisenstein series.

The idea for congruence subgroups of $SL_2(\mathbb Z)$ was recapitulated (at least) in one of R. Godement's articles in the 1965/66 Boulder Conference proceedings (AMS Proc. Symp. IX), which was where I saw it first.

The appendix in Langlands' SLN 544 in which he stitches together several $GL_2$ cases to get minimal-parabolic Eisenstein series for $GL_n$ and $Sp_n$ certainly presumes that the $GL_2$ case is well-known. Apparently these notes were written up by about 1967 even though they were not public until 1976.

My own (then-naive) impression by the mid-1970s was that the meromorphic continuation of $GL_2$ things was a cliche. Selberg's ideas from the 1950's seemed to suffice for rank-one situations without subtle data on the (possibly non-abelian) Levi components. Lectures of G. Shimura at Princeton in the mid-1970's proved things, including meromorphic continuation of Eisenstein series, by looking at Fourier expansions of all sorts of $GL_2$ Eisenstein series, related to applications of Maass-Shimura operators and "nearly holomorphic" automorphic forms on $GL_2$ over totally real fields.

(So my own 1989 book on Hilbert modular forms gave such an argument over totally real fields, maybe mentioning Rankin and Selberg from 1939...)

So, although it might have been reasonable for (e.g.) Iwaniec to mention Kubota, I think it is not accurate to worry that a deserved citation was inappropriately denied. Rather, already by the time of Kubota's book those ideas were perhaps considered "general knowledge", so by the 1990's (and Iwaniec' book) the ideas would have been even more "classical".

Edit (in response to Hugo Chapdelaine's clarifying comments): Selberg's spectral approach seems/seemed limited, if one insists on a full spectral decomposition, to situations where no cuspidal-data or other subtleties enter. My colleague D. Hejhal confirmed (from Selberg's unpublished papers) my suspicion that Selberg had not considered (to whatever purpose) cuspidal-data Eisenstein series, nor corresponding parts of a spectral decomposition for higher-rank groups. Oppositely, so far as I can tell Harish-Chandra's relatively early results on automorphic forms on general semi-simple groups (e.g., 1959) were aimed primarily at the cuspidal part of the spectrum. Especially the residual part of the spectrum, not in the functional analysis sense but in the sense of being a residue of Eisenstein series, is the most ... obscure... part of Langlands' 544.

Yes, the Lax-Phillips 1976 argument that not only do spaces of cuspforms discretely decompose (e.g., for a relevant Laplacian), but even "pseudocuspforms", was a striking point, although I think their literal result was considerably obscured by its context, and a certain unclearness about how much was a physically-based heuristic and how much was really proven/provable. Nevertheless, yes, Colin de Verdiere's 1981 "new proof" of meromorphic continuation of Eisenstein series showed the peculiar potential in that fact, even though (to my recollection) many people were confused about the precise nature of "Friedrichs extensions", and how it could be that certain truncated Eisenstein series (palpably not smooth) could become eigenfunctions for some variant of an elliptic operator. (I recall wrangling between P. Cohen and Selberg at Stanford over such points c. 1980, and at the time I was certainly baffled.)

My perception at the time of the reception of Colin de Verdiere's result was that, mostly, since it reproved a true, known result, was just an "announcement" (rather than detailed paper), and used a tricky feature of unbounded self-adjoint operators, people did not engage with it as much as they might have. Nevertheless, W. Mueller showed in the 1990s that this approach certainly immediately applies to rank-one cases, and H. Jacquet at least sufficiently assured C. Moeglin and J.-L. Waldspurger that the analogous argument would succeed generally, so that they outline a similar argument in their book.

Of course, in all these cases the crux of the matter is that some artfully arranged operator has compact resolvent, therefore discrete spectrum, therefore various things have meromorphic continuations... Thus, in contrast to Roelcke's meromorphic continuation up to $\Re(s)=1/2$, which follows from very general spectral considerations, getting beyond that line when it parametrizes the continuous spectrum is non-trivial.

Yes, in fact, by now, I like the Colin de Verdiere argument better than other forms, since it gives more meaning to poles. In slight contrast, the J. Bernstein reformulation of Selberg's "third" proof (the latter from Hejhal's trace formula volumes) is a general-enough idea that, ironically, it seems not to obviously give sharp information about the case-at-hand, namely, automorphic forms.

One should also mention some experiments done by A. Venkov not only about spectral expansions for automorphic forms, in the usual sense, but also some Schroedinger operators or Dirac operators for automorphic forms...

I think the most serious subtlety is still correct identification of residual (=residue of Eisenstein series) spectrum. Harish-Chandra's SLN 62 from 1968 explicitly doesn't attempt this, and somewhere there's a quote from Langlands that he doesn't remember how that part of his SLN 544 went. For $GL_n$, Jacquet's conjecture from about 1983 (Maryland conference) proven by Moeglin-Waldspurger (1989) is the only (to my knowledge, at this date) systematic result for any family of groups. Of course, various methods can prove that some Eisenstein series have no poles in various cones/half-planes, but that's not the question.

For that matter, residues of maximal-proper-parabolic Eisenstein series are relatively tractable, via Maass-Selberg relations and their implications (as Borel already did in his little $SL_2(\mathbb R)$ book), but iterated residues entail complications that are not trivially dispatched.

  • $\begingroup$ Dear Paul, thanks for portrait that you have brushed. Regarding your comment about Kubota's book I get your point but my feeling is that his book was probably the first comprehensive and accessible reference on $GL_2$-real analytic Eisenstein series for non-experts. Also in the nice portait that you just depicted, one should also probably mention the substantial contributions which came from physics and functional analysis. $\endgroup$ Oct 9, 2015 at 22:50
  • $\begingroup$ I have in mind the book of Lax and Philips who applied scattering theory to the study of automorphic forms. In that book I saw for the first time a truncated version of the hyperbolic Laplacian which has a compact resolvent. Then from a key observation of Colin de Verdiere, one may apply the Fredholm analyticity theorem to obtain the analytic continuation and the functional equation. In any case, thanks again for your answer. $\endgroup$ Oct 9, 2015 at 22:50
  • $\begingroup$ ... oop, and, yes, after writing more, I would certainly want to acknowledge that much "physics-y thinking" went into the Lax-Phillips viewpoint, and certainly into Colin de Verdiere's, as well! I have long been a fan of aggressive heuristics coming from physical considerations. I don't pretend to appreciate E. Witten's more contemporary contributions, but Heaviside's and even more Dirac's are quite striking to me. And "solvable models". Such stuff is very inspiring, even if/when it does not come with proofs! :) $\endgroup$ Oct 9, 2015 at 23:34

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