# Beyond Endoscopy

Shalika volume, Johns Hopkins University Press2004
MathReview: 2058622

Author's comments: This paper has appeared in the Shalika volume, published by the Johns Hopkins University Press. The published paper is not exactly the same as that here because the text was modified by the Press, apparently for reasons of economy. Among other things footnotes were removed and this required somewhat brutal measures. The version in this collection is the preferred version.

Author's comments: This paper is quite informal and I could not immediately reflect on the suggestions it contains. I am grateful to Freydoon Shahidi for suggesting that as an interim measure I write the paper for submission to the Canadian Mathematical Bulletin, where it appeared in volume 50 (2007).

Annales des Sciences Mathematiques du Quebec342010

Author's comments:The most important point for the innocent or inexperienced reader of this paper to understand is that it is the stable trace formula that is here invoked. The stable trace formula, introduced many years ago in the reference [L2], developed and applied in the references [K1], [K2], [K3] and, more recently, in a very systematic way and to extremely good effect in [A2], is what allows the introduction of the Steinberg-Hitchin base and of the Poisson summation formula. I, myself, hope to develop some of the consequences of this, some certain, some as yet only possible, in subsequent papers. The present paper was written together with Edward Frenkel and Ngô Bảo Châu. The paper is expected to appear in Annales des sciences mathématiques du Québec.

Annales mathématiques du Québec37-2pp. 173-253September 2013

Author's comments: This text is provisional from a mathematical point of view, but it may be some time before the obstacles described in the concluding sections are overcome. Serious progress has been made by Ali Altuğ.

It has been easy to misconstrue the principal purpose of this paper and of the previous paper, at least my principal purpose. It was to introduce the use of the Poisson formula in combination with the stable transfer as a central tool in the development of the stable trace formula and its applications to global functoriality. Unfortunately the review in Math. Reviews was inadequate, simply reproducing the abtract, written not by me but by the editors, A transfer similar to that for endoscopy is introduced in the context of stably invariant harmonic analysis on reductive groups. For the group $\mathrm{SL}(2)$, the existence of the transfer is verified and some aspects of the passage from the trace formula to the Poisson formula are examined.'' This transfer is for me a central issue for harmonic analysis on reductive groups over local fields. The problems it raises have, so far as I know, not been solved even over $\mathbb R$ and $\mathbb C$. Its construction for $\mathrm{SL}(2)$ over $p$-adic fields, $p$ odd, was, and remains, for me an interesting application of the explicit formulas of Sally-Shalika for the characters of that group.

Finished in late 2012

Author's comments: There is as yet no text with the title Functoriality and Reciprocity. Begun as preparations for a lecture, the text appearing here was, and remains, a first attempt to come to terms with the two topics of the title, an attempt that is perhaps doomed by its nature and by my years to remain provisional. The attempt demands not only a great deal of reflection and a good number of novel ideas, mine or those of someone else, but also a mastery of several mathematical domains --- algebraic number theory, algebraic geometry, spectral analysis, representation theory, some differential geometry, some mathematical physics --- of all of which I have acquired a smattering of knowledge over the past decades, although without mastering any. As it stands here it is only a part of the Prologue. The present text, A prologue to Functoriality and Reciprocity'', Part I appears, in spite of its failings, in a volume dedicated to the memory of Jonathan Rogawski, Pacific Journal of Mathematics, vol. 260, No. 2, Dec. 2012.

New Haven on Oct. 23, 2013

Author's comments: I took the occasion of the Mostow conference in New Haven on Oct. 23 to discuss my still provisional reflections on the geometric theory, thus for the second part of the Prologue. The best I can offer in the way of a geometric theory with which I would be pleased is a sketch of the principal difficulties to be overcome. There are many. The importance for me is the very strong analytic flavour of the theory I hope to construct or see constructed.

Author's comments. Although these notes were written as a foreword to, or an appreciation of, a book by Qing Zou that is to appear soon, the primary purpose for me was to describe some possibilities in the theory of automorphic representations upon which I believe it is important for specialists to reflect. I was grateful to Qing Zou for the somewhat unexpected request to write an appreciation. The book itself will appear in Chinese; the translated title is From Kummer to Langlands—The history of the Langlands Program''

Author's comments: There are one or two comments I neglected to make in the course of the lecture or matters that were inadequately explained.

(i) The functions to be used in the proof proposed for functoriality are, at least initially, sums of logarithmic derivatives of automorphic $L$-functions,
$-\sum_{\pi^{{\mathrm{st}}}}m(\pi^{{\mathrm{st}}})\frac{L'(s,\pi,\rho)}{L(s,\pi,\rho)}=\sum_{n=1}^\infty\frac{{\mathrm{tr}}_{{\mathrm{st}}}(f_n)}{n^s},$
so that $n$ actually runs over powers $p^k$, $k > 0\in\mathbb Z$, of primes. If the base field is not $\mathbb Q$, it runs over powers of prime ideals $\mathfrak p$. Indeed finitely many $p$, $p_1,\dots,p_k$, or finitely many $\mathfrak p$, $\mathfrak p_1,\dots,\mathfrak p_k$ may be excluded. The function $f_{p^k}$ is the convolution of a spherical function at $p$ and an arbitrary smooth, compactly supported function in $\prod_{k=1}^sG(F_{p_k})$, inserted so that the final result of the limiting process is applicable to individual classes of (cuspidal) automorphic representations. This second function is independent of $n$. The stable multiplicity $m(\pi^{{\mathrm{st}}})$ is the multiplicity of the $L$-packet. The simplest example of this multiplicity is provided by $\mathrm{SL}(2)$ and is discussed in the paper of Labesse-Langlands. The stable trace is that given by the stable trace formula, something whose existence I take for granted here!

(ii) Although I did not insist on it in the lecture, the notion of stable transfer, introduced with the example of $\mathrm{SL}(2)$ in the paper Singularités et transfert, will be, I am convinced, an important feature of the trace formula.

(iii) I observe that no notion of reciprocity is suggested for the geometric theory, neither over a finite field nor over $\mathbb C$. One might ask whether this might exist and might involve one-parameter families of motives. I have never done so!

(iv) I believe I was sufficiently clear in the videos. I repeat here none the less that although the suggestions implicit in (ii) and (iii) of the text oxford.pdf demand a broad understanding of a number of different fields and, for some, a conceptual veering, as well as time and effort, they are not excessively daunting, whereas (i) and (iv) will demand more courage and talent than many of us have. I would hesitate to make, at this point, any very precise assertion, but as I reflect on the lecture the possibility of a more intimate relation than first envisaged between the Yang-Mills connections and the connections constructed from the Hecke eigenfunctions begins to suggest itself. I am still very uneasy with the former. It may be that the relation not only exists but also is well-known in some circles. At the moment, however, I have no reason to think so.

(v) Although Hecke operators appear nowhere in it, I have found the paper of Atiyah-Bott otherwise an excellent, as well as a rich, source of information about the geometry of ${\mathrm{Bun}}_G$ and the connections on the two spaces ${\mathrm{Bun}}_G$ and ${\mathrm{Bun}}_{{}^LG}$. It is not, however, easy to read, and for anyone who does not possess a good deal of geometric and topological facility even more difficult. Moreover, the authors have been intimidated by the algebraic geometers' emphasis on stable bundles, so that some aspects of the geometry that are important in the context of the videos are not discussed. It is, nevertheless, instructive to be forced to reflect on these matters on one's own.

(vi) The purpose of my final comments was perhaps not clear. I am repeating a suggestion made elsewhere, also by me, that what is now, unfortunately and inappropriately, often referred to as the Satake parameter be called the Frobenius-Hecke parameter.

(vii) I draw attention here to an absence in the video, that is also an absence in the program.'' Although I began the study of automorphic forms on listening to the lectures of Steven Gaal on a paper of Selberg's, I was soon led to the papers of Siegel and Harish-Chandra. Among other things, Siegel's papers were often concerned, in one way and another, with theta-series. His techniques were formalized by Weil as the metaplectic group, a topic that was very popular for a while in the sixties and seventies of the last century, partly because it allowed the proof of functoriality (under the unfortunate label lifting'') in a number of special cases. With the appearance of the notion of functoriality and the possibility of using the trace formula to prove it, the metaplectic group has faded into the background, in part, at least in my view, rightly so. It and the associated automorphic forms of half-integral weight are nevertheless there and are not accounted for by the theory of automorphic forms adumbrated in the video. So something appears to be, or may be, lacking.

(viii) For comments on the relation between universality'' and conformal invariance'' from someone who has struggled with these questions in a concrete way and with considerable success I suggest Towards conformal invariance of $2D$ lattice models and other papers of Stanislas Smirnov.

Editorial comment: Please note that the following videos might not play in Mozilla Firefox, but should in Google Chrome and possibly other browsers.

# Letters

## Message to Edward Frenkel on the Mostow Lecture

March 2, 2014
[ frenkel.pdf ]

Author's comments: These terse comments were intended as a suggestion not to read the Mostow lecture or listen to it with preconceptions about the nature of a geometric theory.

## Message to Peter Sarnak

February 18, 2014
[ message-to-Peter-Sarnak.pdf ]

Author's comments (Apr. 6, 2014): The concept Langlands program'' appears in the title of an article by Stephen Gelbart in the BAMS of April, 1984, but Gelbart himself assured me that it was already current, at least orally, before then. He also drew my attention to a phrase of Armand Borel in his Bourbaki seminar of June, 1975, plutôt un vaste programme, élaboré par R. P. Langlands depuis environ 1967.'' I do not recall that I was uneasy with the phrase Langlands program'' in 1984, but it then referred principally to matters on which I myself had long reflected. This has since changed. It has come to refer to a domain much larger than the analytic theory of automorphic forms and its arithmetic applications.

This is reflected in the message to Sarnak. Although this perhaps does not correspond exactly to the historical development, the enlargement can be described in two stages. I discuss them separately. The first is the extension of the theory --- both the established and, in some regards, also the conjectural form --- from a finite extension $F$ of $\mathbb Q$ as the base field to a different kind of base field, to an algebraic function field $F$ over an algebraic curve associated to a field of constants that is either a finite field or the complex number field $\mathbb C$. It is the second possibility that has the novel ramifications. My source of information on the first and on the second enlargement has been various articles of Edward Frenkel. These articles are impressive achievements but often freewheeling, so that, although I have studied them with considerable care and learned a great deal from them that I might never have learned from other sources, I find them in a number of respects incomplete or unsatisfactory.

As I attempted to explain in the Mostow lecture, I believe that the geometric theory as such should be separated from any kind of duality in physics and treated purely mathematically with less emphasis on sheaf theory than is usually met and with a larger dose of classical mathematics'': spectral theory, differential geometry, and algebraic geometry. The algebraic geometry and the spectral theory will, I believe, have to mesh, thus one will be obliged, despite the algebraic geometers, to work with the full classifying space $\mathrm{Bun}_G$. I am hopeful that such a theory can be constructed in a satisfactory and natural fashion, but this demands a mastery of the pertinent mathematics.

It is the third item of the message to Sarnak whose complexity I did not adequately appreciate as I wrote it. Before explaining this, let me comment on the structures implicit in the classical theory and in the geometric theory. In the classical theory, there is a reductive group $G$ over a number field and a reductive group ${}^LG$ over $\mathbb C$. The second group may have several components whose source lies in Galois theory, but that is not the issue here. Their relation is expressed by functoriality. A second aspect of the theory is the relation between the group ${}^LG$ and the Galois group of $F$ or, more conjecturally, the motivic group over $F$. This I refer to as reciprocity, the first manifestation of the pertinent phenomena being the law of quadratic reciprocity. Neither aspect has yet been developed to the extent I believe possible, but both principles have led to strong results whose mathematical importance is undeniable and unlikely to be ephemeral.

In the geometric theory as such there is, so far as I can see, no reciprocity, just functoriality and this manifests itself as a parametrization of automorphic representations by connections, in the sense of differential geometry, with values in ${}^LG$, which is often taken to be connected. The possible construction of such a theory, envisioned in the third item of the message, is one of the problems considered in the Mostow lecture. The marked difference between the geometric theory over a closed nonsingular Riemann surface and the arithmetic theory is that in the geometric theory $G$ and ${}^LG$ are groups of the same type, namely effectively groups over $\mathbb C$. This leads to a third possibility in addition to functoriality and reciprocity and to a second stage, namely duality. Here we meet problems outside the domain of pure mathematics.

One popular introduction to the topic is Frenkel's Bourbaki lecture, Gauge theory and Langlands duality. On the first page, he describes electro-magnetic duality as an aspect of the Maxwell equations and their quantum-theoretical form or, more generally, as an aspect of four-dimensional gauge theory. This duality is quite different than the functoriality and reciprocity introduced in the arithmetic theory. It entails a supplementary system of differential equations. Moreover, it has to be judged by different criteria. One is whether it is physically relevant. There is, I believe, a good deal of scepticism, which, if I am to believe my informants, is experimentally well-founded. Although the notions of functoriality and reciprocity have, on the whole, been well received by mathematicians, they have had to surmount some entrenched resistance, perhaps still latent. So I, at least, am uneasy about associating them with vulnerable physical notions. On the other hand, as strictly mathematical notions this duality and various attendant constructions, such as the Hitchin fibration, appear to have proven value, especially for topologists and geometers. Whether it is equal to that of functoriality and reciprocity is open to discussion.

In contrast to what I foresaw when describing item (iii), the considerable reflection on duality contemplated there will demand a more sophisticated understanding of topology, geometry, and the relevant physics than I can ever hope to possess.