I am interested what kind of (different) proofs of meromorphic continuation Eisenstein series (for general parabolic subgroups) exist in the literature? The only one I understand well, is Bernstein's proof using his "continuation principle" (it is probably unpublished) but apparently there are many others (especially I am interested in proofs which are different from the one in Langlands book)
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asked Mar 14, 2011 at 5:59
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4 Answers
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A belated response: so far as I can tell, Selberg's idea, taken literally, does not apply at all in rational rank above 1. One should note that Avakumovic and Roelcke had similar ideas, which also did not anticipate the complications of higher rank. Langlands' 544 (notes written in the mid-60s, not public until mid 70s) were extremely novel in their recognition of complications in higher rank, e.g., cuspidal-data Eisenstein series in the first place, and non-constant residual automorphic forms (e.g., Speh forms). Colin-de-Verdiere's argument works well in rank 1, but, in its nascent form, has the same limitations as Selberg's 1950s viewpoint. Moeglin-Waldspurger cite Langlands and others, but do not give proofs of several analytical points.
Bernstein's apocryphal proofs of meromorphic continuation are reputed to be instantiated last fall... but one should not be over-optimistic, given the possibility of people finding other priorities.
Around 2001, I tried to rewrite notes on Bernstein's idea, with help from notes obtained thanks to Hejhal and Sarnak. I think it is fair to say that there are several confusing points, even if other potentially confusing points can be cleared up by "standard mathematics".
In the last few years, there has been interest in supposedly applying Bernstein's method [sic] to not-cuspidal-date Eisenstein series... My personal reaction, based on some experience, is skeptical.
I would like to see (and may try to write it myself) actual proofs for cuspidal-data Eisenstein series. :) A significant caution is that various spaces of automorphic forms meeting growth conditions are not representation spaces for the relevant group G, so reasoning that implicitly assumes this is dangerous. Of course, one often needs less...
Lisa Carbone and Howard Garland have recently written some things about Eisenstein series on not-classical-groups (e.g., loop-or-something)... that seem to succeed, based aesthetically/morally on the Bernstein-Selberg arguments.
If anyone wants further technical information about my assessment of the situation, I welcome email about meromorphic continuation of Eisenstein series. :)
Edit (15 April '12): [By the way, deleted my last year's comment on mis-spelling in the question, which I could not correct last year...!] Disregarding the relatively special cases where arguments based on Poisson summation can succeed, as far as I can tell, all other approaches need some compactness or finite-dimension assertion at some crucial juncture. (The usual way to prove some space is finite-dimensional is to exhibit it as a non-zero eigenspace for a compact operator!) Granting some such assertion, the remainders of the arguments are relatively formal. E.g., inside Colin de Verdiere's argument is an essential Rellich-like compactness assertion about a resolvent, in a form due to Faddeev-Pavlov and Lax-Phillips. I suspect that the essential "confusing" or "mysterious" aspects of proof-sketches reside in problems about this sort of point. Not that incorrect conclusions are reached, but that the complexity of the set-up often gives an impression that one has done sufficient work to have completed a/the proof, and "surely" an "auxiliary" finite-dimensionality statement oughtn't be critical? That is, it's not that various "confusing" argument are incorrect, but, rather, perhaps incomplete. Correctibly so, indeed, but perhaps not trivially.
answered Apr 4, 2011 at 23:14
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2$\begingroup$ I found your answer itself a bit apocryphal, although that might be a language thing (e.g. I never heard about the word "instantiate" before). Are you saying that there are no complete proofs available for the meromorphic continuation of Eisenstein series and the spectral decomposition of $L^2(G(k)\backslash G(\mathbb{A}))$? What's the status of $G=\mathrm{GL}_n$? $\endgroup$ Commented Apr 5, 2011 at 0:48
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$\begingroup$ @GHfromMO, sorry to be so late in responding, but, indeed, if I could usefully clarify, what might it be? $\endgroup$ Commented May 8, 2015 at 0:32
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You'll find proofs of analytic continuation (and functional equation, both generally come together) in Weil's "Basic number theory", Bump's "Automorphic forms and representations", in Godement's "Séries d'Eisenstein" (available here), in Hida's "Elementary theory of $L$-functions and Eisenstein series", Garrett's "Holomorphic Hilbert modular forms" and a host of other places...
answered Mar 14, 2011 at 6:51
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$\begingroup$ The question seems to be about the case of a general reductive group. It is certainly not in Weil, not in Bump (GL_2 only), not in Hida (GL_2 only), not in Godement (Siegel modular forms only, and not in representation-theoretic language), and I don't know about Garrett (but I doubt it, since the title indicates a very restricted case). $\endgroup$ Commented Mar 14, 2011 at 7:52
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$\begingroup$ I took the question as meaning "I have other references for the same thing, but can't wrap my head against them"... so I tried to give a larger view. The references I gave have different proofs (or sometimes the same but with slightly different explanations) in different settings -- which should still help, since the question says "The only one I understand well" and "proofs which are different from the one in Langlands book". $\endgroup$ Commented Mar 14, 2011 at 8:11
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$\begingroup$ I actually did mean proofs that work for arbitrary $G$ and arbitrary parabolic $P$. The case of $GL(2)$ is more or less trivial anyway (and more generally when we talk about Eisenstein series induced from Borel subgroup). $\endgroup$ Commented Mar 16, 2011 at 13:50
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I'm not familiar enough with the proofs to say if they are more than superficially different, but here is something:
Moeglin-Waldspurger prove continuation crediting Jacquet (see p. xix), instead of Langlands. They say it is similar to that given by Efrat, in his treatment of the Hilbert-modular ($PSL_2$ over a totally real field) case. Here, Jacquet credits M-W's proof to Colin de Verdière's "new and strikingly brief and elegant proof" (from the MR review) for $SL_2({\mathbb R})$ (extended here). I couldn't find any extensions of Colin de Verdière's argument to higher-rank cases, but that may be because it was done in Moeglin-Waldspurger.
Muller also has a proof in the rank-one case.
Wong gave a proof using integral equations.
Everyone listed credits Selberg with their ideas.
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$\begingroup$ The proof of Colin de Verdière is indeed very nice, and it would be very interesting to see how far it can generalize (it is written for subgroups of $SL_2(\mathbf{R})$ with a single cusp; the paper is available numdam.org/numdam-bin/fitem?id=AIF_1983__33_2_87_0 on Numdam.) $\endgroup$ Commented Mar 17, 2011 at 5:43
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$\begingroup$ Looking around a bit more, Jacquet attributes Moeglin-Waldspurger's proof to Colin de Verdière, so that may be our answer. Thanks for the working link! $\endgroup$– B RCommented Mar 17, 2011 at 8:10
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$\begingroup$ Also, it's not clear to me that these proofs don't implicitly prove the critical component needed to invoke Bernstein's continuation principle (that the parametrized system of equations has a locally finite-dimensional solution space). $\endgroup$– B RCommented Mar 17, 2011 at 17:33
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$\begingroup$ Actually, in the situation I have to deal with, the "critical component" of Bernstein's proof turns out to be the property that in some domain the system has unique solution. In any case, Moeglin-Waldspurger prove more than Bernstein - they give you some information about poles which Bernstein's proof has no chance to give. Thank you all very much! $\endgroup$ Commented Mar 18, 2011 at 2:15
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$\begingroup$ Also I tried to read Selberg's original proof in his ICM paper and didn't understand it. I think that Sarnak couldn't understand it either and he rewrote it a little (using slightly different idea). Selberg's original proof remains a mystery for me. $\endgroup$ Commented Mar 18, 2011 at 2:16
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I think that in particular Erez Lapid has done a nice job with these two slides
http://www.math.clemson.edu/~jimlb/ConferenceTalks/ColumbiaWorkshop2006/lapid1.pdf
http://www.math.clemson.edu/~jimlb/ConferenceTalks/ColumbiaWorkshop2006/lapid2.pdf
Have a look in particular on page 10 in the first slide session for Bernstein's prinicple, and a proof of it is on page 11. He has also states it in for $SL(2)$ on page 9, which is always helpful for me before seeing a general statement. The second slides focus on the higher rank situation.
By the way, the whole side is great: http://www.math.clemson.edu/~jimlb/coursenotes.html
Perhaps a remark about Eisenstein series: I think that at least in a congruence setting, the analytic continuation is in some sense equivalent to analytic continuation of automorphic L function. The Langlands-Shahidi http://en.wikipedia.org/wiki/Langlands-Shahidi_method method deduces from the analytic continuation of Eisenstein series the analytic continuation of automorphic $L$ functions. So every new proof of analytic continuation for Eisenstein series yields a new proof for the analytic continuation of automorphic $L$ functions.