Author's comments: There are two, related parts to this thesis: one on representations of Lie semi-groups and one on operators associated to representations of Lie groups. The first part was published in the Canadian Journal of Mathematics, but the second was published only as an announcement in the Proceedings of the National Academy of Sciences of the USA. It nevertheless had the good fortune to be taken seriously by Derek Robinson, who incorporated some of the results into his book on Elliptic Operators and Lie Groups.

Examining again, after forty years, the verification of the basic estimates of the second part, I found a large number of misprints, so that it would have been difficult if not impossible to follow my arguments line by line. I have tried in the present version to correct the misprints, but cannot be certain to have fully succeeded. The notation sometimes takes a few minutes to decipher but appears to be comprehensible.

The thesis remains, to my regret, my only active encounter with partial differential equations, a subject to which I had always hoped to return but in a different vein.

Added April, 2017. This thesis was written by me, with only a formal advisor, in 1959 and typed by my wife, Charlotte. I have examined it again, but find a good number of lines and formulas impossible to understand precisely. I would advise anyone who wishes to examine the material, at least the material on holomorphic semi-groups, to consult the book of Derek Robinson, in which there are several precise references to the thesis:

Elliptic operators and Lie groups. Clarendon Press, Oxford u. a. 1991, ISBN 0-19-853591-0.

Added January, 2018. The author and the editor are grateful to Derek Robinson for two very important contributions to this section of the site. The first was a list of the many misprints and/or small errors in the original text. The second is a clear, although brief, description of the context in which the thesis was written, or rather the context in which it is best to consider it. I add a few autobiographical remarks.

My graduate education was in two stages: a year at UBC, for a master's degree was followed by two years at Yale for a doctor's degree. In neither case was my formal supervisor anything more than formal. At UBC, the thesis was on commutative algebra. The topic I chose myself, but after a seminar with Prof. Douglas Murdoch from Northcott's brief text Ideal Theory. The thesis was undoubtedly not well-written and could be understood by no-one. Moreover, I myself discovered very soon after submission an error in the arguments. It was nevertheless decided to award me the degree, presumably so that I could profit from my successful application to Yale. This was generous of the mathematics faculty at UBC and, of course, decisive for my life.

At that time, functional analysis was the principal topic at Yale, the principal texts being the first volume of Dunford-Schwartz and the tome Functional Analysis and Semi-Groups by Hille-Phillips. Hille was trained as a classical analyst and their book, certainly thick, was, in spite of the title, a rich introduction to many topics of classical analysis. I spent a good deal of time with it, but also with many of the paper-back reprints available at the time. Analysis as such was represented by Felix Browder, whose course on partial differential equations was given extemporaneously, although seldom clearly, but none the less instructively. I spent a good deal of time transforming his repeated attempts, not always successful, to reach an end with the proof of this or that assertion into lecture notes that I could understand. There are traces of all this in my thesis, written, I think I can assert truthfully, quite independently. Once again, there was, oddly enough, no-one to understand it, but as I know from a conversation overheard in a stairway, Browder was quite firm in defending me and my thesis in the face of another faculty member, whose stated grounds for rejecting it were solely that no-one could read it.

Although, by the end of my time at Yale, I had already, as a result of listening to lectures by Steven Gaal on automorphic forms and the work of Selberg, begun to think of other things, I had become an analyst, even a functional analyst, and, I believe I can assert, have never stopped being one. This has many advantages, even in the context of more classical branches of our science, especially that of independence.

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Author's Comments: Although the principal purpose of this paper was to review how the Selberg trace formula is combined with character formulas to calculate the dimension of various spaces of automorphic forms. it is included as a paper on representation theory because the most influential observation in the paper was the description of a possible realization of the discrete series representations on spaces of \(L^2\)-cohomology.

Editorial comments: This was written in 1973. It first appeared as a preprint distributed by the Institute for Advanced Study, and was later (1988) published by the A.M.S. in Math. Surveys and Monographs 31.

Editorial comments: This has not been published before. It was written around 1977, just after A. Knapp and G. Zuckerman had announced their results on reducible unitary principal series, subsequently explained in a talk at the A.M.S. 1977 summer school in Corvallis (pp. 93-105 of the published proceedings of that conference.)

Author's comments: This paper, or some aspects of this paper, have been called into question in

https://mathoverflow.net/q/144419

A good many years have passed since I wrote the paper, so that I cannot easily verify the validity of the criticisms in the above reference. They appeared several years ago and are very likely valid, but anyone who is currently concerned with these matters can decide for himself. I thank Anthony Pulido and Yvan Saint-Aubin for drawing my attention to the paper and to the criticisms.

Editorial comments: The letter to Weil included a number of striking conjectures which eventually changed much of the direction of research in automorphic forms. Some of their consequences were explained in a graduate course given at Princeton in the spring of 1967, and then things were put in a somewhat wider context in a series of lectures at Yale later that Spring. These notes were previously published as the first of the Yale Mathematical Monographs.

Author's comments: This monograph was based on lectures given early in April of 1967 at Yale University, thus several months after the letter to Weil. Nonetheless it is reticent about the conjectures formulated in that letter. Results are formulated in terms of the dual group introduced there, which could for the groups of the lectures be introduced without any reference to the Galois group because only split groups are treated. There is, however, only the slightest of allusions to any generalization of class-field theory: the observations that what can be done for one reductive group should be done for all and that the identification of an automorphic \(L\)-function with an Artin \(L\)-function or with a Hasse-Weil \(L\)-function is tantamount to a reciprocity law. These two observations underline that functoriality arose in an attempt to find a nonabelian class-field theory under the influence of the view, which arose in the early sixties, that much of the theory of automorphic forms could and should be treated in the context of group representations. The major technical impulse was the need for a concisely defined general class of Euler products that included those arising from the theory of Eisenstein series.

The formula (6), which is established in sufficient generality to verify the convergence of the Euler products, is not established in general, although it is surmised that it is generally true. This was, indeed, proved a little later in complete generality and, so far as I know, quite independently by Ian MacDonald (Spherical functions on a group of \(p\)-adic type). I had heard of his result, even though his monograph was not yet available, by the time Problems in the theory of automorphic forms was written, so that I could simply invoke it. The formula now carries, quite rightly, his name.

The formula referred to as the formula of Gindikin-Karpelevich was, indeed, proved in general by them, but had first been discovered by Bhanu-Murty and proved by him for the special linear group over \(\mathbf R\) in мера Планшереля для фактор-пространства \(\mathrm{SL}(n,\mathbf R)/\mathrm{SO}(n,\mathbf R)\), ДАН 133 (1960).

Although the notes for the lectures were available as a preprint at the time they were delivered or shortly thereafter, the monograph did not appear until 1970. Apart from the addition of one or two footnotes and the correction of misprints and slips of the pen, there were no alterations.

Gelbart and Shahidi have written a useful survey of the theory of automorphic \(L\)-functions, Analytic properties of automorphic \(L\)-functions. I recommend it to the reader of Euler products.

Editorial comments:

• This was written in 1964 and distributed for many years in a famous purple mimeographed document by the Yale University Mathematics Department. It was later published in the Springer Lecture Notes series (volume 544).
• (12/2/2015, AVP) We would like to note that we have done our best to clarify several arguments, sometimes without having fully reconstructed them. Many thanks also to Allan Adler for a long list of corrections.
• (7/22/2016, AVP) We compared several charts and text from appendix III to the author's original handwritten notes and adjusted this online text to match them.

Editorial comments: This originally appeared as a supplement to an article by A. Borel and H. Jacquet in Automorphic forms, representations, and L-functions, Proceedings of Symposia in Pure Mathematics XXXIII, AMS, 1979.

Author's Comments: This letter was written from Berkeley.

Editorial comments: The letter to Weil that saw the birth of the \(L\)-group was written in January, 1967. Somewhat later that same year, Roger Godement asked Langlands to comment on the Ph. D. thesis of Hervé Jacquet. His reply included a number of conjectures on Whittaker functions for both real and \(p\)-adic reductive groups. These were later to be proven, first in the \(p\)-adic case by Shintani for \(\mathrm{GL}_n\) and Casselman Shalika in general, and much later in the real case by a longer succession of people.

Author's comments: This letter, a report on Jacquet's thesis, is undated, but a letter from Godement dated May 12th, 1967 asks that the report be submitted before the end of May. I assume it was sent from Princeton so as to arrive in Paris before the date requested.

The notation may cause the reader some difficulties. Some symbols, for example \(\chi\), have meanings that change (sometimes explicitly but sometimes only implicitly) in the course of the letter. There is a particularly dangerous lapse in regard to \(\xi\). Other symbols, sometimes the same, are employed in ways that have become uncommon. The symbol \(\pi\) appears, for example, as a representation of a compact group. The notation \(\langle a, \alpha\rangle\) for the value of the multiplicative function \(\alpha\) at the group element \(a\) is particularly disconcerting.

References to pages either in Jacquet's thesis or in the handwritten letter have been allowed to stand.

The formula for Whittaker functions for unramified representations suggested in the letter was proved by Casselman and Shalika.

It appears from the Institute records that Godement visited Princeton early in March of 1967. It must have been then that I spoke to him. The lectures at Yale were given early in April of 1967 and appeared later as the monograph Euler Products (included just above).

Editorial comments: In January of 1967, while he was at Princeton University, Langlands wrote a letter of 17 hand-written pages to Andre Weil outlining what quickly became known as `the Langlands conjectures'. This letter even today is worth reading carefully, although its notation is by present standards somewhat clumsy. It was in this letter that what later became known as the `\(L\)-group' first made its appearance, like Gargantua, surprisingly mature. Because of its historic importance, we give here two versions of this letter, as well as a pair of supplementary notes accompanying it. A typed copy of this letter, made at Weil's request for easier reading, circulated widely among specialists in the late sixties and seventies. The covering note from Harish-Chandra has been helpful in establishing a date for the letter, which is itself undated.

In reply to a question asked by many: there was no written reply from Weil.

Author's comments: The letter to Weil is undated. However, thanks to David Lieberman, I was able to discover that Chern's talk in the IDA Mathematics Colloquium was held on January 6, 1967. Thus the letter was written between then and the date January 16 that appears in the note of Harish-Chandra.

In order to make it easier for Weil to read, the handwritten note was typed some days later. The four footnotes were then added and one or two phrases were modified for the sake of clarity. These modifications are incorporated into the present version. Otherwise the letter has been allowed to stand as it was. Even unfortunate grammatical errors have not been corrected.

The emphasis on explicit, concrete reciprocity laws may surprise the reader. The note A little bit of number theory will clarify what I had in mind.

Author's comments: This note Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele was written as commentary to accompany the original letter in a collection of documents on reciprocity laws and algebraic number theory, to appear shortly.

There is a curious ambiguity in the fifth section regarding the location of my office in the old Fine Hall and of a small seminar room. I describe them both as being on the right of the principal entrance, but for my office it is to the right on entering the building, for the seminar room to the right on leaving it. Since I observed this unconscious aspect of my relation to the two rooms only after the article had been published in a book edited by Della Dumbaugh and Joachim Schwermer, I prefer not to make any changes in the article itself.

(July 5, 2015) I add the following letter from James Milne, correcting a careless and incorrect attribution of mine. Unfortunately, it is again too late to correct the article itself, at least as published.

Dear Langlands,

In your article in the Dumbaugh/Schwermer volume, you again credit Borovoi with the proof of Shimura's conjecture. In fact your sentence (p. 205), "Borovois endgültige allgemeine Konstruktion aller Shimuravarietäten war auch von diesem Bericht beeinflusst," makes no sense at all. Borovoi attempted (unsuccessfully) to prove Shimura's conjecture directly. I was certainly the one to prove it via the conjecture in your Bericht.

As I write in an article recently posted on my website (The Riemann Hypothesis....), "Concerning Langlands's conjugacy conjecture itself, this was proved in the following way. For those Shimura varieties with the property that each connected component can be described by the moduli of abelian varieties, Shimura's conjecture was proved in many cases by Shimura and his students and in general by Deligne. To obtain a proof for a general Shimura variety, Piatetski-Shapiro suggested embedding the Shimura variety in a larger Shimura variety that contains many Shimura subvarieties of type $A_{1}$. After Borovoi had unsuccessfully tried to use Piatetski-Shapiro's idea to prove Shimura's conjecture directly, the author used it to prove Langlands's conjugation conjecture, which has Shimura's conjecture as a consequence. No direct proof of Shimura's conjecture is known. Regards,"

Editorial comments: The conjectures made in the 1967 letter to Weil were explained here more fully. This appeared originally as a Yale University preprint, later in the published proceedings of a conference in Washington, D.C. Lectures in modern analysis and applications III, Lecture Notes in Mathematics 170, Springer-Verlag, 1970. The lecture is dedicated to Salomon Bochner. 

Author's comments: The lecture in Washington, D.C. on which these notes were based (they were presumably written shortly thereafter) was, I surmise, delivered sometime in 1969, thus more than two years after the letter to Weil. They were the first published account of the conjectures made in the letter. In the meantime, a certain amount of evidence had accumulated.

The letter had been written, I believe, only a few days or at most weeks after the discoveries it describes. They were not mature. The local implications appear not to have been formulated, and the emphasis is not on the reciprocity laws as a means to establish the analytic continuation of Artin \(L\)-functions but on concrete, elementary laws, for which groups other than \(\mathrm{GL}(n)\) are important because they admit anisotropic \(\mathbf R\)-forms. The coefficients of automorphic \(L\)-functions attached to groups anisotropic over \(\mathbf R\) can be interpreted in an elementary way as in A little bit of number theory. In addition, I was not aware of Weil's paper on the Hecke theory or of the Taniyama conjecture. Indeed, not being a number theorist by training (and perhaps not even by inclination) I was well informed neither about Hasse-Weil \(L\)-functions nor about elliptic curves.

After the letter had been transmitted, I learned from Weil himself both about his paper and about the Weil group. This is implicit in the lecture and accounts in part for its greater maturity. First of all, encouraged by Weil's re-examination of the Hecke theory, Jacquet and I had developed a theory for \(\mathrm{GL}(2)\) with some claims to completeness both locally and globally, although at both levels the major questions about reciprocity remained unanswered. With the local theory for \(\mathrm{GL}(2)\) came \(\epsilon\)-factors and the correspondence of the letter then required that such factors also exist for Artin \(L\)-functions. One achievement of a year spent in Turkey was the proof that these \(\epsilon\)-factors exist. One achievement of the following year, accomplished in collaboration with Jacquet, was a complete proof of the correspondence between automorphic forms on \(\mathrm{GL}(2)\) and on quaternion algebras. This correspondence had, of course, already appeared classically. Our achievement was, I believe, local precision, in particular the understanding that there were local phenomena of importance, and generality.

Although specific attention is drawn in the lecture to the case that \(G'\) is trivial and the automorphic \(L\)-functions attached to it therefore nothing but Artin \(L\)-functions, it is not at all stressed that functoriality entails the analytic continuation of the Artin \(L\)-functions. It is of course evident, but I had not yet learnt the advantages of underlining the obvious. The other examples of functoriality may or may not appear well chosen to a number theorist in 1998. In 1967, however, it was rather agreeable to see the recently established analytic theory of Eisenstein series fitting so comfortably into a conjectural framework with much deeper arithmetical implications.

The question about elliptic curves appearing toward the end of §7 is nothing but a supplement to the conjecture of Taniyama-Shimura-Weil, but a useful one: a precise local form of the conjecture, that is now available, thanks to Carayol and earlier authors, whenever the conjecture itself is. At the time, what was most fascinating was, as mentioned in the comments on the letter to Serre, the relation between the special representation and the \(\ell\)-adic representations attached to elliptic curves with nonintegral \(j\)-invariant.

The observation about \(L\)-functions and Ramanujan's conjecture has, I believe, proved useful.

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Editorial comments: This first appeared in mimeographed notes dated 1968 available from the Mathematics Department of Yale University. It was reprinted in the issue of the Pacific Journal of Mathematics dedicated to the memory of Olga Taussky-Todd (volume 181 (1997), pp. 231-250).

Editorial comments: 

• This is a short note written to illustrate some examples of how the conjectures worked out in very explicit examples.
• (3/16/2017, AVP) At the author's suggestion, Anthony Pulido worked through theorem 2 for $p = 3 \equiv 3 \pmod{4}$ and $p = 5 \equiv 1 \pmod{4}$. This is the note titled Examples. This exercise led to writing a short Haskell program, lbnt.hs, which verifies theorem 2 for the first 100 odd primes.

Author's comments: I am not sure exactly when this text was written. Internal evidence and memory together suggest that it was early in 1973. The internal evidence cannot be interpreted literally, as I was unlikely to be sure even in 1973 exactly when the letter to Weil was written.

The examples are of the type I had in mind when writing that letter. I had not, however, at that time formulated any precise statements. Indeed, not being aware of the Shimura-Taniyama conjecture and not having any more precise concept of what is now known as the Jacquet-Langlands correspondence than that implicit in the letter, I was in no position to provide the examples of the present text, some of which exploit results that had become available in the intervening years. The formulas are as in the original text. I did not repeat the calculations that lead to them.

I have never found anyone else who found the type of theorem provided by the examples persuasive, but, apart from the quadratic reciprocity law over the rationals, explicit reciprocity laws have never had a wide appeal, neither the higher reciprocity laws over cyclotomic fields nor simple reciprocity laws over other number fields (Dedekind: Über die Anzahl der Idealklassen in reinen kubischen Zahlkörpern).

The conjecture referred to in the text as the Weil conjecture is now usually referred to as the Shimura-Taniyama conjecture.

Editorial comments: Langlands spent 1967-68 visiting in Ankara, Turkey, and while there wrote this letter to Serre. In it occurs for the first time the question of how to account for `special' representations of the Galois group, such as at primes where an elliptic curve has unstable bad reduction, corresponding to special representations of \(\mathrm{GL}_2\). This correspondence was later expanded to the Deligne-Langlands conjecture, proven eventually by Kazhdan and Lusztig.

Author's comments: This letter is a response to a question of Serre about the gamma-factors appearing in the functional equations of automorphic \(\ell\)-functions. Fortunately Serre's letter to me was accompanied by several reprints, among them apparently the paper Groupes de Lie \(\ell\)-adiqes attachées aux courbes elliptiques that appeared in the volume Les tendances géométriques en algèbre et théorie des nombres.

Although the letter promised in the last line was never written, it is clear what I had in mind. Sometime soon after writing the letter to Weil, perhaps even at the time of writing, I was puzzled by the role of the special representations. The solution of the puzzle was immediately apparent on reading Serre's paper which treated the \(\ell\)-adic representations associated to elliptic curves whose \(j\)-invariant was not integral in the pertinent local field. The special representations of \(\mathrm{GL}(2)\) corresponded to these \(\ell\)-adic representations. The connection between non-semisimple \(\ell\)-adic representations and various kinds of special representations is now generally accepted. The theorem of Kazhdan-Lusztig is a striking example.

Editorial comments:

1. The collection of problems compiled by Jean Dieudonné referred to in the letter appeared in Problems of present day mathematics, Proceedings of symposia in pure mathematics, XXVIII, 1976, pp. 35-79. The symposium was held in Dekalb, Illinois at North Illinois University, May 1974.
2. (AVP, 4/3/2019: An appendix has been added.)

Author's comments: Comments on this letter as on many of the others about functoriality and related matters are at the moment (2009) necessarily provisional, for I hope we shall soon arrive at a stage where we can weigh with more confidence the significance of a number of specific contributions to the theory of automorphic forms and its connections to Galois theory.

Taking, for the purposes of these personal comments, as the subject's beginning the letter to Weil in January, 1967, there are up to the time of this letter three periods or phases, the third emerging, in part, from the second. In the second two quite distinct currents appeared, sometimes merged, sometimes remaining quite separate. During the first, initial period the local consequences or analogues of the global notions were formulated, and in the case of \(\mathrm{GL}(2)\) collated, developed, and compared with the available material. Moreover, some significant, specific features of general and long-term importance were discovered: the existence of the \(\epsilon\)-factor; the role of the special representation, at first for \(\mathrm{GL}(2)\), but implicitly in general (see the 1967 letter to Serre in Group 5) although there was no urgent need for general formulations; a first use of the trace formula to establish a significant case of functoriality, the correspondence, local and global, between representations of \(\mathrm{GL}(2)\) and representations of its inner forms.

The second period began, for me, with two matters. The first was the introduction, as recorded in the letters to Lang of December, 1970, of the Galois representations of general Shimura varieties as an enlargement of the conceptual frame, partially implicit, partially explicit in the letter to Weil of January, 1967. The letters were written a couple of months before the Bourbaki talk of Deligne in which he gave a uniform reformulation of Shimura's theory that was tremendously valuable to me and, indeed, to every student of Shimura's papers. If that reformulation had been available the letters might have been briefer, but it was perhaps more amusing to discover the existence of the necessary representations of the \(L\)-group experimentally. It was also perhaps instructive to follow, at least at first, the longer road traversed by Shimura. I began fairly soon to use the designation Shimura variety and introduced it formally, I believe for the first time, in the 1974 paper Some contemporary problems with origins in the Jugendtraum.

The second matter was what came to be called endoscopy, a term created by Avner Ash in response to an appeal of Diana Shelstad. As is apparent from the discussion in the first letter to Lang in which the Blattner conjecture is invoked, the ideas of that letter led immediately to complications caused by the unequal multiplicities of what I referred to as \(L\)-indistinguishable representations, a term improved on by the that of \(L\)-packet introduced, I believe, by Borel later. The problems raised could be attacked in two different contexts: real groups, where Harish-Chandra's theory was available; and \(\mathrm{SL}(2)\) where they were of an elementary nature. Shelstad, in her thesis and later, clarified the real theory completely. I undertook, jointly with Labesse to whom I had described the problems, the study of the group \(\mathrm{SL}(2)\), much more elementary but also very illuminating. Endoscopy is now, thanks to Ngo Bao Chau, Laumon, Kottwitz and others a subject of independent interest and importance, although still central to the theory of automorphic forms. I tried to underline the significance of Kottwitz's contributions to endoscopy and Shimura varieties in the comments on the paper Representation theory and arithmetic that appears in this section.

The third phase began for me with the paper on base change Base change for \(\mathrm{GL}(2)\), but it is well to remember that base change had an earlier history that is mentioned already in this letter, namely the work of Doi and Naganuma, as well as that of Jacquet, on what would now be called quadratic base change for \(\mathrm{GL}(2)\). The subsequent work by Saito and by Shintani, who were influenced by the results for quadratic base change had of course a decisive and direct influence on me. Base change for \(\mathrm{GL}(2)\) together with later work by Arthur and Clozel for \(\mathrm{GL}(n)\) has played a major role in the fusion of the general theory of automorphic forms on one hand and the study of \(\ell\)-adic Galois representations on the other. The decisive factor was the role played by base change in the proof of some previously inaccessible cases of the Artin conjecture that were invoked by Wiles in his proof of Fermat's Last Theorem. Later, similar ideas were used by Richard Taylor and collaborators in the proof of the Sato-Tate conjecture, but as I have already intimated elsewhere -- as indeed the present letter to Deligne foreshadows -- I do not believe this is where long-term importance of Wiles's ideas lies. Indeed, in this letter in the first of the "two vague problems", I already describe why and how the Sato-Tate conjecture, in the particular form in which it first appeared and in a general form, is to be regarded as an immediate consequence of functoriality. The vagueness is considerably reduced if functoriality is regarded as including "Arthur's conjectures", appearing in the two papers Unipotent automorphic representations: conjectures (1989) and Unipotent automorphic representations: global motivation (1990).

The uncertainty, on the one hand, in the present status of functoriality, or rather of its status in the next few years, and, on the other, of the relation of the theory of automorphic forms to algebraic geometry -- in the sense of the intimations of Grothendieck -- and to Galois representations makes any precise speculations about the development of these subjects rash. There are more grounds for confidence in general expectations than there were in 1974 but circumspection in specifics is still wise. My own hope is that we shall soon be on the road to a proof of functoriality and by methods of (nonabelian) harmonic analysis. So the largest unknown may become after not too many years the relation between arithmetic (thus motives!) and the analytic theory.

The questions in the letter were callow but not premature. The question for function fields was not much to the point. The work of Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, clarifies considerably what to expect, but I do not believe much has yet been done beyond the group \(\mathrm{GL}(n)\). Whatever is true, the question formulated for function fields towards the end of this letter does not seem to be now particularly perceptive. Whatever is valid along the lines of the not adroitly formulated question would presumably be a consequence of Lafforgue's work and functoriality.

For number fields, the "obvious guess" at the end of the letter is still very much just that. The question of showing that the automorphic \(\pi\) over a number field that correspond to motives are characterized by their infinite components remains as before and not much has been done, beyond some cases of Artin's conjecture for two-dimensional representations and, of course, the theorem of Deligne-Serre, to clarify it. It remains a central question.

It is not, however, the primary problem. This is to show that every motive over a number field corresponds to an automorphic form. I have since reflected in various ways on the question, first in the paper Automorphic representations, motives, and Shimura varieties. ein Märchen, in which the Taniyama group was introduced. In particular there was shown to be a canonical homomorphism from the Weil group to this algebraic group, or rather to its finite-dimensional quotients. The Weil group can be said, as a consequence of the paper Representations of abelian algebraic groups to be the "Galois group" for automorphic forms on tori. Thus a complex homomorphism of the Taniyama group to the \(L\)-group of a torus defines on composing it with the homomorphism from the Weil group to the Taniyama an automorphic form on the torus. Although the remarks in the introduction to the Märchen suggest that I was aware, in some sense, while writing it that the Taniyama group was related to motives of CM-type, I do not think my ideas were very precise. A precise theorem was formulated and proved by Deligne. (See the book Hodge cycles, motives, and Shimura varieties with papers by Deligne, Milne, Ogus, and Shih.) Using a provisional, but acceptable and, so far as I can see, logically impeccable, notion of motive, Deligne proves that the Taniyama group is isomorphic to the "Galois group" for motives of potentially CM-type. Any theorem for motives in general will have to be compatible with this result.

In the Märchen, I was too strongly influenced by the categorical constructions found for example in the book Catégories Tannakiennes of Saavedra Rivano, a theory explained again, with mistakes corrected, by Deligne and Milne in the collection of papers mentioned. For example, I introduced, for automorphic forms on \(\mathrm{GL}(n)\), direct sums, which exist thanks to the theory of Eisenstein series, and products, whose existence is, so far as I know, still only partially established. That was fine, but I was also attached to the notion of a fiber functor for automorphic representations. I am now inclined to suppose that this was misguided. As I explained in the article Reflexions on receiving the Shaw Prize in section 12, it may be simplest, once functoriality has been established along the lines of Beyond endoscopy, if that is possible, to construct the "automorphic Galois group" by hand by patching together groups corresponding to the "thick representations" of Reflexions. This would be a large group, involving inverse limits of reductive groups, but in fact any concrete meaning it had would undoubtedly be at a finite level. The groups would only be defined over \( \mathbf C\).

I observe in passing that although the adjective "thick", as in "thick description" has met with considerable success among historians and social scientists, mathematicians seem reluctant to employ it. An alternative would be "hadronic", taken from the Greek, well-known from elementary particle theory, and meaning exactly "thick".

The Märchen was written in the late 1970's when I was still relatively young and impressionable. Having now lived for some decades with various ideas that were new to me then and having had many more years to reflect on the theory of automorphic representations and related matters, I am now inclined to think that although Tannakian categories may ultimately be the appropriate tool to describe the basic objects of algebraic geometry, automorphic representations have a different structure, best expressed by functoriality, in which of course statements formulated in terms of the finite-dimensional representations of the \(L\)-group are central. Among the less well-known, but still striking, statements of this kind are predictions of the multiplicity with which a given automorphic representation appears in the space of functions on \(L^2(G(F)\backslash G(\mathbf A_F)\). Some examples to which I draw the reader's attention have been found by Song Wang, Dimension data and local versus global conjugacy in reductive groups.

One expects the global correspondence to define and to be defined by a homomorphism of the "automorphic Galois group" \(\mathcal G_{\mathfrak A}\) onto the "motivic Galois group" \(\mathcal G_{\mathfrak M}\), but only as a group over \( \mathbf C\). Of course we are still a long distance from the global correspondence, as the contributions to the footnote to the review in section 12 of Hida's book \(p\)-adic automorphic forms on Shimura varieties make clear. Nevertheless the possibility of such a correspondence and such a homomorphism influence, often in a very concrete way, the thinking of many mathematicians. The existence of the Tate motives defined by projective spaces means that there has to be a homomorphism of the "motivic Galois group" to \(\mathrm{GL}(1)\); the existence of a "degree" or "weight" for motives, just as there is for cohomology, would mean that there was a homomorphism of \(\mathrm{GL}(1)\) to the motivic Galois group. Something similar is available for automorphic representations: the automorphic representations of \(\mathrm{GL}(1)\), especially those defined by powers of the norm, correspond to the Tate motives. Thus there may be something like a Tate-twisting. There may also be an analogue of the weight, or, more generally of one-dimensional motives. For automorphic forms these are best thought of as characters of the group of idele classes over the ground field, so that we expect that a homomorphism of this group, or some modification of it, perhaps as an inverse limit, into the "automorphic Galois group" \(\mathcal G_{\mathfrak A}\) exists. These two homomorphisms, of an abelian group \(\mathcal T_{\mathfrak A}\) into \(\mathcal G_{\mathfrak A}\) and of \(\mathcal G_{\mathfrak A}\) to \(\mathcal T_{\mathfrak A}\) allow the introduction of various twistings of any homomorphism of the automorphic Galois group \(\mathcal G_{\mathfrak A}\) to the motivic Galois group \(\mathcal G_{\mathfrak M}\).

That the homomorphism of the "automorphic Galois group" to the "motivic Galois group" can only be defined over a field large enough for the definition of both, thus, perhaps, only over \( \mathbf C\) troubles some specialists. As I observed in the comments on the letter to A. Gee, some are also troubled by the circumstance that in the theory of Shimura varieties the automorphic representation that defines the cohomology groups from which the Galois representation is constructed is not the automorphic representation to which it corresponds by the (Langlands) correspondence -- at least not if one uses the local correspondence introduced by me, a correspondence with, in my view, much to be said for it. There are even those who would like to modify the definition, by isolating a collection of automorphic representations that define the image of the motives and that, in contrast to a larger collection of automorphic representations, permit, perhaps after a twisting of the kind just described, the introduction of an "automorphic Galois group" over, say, the field of algebraic numbers. L. Clozel has, in the paper Motifs et formes automorphes examined the question carefully.

Only motives over number fields can correspond to automorphic representations. Nevertheless we are certainly hoping to establish sooner or later a theory for general motives, say over \( \mathbf C\). One is very quickly led to ask, what will be the function of a richer theory over \(\overline{\mathbf Q}\), in which the relation with automorphic representations appears, to the theory over \( \mathbf C\). I do not know of any papers in which this question has been broached, say in relation to the Hodge conjecture.

The letter does not indulge in any real mathematics and, when it does, the explanations are obscure to the point of incomprehensibility. My explanation of the expected local correspondence, in the section labelled \(\ell\)-adic motivation is certainly somewhat embarrassing! I was trying to express whatever understanding I had of the \( \ell\)-adic Galois representations associated to algebraic varieties -- not knowledge, which I lacked almost completely. An informed, current survey can be found in the first section of R. Taylor's article Galois representations. in Annales de la Faculte des Sciences de Toulouse 13 (2004), 73-119. The embarrassing discussion in the letter is, however, purely local, a matter of explaining how appropriate \( \ell\)-adic representations correspond to a pair \((\psi,Y)\).

The loose phrase "such that \(\psi(\sigma)\) is semi-simple if \(\sigma\) projects to the Frobenius" is inappropriate and (surely?) not what was intended. Its presence in the letter is presumably a result of haste and carelessness, not to speak of some real ignorance. I have left it only for the sake of historical veracity. The assumption should be, and was implicitly, that \(m(\sigma)\) is semi-simple. I neglected, moreover, to state that the residual characteristic of the local field \(F\) is supposed prime to \(\ell\). Anyhow, the purpose is clearly to explain to myself---not to the recipient---how an \(\ell\)-adic representation leads to a pair \(\phi, Y\). The argument is not only brief and hurried, but also fundamentally incomplete, although not fundamentally incorrect. It is clear from the discussion of de Rham representations in Taylor's report that in 1974, at least, I was in to position to explain adequately what I had in mind. There is, none the less, something to be said for the use of the Jacobson-Morosow theorem, implicitly invoked by the phrase "as you know", for it leads to the replacement of the unipotent parameter by an imbedding of \(\mathrm{SL}(2)\) in the \(L\)-group.

I was, by the way, surprisingly -- and inappropriately -- optimistic in the letter about the local conjectures, although serious inroads were made within twenty-five years, so that the time scale of five to ten years was not completely out of order. My suspicion now is that the decisive insights, namely for all groups and complete from the point of view of harmonic analysis, will appear for the global and local problems simultaneously

I add finally that the list of problems suggested by Dieudonné did appear, but so far as I can tell this letter did not influence it in any way.

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Author's comments: These two letters were, in fact, electronic messages and were written very recently, in December, 2013. They were inspired by a conversation with Julia Mueller and Michael Volpato, in which I tried to explain to them the origins of the general definition of automorphic \(L\)-functions. I had difficulty, not because I had forgotten that these were in the theory of Eisenstein series, but because I had not recognized in 1966, when I discovered after many months of unsuccessful search a promising definition of automorphic \(L\)-function, what a fortunate, although, and this needs to be stressed, unforeseen by me, or for that matter anyone else, blessing it was that it lay in the theory of Eisenstein series. This, I try to explain in the messages. There is something to add to them.

I begin by suggesting that the modern analytic theory of automorphic forms has its origins largely in the work of Hecke and Siegel, and that one decisive step was the extension of the reduction theory to general reductive groups by Borel and Harish-Chandra in Arithmetic subgroups of algebraic groups Ann. Math. (1962). The general \(L^2\)-theory, thus the theory for general reductive groups, especially the notion of cusp form, seems to have had its origins in two papers, one by Godement (in Sém. H. Cartan, 1957/58, Ex-. 8, pp. 8-10) and the other by Harish-Chandra (Automorphic forms on a semisimple Lie group, 1959). The central analytic problem of establishing the spectral theory, continuous and discrete was, on the other hand, broached first, but only in the case of \(\mathrm{SL}(2)\) by Maaß and, through his influence but with stronger results, by Selberg.

Some authors refer to the philosophy of cusp forms introduced by Harish-Chandra but only later, when he was well on his way to the general Plancherel formula, thus some time after he had constructed the discrete series in general and, in particular, after he had the material of the papers Discrete series for semisimple Lie groups, I, II in hand and about the time, 1966/67, that he was lecturing on the theory of Eisenstein series for cusp forms. The papers had appeared in 1965/66; my notes on Eisenstein series earlier. Although I cannot be sure, I believe he undertook the lectures in response, a very generous response, to doubts expressed by various mathematicians about the validity of my paper, which I had delayed publishing in the hope of improving the exposition. So far as I know, he was not moved to express the similarity of the local and global spectral theories as a philosophy until he understood the technical details of both well. It has certainly been of great importance that the local theory, thus the representation theory, and the theory of automorphic representations have developed in tandem since the 1960's and that they have many features in common, but when examined closely, there is also a striking difference: the presence of non-tempered representations in the global theory. The ``philosophy'' as such was appealing when first introduced, and remains so, especially as an expression of Harish-Chandra's personality, but it is not of much technical significance and, although certainly valid for nonarchimedean fields, has not been the key to treating them.

I, myself, came to the modern theory of automorphic forms by reading first Selberg, then Siegel, and then the earlier papers of Harish-Chandra, and for the theory of Eisenstein series in one variable was at first largely concerned with methods with their direct source in papers of Siegel. In particular, I had not noticed the papers of Godement and Harish-Chandra to which I just referred or the importance of the notion of cusp forms for the spectral theory on the quotient \(\Gamma\backslash G\). So the lecture of Gelfand at the 1962 ICM was a revelation to me. It suggested, in recollection at least, almost immediately, a possible way of establishing the analytic continuation of Eisenstein series in general. There would be three steps: (i) the continuation for the rank-one series associated to cusp forms; (ii) the continuation for the series in several variables associated to cusp forms; (iii) the continuation for series in any number of variables associated to forms that are not cuspidal. The first step would be accomplished by the method used by Selberg for discrete subgroups of \(\mathrm{SL}(2)\) or more generally for discrete subgroups of reductive groups of rank one. For the second step, as I had discovered as a graduate student at Yale, standard methods from the theory of functions in several variables that, by a stroke of luck, I came across in my reading for a seminar of Browder/Kakutani on functions of several complex variables, which ultimately did not take place, allow one to pass from groups of rank one to groups of higher rank. I, of course, was initially thinking in very concrete terms. The argument is explained in a somewhat more sophisticated number-theoretical context in the first appendix to the Springer notes (544) on Eisenstein series, included in Part 3 of the present collection. The appendix itself was written I believe during my first one or two years at Princeton and reflects the algebraic number theory that I learned after arriving there. The method was presumably also used by Selberg, but I never discussed this with him.

It was the third step, the construction of the full spectrum from residues of Eisenstein series, closely related to what are now referred to as Arthur packets, that caused me the most difficulty. So far as I know, Gelfand, for example, was not aware of its necessity. Although I began the project that led to the text in the Springer notes not long after reading the Gelfand lecture, thus in the academic year 1962/63, which I had spent at the IAS, I did not finish it until the spring of 1964. I was exhausted. I found the correct induction assumptions for the proof only after many false tries. The next academic year I spent at Berkeley. As I confess in the essay Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele in Part 3, that year was mathematically disappointing. The following academic year, 1965/66 was worse. As I recall in the same essay, I was trying, with no success, to find on the one hand the correct generalization of the Hecke \(L\)-functions to the theory of automorphic forms on general reductive groups and on the other hand some form of a nonabelian class field theory that I could accept as the correct form. I was not succeeding, although about the same time or perhaps a little later, Tamagawa and Godement were beginning to study forms of the standard \(L\)-function associated to \(\mathrm{GL}(n)\), a function that can be treated with the help of the Poisson summation formula, thus by classical methods. Their proposals were basically correct, but Tamagawa had confined himself, presumably for technical reasons, to the multiplicative groups of division algebras and Godement, perhaps also for technical reasons, perhaps because of a taste for functional analysis, wanted to introduce an auxiliary parameter.

I, myself, was searching for a general notion and had despaired. By the fall of 1966, I was prepared to abandon mathematics and to turn to some other life, a first step being a year or two in Turkey with my wife and children, as a prelude to an existence whose exact form was undetermined. I, who had never been anywhere outside of English-speaking North America, returned to the study of Russian and began the study of Turkish, frivolously daydreaming of a trip to Turkey --- with wife and four small children --- through the Balkans or through the Caucasus. In the end we arrived in Ankara by a more banal route. Even with the Russian and Turkish, I had time to spare and began, as an idle amusement, to calculate the constant term of the Eisenstein series for various rank-one groups. I had, curiously enough, never done this before. I discovered rather quickly a regularity of which I had been unaware. It was described in the lectures delivered at Yale some months later and included in Part 3 of this collection. The constant term, or rather the second part of the constant term, the part that expresses the functional equation was there denoted \(M(s)\) and given at the very end of §5 as a product that I write here as

\begin{equation} \prod_{i=1}^r \frac{\xi_i(a_is)}{\xi_i(a_is+1)} \tag{1} \end{equation} \(r\) being a small integer, often \(1\), and \(a_i\) being a positive number. Suppose, in order not to confuse the explanations, that \(r\) is \(1\). The issues arising in the general case are treated in the references. It is the relation expressed by (1) that suggests and allows the passage from the theory of Eisenstein series to a general notion of automorphic \(L\)-function that can accommodate not only a non-abelian generalization of class-field theory but also, as it turned out, both functoriality and reciprocity. It was the key to the suggestions in the Weil letter.

The Yale notes were written a long time ago and were hardly exemplary expositions. I have no desire at the moment to recall the details or to improve their presentation --- the reader is encouraged to consult the writings of Shahidi, for example the book Eisenstein series and automorphic \(L\)-functions --- but there are a number of points to which I would like to draw attention, and it is more convenient to refer to my own notes. I repeat, first of all, that (1) refers to rank-one parabolic subgroups, thus to Eisenstein series arising from maximal proper parabolic subgroups, so that it does not require the second or the third steps and is analytically at the level of Selberg's original arguments. Algebraically, however, one has to be thinking at the level of the theory of reductive groups. If I had not searched assiduously for a general form of the theorems of Hecke and of the founders of class field theory, or had not been familiar with various principles of nonabelian harmonic analysis as it had been developed by Harish-Chandra, in particular with the theory of spherical functions, I might have failed to recognize the importance or value of (1). At all events, between the discovery of (1) and the Yale lectures, I had no time for my linguistic undertakings, which were temporarily set aside, and by the summer of 1967, after the letter to Weil and the Yale lectures, I was again exhausted and was content to rest before departing with my family for Ankara.

Although the letter to Weil is not mentioned in those lectures, the critical discoveries that led quickly to the conjectures in that letter are described. They are all related to the formula (1). Before reviewing them, I recall that in the early sixties, thus before the letter to Weil and before the Yale lectures, a number of mathematicians had created a structure theory for groups over \(p\)-adic fields. These were described in the very successful Boulder conference organized by Borel and Mostow. Some of the representation theory for real groups, created by Harish-Chandra, by the Russian school and, to a lesser extent, by others, had been extended to groups over \(p\)-adic fields, in particular the theory of spherical functions to which among others, Satake had contributed. This was perhaps not very difficult, but it was certainly necessary. Satake describes the contribution of his paper, Theory of Spherical Functions on reductive algebraic groups over \(\frak p\)-adic fields, Publ. Math. IHES, 18 (1963), in the following words, ``Then our main theorem asserts that \(\mathcal L(G,U)\) is isomorphic to the algebra of all \(W\)-invariant polynomials functions on \(\mathrm{Hom}(M,\mathbf C^*)\simeq\mathbf C^\nu,\dots\); thus \(\mathcal L(G,U)\) is an affine algebra of (algebraic) dimension \(\nu\) over \(\mathbf C)\).'' The algebra \(\mathcal L(G,U)\) is the algebra of spherical functions. This is a clear, precise statement of an indispensable theorem, or lemma, a lemma that is, however, suggested immediately by the analogous lemma for spherical functions over the real field. The step from it to what I have called the Frobenius-Hecke conjugacy class rather than the Satake parameter --- a term often used by others --- in order to emphasize the importance of the contributions of these two outstanding mathematicians to the theory of automorphic \(L\)-functions, is technically minute, but entails a fundamental conceptual change that arises directly from the expression (1). To take the step it is only necessary to be aware that the space of coweights of a reductive group \(G\) is the space generated by the weights of a second group, again a reductive group and naturally associated to \(G\). Without understanding the formula (1), there is, however, no reason for taking it. Even serious mathematicians, for example, Benedict Gross --- see his exposition, On the Satake isomorphism, in Galois representations in arithmetic algebraic geometry, London, Math. Soc. Lecture Notes, 254 --- fail to appreciate this. I tried in the two messages to Mueller and Volpato to explain how unexpected --- and perhaps undeserved --- it was that the Eisenstein series and their constant terms, once calculated, suggested the step and led to the introduction of the \(L\)-group and of the Frobenius-Hecke class into the theory of automorphic forms. At all events, the \(L\)-group and the Frobenius-Hecke conjugacy class do not appear in Satake's paper nor does the isomorphism of the Hecke algebra with the representation ring of \({}^LG\) as such, only with a ring to which it is isomorphic. There was no reason, in the context of the paper, that they should!

I have recalled the context in which they appeared. After having been introduced, in one way or another in the early 1960's to the papers of Siegel, Selberg, Hecke, and Harish-Chandra and to class-field theory and after having passed a free, but disappointing year 1964/65 in Berkeley, where I made an unsuccessful attempt to learn algebraic geometry and spent a good deal of time to not much purpose thinking about spherical functions in general and their relation with the classical hypergeometric functions, functions whose integral representations had intrigued me, I participated in the Boulder conference and learned, somewhat belatedly, to think in terms of reductive algebraic groups. So I had the background to reflect on the possibility not only of a non-abelian class field theory but also of attaching \(L\)-functions to automorphic forms or, better representations in general. My initial reflections were none the less, as already recalled, not successful.

Luckily I had not forgotten the problems and, when in the fall of 1966, as a pastime, I began to calculate the constant term of the Eisenstein series for rank-one groups, basically one group or one class of groups at a time, I noticed almost immediately expressions like formula (1) appearing. What is their value or significance? First of all, as they arise from Eisenstein series which are meromorphic in the whole \(s\)-plane, they themselves are meromorphic in the plane. Secondly, at least when \(r=1\), if the quotient \(\xi(as)/\xi(as+1)\) is meromorphic, then so is \(\xi(s)\). As observed, the cases in which \(r\neq 1\) can be treated by supplementary arguments. Moreover, \(\xi(s)\) is an Euler product and it is associated to the automorphic representation defining the Eisenstein series. Running through the pairs \((H,G)\) for which \(H\) was the Levi factor of a maximal proper parabolic subgroup of \(G\), I found that for all but three classes of simple \(H\) (at first I focussed on split groups) one could construct Euler products with meromorphic continuation. Clearly at this initial stage, after my earlier lack of success, meromorphic continuation was a triumph! Had I really arrived at the series for which I had been searching for so long? Was there a direct definition of these Euler products that depended only on the automorphic representation not on the construction of the Eisenstein series? I made a list of the groups \(H\), many of them classical groups, with familiar definitions and of the degree of the Euler products that arose from the several ways that \(H\) could appear as a Levi factor. The lists, published in the notes from the Yale lectures, revealed that the degrees were degrees not of representations \(\rho\) of \(H\), but of the group \({}^LH\) dual to it in the classification of semisimple groups or, more generally, reductive groups. The local factors were moreover immediately seen to be \(L(s,\pi,\rho)=1/\det(1-\rho(\gamma_p)/p^s)\), where \(\gamma\) was the parameter associated to the prime \(p\) and the unramified representation \(\pi_p\) of the local group \(H(\mathbf Q_p)\) (I was working over \(\mathbf Q\).) when one interprets the Satake isomorphism in terms of this duality. Here \(\rho\) was a representation of the group of \({}^LH\) that depended on the pair \((H,G)\). That this was not the way the isomorphism was interpreted by Satake is less important than the miracle of the appearance of these Euler products in the theory of Eisenstein series, although in a manner not immediately evident. Once their appearance was discovered, not only the resemblance to the Artin \(L\)-functions and the possibility of a non-abelian class field theory but also the generalizations of it envisaged in the Weil letter were evident. The miracle remains and it is this that I was trying to explain in the accompanying messages to Mueller/Volpato.

I add, as a supplementary remark, that I had calculated the factors of the Euler product for one pair \((H,G)\) at a time, and it was by no means a foregone conclusion that their degree would be given by the dimension of a representation \(\rho\) of \({}^LH\). Indeed, initially this was an empirical observation that led to the formal introduction of the group \({}^LH\). It was Tits who pointed out when I gave, during the lectures, the complete list of pairs \((H,G)\) and the associated \(\rho\) that a case-by-case verification was unnecessary, that \(\rho\) was the representation of \({}^LH\) on the unipotent radical of the dual pair \({}^L\mathfrak h \subset {}^L\mathfrak g\).

As a second comment, I also add that a question of convergence arose for the Euler products defining \(L(s,\pi,\rho)\) for an automorphic \(\pi\). To deal with it, it was necessary to give a formula for the spherical function associated to the local \(\pi_p(g)\), but the formula did not need to be everywhere valid. The formula with the necessary domain of validity was proven in my Yale notes. The same formula was later discovered independently and in a different context by Ian Macdonald, was proved by him for all \(g\), and is known as Macdonald's formula.

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Author's comments: Some surprise has been expressed that the notes of Jacquet-Langlands have been placed in the same section as the notes on the \(\epsilon\)-factor. There is a good reason for this. Although the notion of functoriality had been introduced in the original letter to Weil, there were few arguments apart from aesthetic ones to justify it. So it was urgent to make a more cogent case. One tool lay at hand, the Hecke theory, in its original form and in the more precise form created by Weil.

The theory as developed in terms of representation theory, both local and global, suggested the existence of the \(\epsilon\)-factors in the context of Galois representations. Moreover the existence of these factors was an essential ingredient in the application of the converse theorem, as formulated in the Jacquet-Langlands notes, to establish that the Artin conjecture in its original form could be valid for two-dimensional representations only if the stronger version was also valid, that to every two-dimensional complex representation of the Galois group was associated, as predicted by functoriality, an automorphic form on \(\mathrm{GL}(2)\). Thus the proof of the existence of the \(\epsilon\)-factor and the use of the converse theorem to provide solid evidence for functoriality are for me intimately linked.

It is this confirmation of functoriality in its relation to the Artin conjecture and the introduction of the local correspondence that is, in my view, one of the two principal contributions of Jacquet-Langlands to a clearer, more mature formulation of functoriality and to a more solidly based confidence in its validity. The other is the formulation of the correspondence between automorphic forms on \(\mathrm{GL}(2)\) and on the multiplicative group of a quaternion algebra. This correspondence as such was not new and had appeared in work of Eichler and of Shimizu, but not in complete generality, not with the necessary precision, and not in both a local and global form. With this correspondence well in hand, the special role of quasi-split groups in functoriality became clear, as it had not been before.

There was one letter to Weil on the Hecke theory in an adelic, group-theoretic context and another, later letter to Jacquet. Although called letters, they were long and written in the form of essays, intended, perhaps, as first drafts of papers. They were, however, handwritten documents, not intended for publication. In particular, no attention was given to problems of typing or typesetting. The present typed version comforms as closely as possible to the original handwritten letter.

So far as I can tell from the evidence available, the first letter was written in two parts, chapters 2 through 5 in Princeton in late spring or early summer of 1967 and chapters 1, 6, and 7 in Ankara, presumably in August and September. There is an acknowledgement from Weil extant, dated Sept. 20 and a substantial difference in the quality of the xerox copies of the two parts.

The first letter was originally intended as a response to a question of Weil, who was having trouble extending his original paper on the Hecke theory to fields with complex primes, but it began to take on a different shape as the possibility for verifying some simple consequences of an earlier letter, on what is now referred to as functoriality, presented itself. In that letter the suggestions were entirely global, whereas in the published lecture Problems in the theory of automorphic forms the global conjectures had local counterparts. It was the study of \(\mathrm{GL}(2)\) that first permitted some confidence in the local conjectures.

The first letter did not fully deal with the nonarchimedean places. This was not possible until at some point during the year in Ankara I stumbled across, in the university library and purely by accident as I was idly thumbing through various journals, the article of Kirillov that contained the notion referred to in the notes of Jacquet-Langlands as the Kirillov model. With the Kirillov model in hand, it was possible to develop a complete local theory even at the nonarchimedean places. This is explained in the second letter. The date of this second letter can be inferred from the collection of short notes to Jacquet, as can the approximate date for my first acquaintance with the Kirillov paper. These letters, as well as two letters to Harish-Chandra and one to Deligne, document -- for those curious about such matters -- the path to the conviction, far from immediate, that there were more representations over fields of residual characteristic two than at first expected. I myself was surprised to discover, on reading the long letter to Jacquet, that as late as January, 1968 I still thought that the Plancherel formula for \(\mathrm{GL}(2)\) for such fields would not demand any more representations than for fields of odd residual characteristic. Lemma 5.2 of that letter, for which the proof was supposed to come later, is not, as we know very well today, correct for residual characteristic two.

Real conviction in the matter demanded the existence of the local \(\epsilon\)-factor for Artin \(L\)-functions and, as appears from the letters to Harish-Chandra and Deligne, this took some time to establish.

There is little in the two long letters that does not appear in Jacquet-Langlands, except the proofs, which are more naive than many of those appearing in those notes and to which I am sentimentally attached. That is the main reason for including the letters in this collection. The others are included principally to establish the sequence of events. I have taken the liberty of correcting a number of grammatical errors in the letter to Deligne.

Editorial comments:

• Although a part of these notes have circulated as a rather bulky preprint, they remained, for reasons to be described, incomplete, and even the parts completed were never all typed.
• (2024-01-02): Dwork's thesis referred to below is available here: http://alpha.math.uga.edu/~lorenz/articles.html

Author's comments: One project that was formulated after writing the letter to Weil and that was suggested by his 1967 paper on the Hecke theory was to establish a representation-theoretic form of it and to acquire thereby a clearer notion of the implications of the conjectures. In particular, I suppose although I have no clear memories, it was only after writing the letter that the possibility of local forms of the conjectures, over the reals, the complexes, and nonarchimedean fields, presented themselves. As the theory for \(\mathrm{GL}(2)\) worked itself out, with precise product formulas for the factor appearing in the functional equation, it became clear that, as a consequence of the conjectures in the form they were taking, there would have to be a similar product formula for the analogous factor in the theory of Artin \(L\)-functions.

My office in Ankara was next to that of Cahit Arf, and when I mentioned the question to him, he drew my attention to a paper of Hasse that had appeared in a journal not widely read, the Acta Salmanticensia of 1954. He fortunately had a reprint. So I could begin to think seriously about the matter. The critical idea came in April 1968 in a hotel room in Izmir, where I had gone to deliver a lecture. It was the understanding that all identities needed were consequences of four basic ones, formulated in the notes as the four main lemmas. Once this is understood and basic facts about Gauss sums are understood, as in the papers of Lamprecht and Davenport-Hasse, three of these four identities are not so difficult to establish. The second main lemma turned out, on the other hand, to be a major obstacle. Fortunately while leafing idly through journals in the library, either in Ankara or later in New Haven (I no longer remember), I came across Dwork's paper in which the first and the second main lemmas were proved. Dwork had indeed tried to establish a product formula for what has come to be called the \(\epsilon\)-factor but, without the insight that came from the adelic form of the Hecke theory and the conjectured relations of that to Artin \(L\)-functions, did not appreciate the need to introduce the factor \(\lambda(E/F,\psi_F)\) in condition (iii) of Theorem A. So he fell short of the goal, but fortunately not before he had established these two lemmas, which are indeed far more than lemmas, the proof of the second being a magnificent tour de force of \(p\)-adic analysis. Unfortunately he did not publish a proof, and the only material I had available when writing these notes was the thesis of K. Lakkis which reproduced Dwork's arguments, but only up to sign, and this is of course not enough. Nonetheless although many of the calculations are there, I was never able to work my way through them or put them in a form that was at all publishable. What I put down on paper from my attempts to understand the arguments of Dwork as reproduced by Lakkis is included here as fragmentary Chapters 12 and 13. They are included for what they are worth. Chapter 10, in which the proof of the first main lemma is completed, is also missing. Either it was never written or was misplaced. In any case, the material of Chapters 7, 8, and 9 at hand, the proof of the first main lemma is neither long nor difficult. With the exception of Chapters 10, 11, and 12, and perhaps some easy material that was to have been included in Chapters 8 and 9, the notes are complete. The proof is complete if one accepts the two lemmas of Dwork. Whether the complete proofs, which certainly existed, appeared in his thesis, I do not know, nor do I know whether his notes are still extant.

I abandoned my attempt to prepare a complete manuscript when Deligne observed that it is an easy matter to reverse the arguments and to proceed from the existence of the global \(\epsilon\)-factor, known to exist since Artin introduced the \(L\)-functions, to the existence of the local factors. It suffices to be clearly aware of their defining properties. Since these had escaped a mathematician of Dwork's quality, they cannot be regarded as manifest, or in the words of an eminent French mathematician "peu de chose"! Perhaps he was misled once again by partisan sentiments.

What of any possible use remains of the arguments here? First of all a general lemma about the structure of relations between induced representations of nilpotent groups that is conceivably of interest beyond the purposes of these notes, but that has never, so far as I know, found application elsewhere. Perhaps of more importance: although the local proof, which could be reconstructed from Dwork's notes and the material here, is far too long, a global proof of a local lemma is also not satisfactory. So the problem of finding a satisfactory local proof remains open.

The local \(\epsilon\)-factor is often incorporated into characterizations of the local correspondence for \(\mathrm{GL}(n)\). This is also unsatisfactory. The only real criterion for deciding whether a local correspondence is correct is that it be compatible firstly with the global correspondence and secondly with localization for representations of the Galois groups on one hand and automorphic representations on the other. Such a local correspondence established, the existence of the \(\epsilon\)-factor is immediate. At present, however, all aspects of the theory are rudimentary and inchoate. What may ultimately happen---I am not inclined to predictions in the matter---is that the existence of the local correspondence and of the \(\epsilon\)-factor will be established simultaneously, and that some of the arguments of these notes will reappear, but supplemented with information about the representations of \(\mathrm{GL}(n)\) over nonarchimedean fields.

I stress that these notes were written about 1970. I have not examined them in the intervening years with any care. There may be slips of the pen and even small mathematical errors.

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Auhor's comments: Although I have the feeling of having left unfinished almost every mathematical project undertaken, the study of endoscopy and the stabilized trace formula was, in this respect, one of the most unsatisfactory of all. It went on for a very long time without reaching any very cogent conclusions. This now seems with hindsight to have been inevitable. The efforts of a number of excellent mathematicians make it clear that the problems to be solved, many of which remain outstanding, were much more difficult than I appreciated. In particular, the fundamental lemma which is introduced in these notes, is a precise and purely combinatorial statement that I thought must therefore of necessity yield to a straightforward analysis. This has turned out differently than I foresaw.

Without the kind invitation of Marie-France Vignéras to deliver lectures at the École normale supérieure de jeunes filles, I would never have attempted to communicate the inchoate results at my disposition and I would have continued, no doubt unsuccessfully, to struggle with problems, both local and global, that were beyond me. The lectures were an occasion to clarify and organize the few ideas that I had, and have served as a stimulus to other, more competent, investigators.

Author's comments: This note was written for Mathematical Reviews, but it is of more interest for me than a simple review would have been because I came to understand on writing it and reflecting on its contents that perverse sheaves are likely to have much more import for nonabelian harmonic analysis in the sense of Harish-Chandra than I had previously appreciated.

Editorial Comments: These are the notes for a lecture given at the Institute for Advanced Study on March 30, 2000. It is a preliminary version of a much more carefully written account of essentially the same material published in the volume dedicated to Joseph Shalika, which will also appear on this site eventually.

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).

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.

Author's comments (2021-06-22): The paper as presented here is not the published paper. That was unfortunately modified, namely slightly abridged, by the editors without consulting the author and without his approval. The present paper, the original paper, is the preferred form.

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 \(\mathbf R\) and \(\mathbf C\). 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.

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.

Editorial comments: This essay was first posted here on December 20, 2013. The latest version dates to May 16, 2014.

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: 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.

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 \(\mathbf Q\) 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 \(\mathbf C\). 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 \(\mathbf C\). 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 \(\mathbf C\). 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.

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Author's comments: Although this paper has some merit, it also leaves a lot to be desired, and suffers from the author's inexperience with the material. For a better treatment of more general questions and corrections to the assertions of this paper, the reader is advised to consult a paper of Thomas Zink, Über die schlechte Reduktion einiger Shimuramannigfaltigkeiten, Comp. Math. 45 (1981).

Author's comments: It has been pointed out by J. Milne, by M. Pfau and by H. Reimann that there is a blunder in this paper in the treatment of gerbes. For a description of the error, for necessary corrections, and for further references, see the notes of Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lect. Notes Math. 1657.

Author's comments: It is likely that these two letters to Serge Lang, like my earlier letter to Weil on problems in the theory of automorphic forms, were never read with any attention by the recipient. Moreover, the earlier letter is a model of clarity compared with these two. Besides, there is, in retrospect, no reason to think that either Lang or Weil had the necessary background in the theory of semisimple groups and certainly not in the theory of their infinite-dimensional representations. I suppose that, spending a year in Bonn, where, even though I had many occasions to discuss mathematics seriously with Günter Harder, the environment was novel and the impulse to communicate with other mathematicians less easily satisfied than at home, I was impelled to write Lang, perhaps partly because I was venturing into algebraic geometry for the first time, perhaps partly because I had had some months earlier some agreeable conversations with him, when he was in the course of leaving Columbia and had fixed upon Yale as a possible alternative.

The interest of the letters is not in the details themselves. It is in the origins of endoscopy and in the beginnings of the reciprocal influence of nonabelian harmonic analysis and the algebraic geometry and arithmetic of moduli varieties.

In the months before coming to Bonn in September 1970, I had already undertaken the study of Shimura's papers, although I was ill-prepared, having mastered neither Weil's Foundations, the principal technical resource for Shimura, nor the modern language of Grothendieck. Weil's book I had at least attempted to read, but Grothendieck's geometry was not a topic to be learned in the sixties in Princeton and certainly not at Yale or, indeed, anywhere in the western hemisphere outside of Cambridge, MA. At Princeton his name was seldom pronounced and not always favorably. It was first in 1972 that I began, awkwardly, to try, with the help of an unlikely text, to try to learn something about etale cohomology. In Bonn, I proposed, somewhat presumptuously, to lecture on the work of Shimura, beginning with curves, even with subgroups of \(\mathrm{SL}(2,\mathbf Z)\) and the upper half-plane. Although the clarity of my lectures was not improved by the decision to profit from them to learn German, in the end, they were, at least from my point of view, a success, both linguistically and mathematically and the audience, although small, was tolerant. I developed an attachment to the German language and literature that has been a source of great pleasure and profit in the subsequent decades.

As for the mathematics, I began to reflect on Shimura's results, but on the basis of my own experience and knowledge. My thoughts were informed on the one hand by the principles enunciated in the letter to Weil, on the other, by the newly created theory of the discrete series. This theory, apart from its beginnings in the hands of Bargmann, the work of a single mathematician, Harish-Chandra, is, in my view, one of the great mathematical creations of the second half of the twentieth century, not sufficiently appreciated in its time and not yet today. Although the study of the zeta-functions of Shimura varieties demands inevitably also a great deal from algebraic geometry and number theory, those number theorists or algebraic geometers who attempt to develop it in ignorance of the discrete series and other pertinent aspects of nonabelian harmonic analysis are in danger of condemning themselves, whatever the immediate advantages, to ultimate irrelevance.

The present two letters are an attempt to explain the discovery central to my own first efforts at understanding Shimura's investigations, a discovery made, I observe, before the many results, appearing in a long sequence of papers, of Shimura's investigations were formulated simply, elegantly, uniformly and more generally by Deligne in his, I think one can say, celebrated Bourbaki seminar, which, because of conceptual transparency, has for better or worse eclipsed the original papers of Shimura. The basic coweight in terms of which Deligne formulates his definitions and which was introduced by him at one stroke is in the first of these letters calculated laboriously group by group, but with a particular aim in mind, compatibility with the cohomological information provided by W. Schmid's construction of the discrete series. It was an agreeable surprise to see it re-appear a month or two later, when I had the occasion to listen to a repetition of the Bourbaki lecture by Deligne himself in Bonn.

The major revelation, which arrived as I was standing smoking a cigarette, an unfortunate habit long abandoned, near the mathematical institute in Bonn, just at the intersection — or junction — of Beringstraße and Wegelerstraße, was that – in normal circumstances – each element of the discrete series of the appropriate weight contributes a one-dimensional subspace to the cohomology of the appropriate sheaf on the Shimura variety (a term that I introduced only later and that was imposed only because of some insistence on my part.) Let the Shimura variety be associated to a group \(G\). The number of different discrete series associated to a given weight is, typically, the index \(d=[\Omega_G:\Omega_K]\) of the Weyl group of the maximal compact subgroup of \(G\) in the Weyl group of \(G\). So one is led to reflect along the following lines. The cohomology groups are defined topologically or \(\ell\)-adically, but the two are normally of the same dimension. An automorphic representation is written \(\pi=\pi_\infty\otimes\pi_f\), where \(\pi_f\) is the product over the nonarchimedean places of \(\pi_v\) In some sense, \(\pi_\infty\) determines what cohomology is attached to \(\pi\) and \(\pi_f\) determines the associated the \(\ell\)-adic representation. If, as is at first suggested, one element of the discrete series of a given weight is matched with a given \(\pi_f\), then so are all \(d\) of them, say \(\pi^{(1)}_\infty,\ldots,\pi^{(d)}_\infty\). These \(d\) representations should be taken as a packet and the packet determines a subspace of the cohomology of dimension d. It should correspond to an \(\ell\)-adic representation of dimension \(d\) and one supposes, along the lines of the Eichler-Shimura relations, that the \(L\)-function of this representation is equal to an automorphic representation \(L(s,\pi,\sigma)\), where \(\sigma\) is a representation of the \(L\)-group of \(G\). So one predicts that for each \(G\) to which is attached a Shimura variety, there is associated a natural representation \(\sigma\) of degree \(d\) of the group \({}^LG\). The existence of this representation is by no means obvious and was proven by a case-by-case examination of the groups to which Shimura varieties are attached. The necessary calculations appear in the first letter of these two letters and form about the first third of it.

Reflection leads, however, to the conclusion that if one element of the appropriate discrete series appears matched with \(\pi_f\) by no means do all other elements of the same packet (a term not invented at the time) necessarily also appear matched with the same \(\pi_f\). Typically, however, they do. So the failure of the full matching is anomalous and needs a particular investigation, an investigation that began with \(\mathrm{SL}(2)\), and has come to be referred to as endoscopy, one of whose elements is the stable trace formula. The shape of the theory revealed itself only slowly, for \(\mathrm{SL}(2)\) in collaboration with Labesse, and for real groups, where one had the nonabelian harmonic analysis of Harish-Chandra at one's disposal, in the papers of Diana Shelstad. One of the critical features of endoscopy is endoscopic transfer, whose possibility was credible only thanks to her efforts.

In 1980, in a series of lectures in Paris, published as Les débuts d'une formule des traces stable, I sketched the theory as it had developed by then: introduction of the notions of transfer factor and of stabilization and a statement of the fundamental lemma. Even a cursory examination of the text shows that important details were lacking, above all a precise definition of the transfer factor. At the time of the lectures, I expected that the fundamental lemma, an apparently elementary combinatorial statement, would be quickly proved. This was not to be so and it yielded, after initial exploratory efforts by myself, J. Rogawski and others over a full but discouraging decade only slowly to much more sophisticated attacks by Kottwitz, Hales, Waldspurger, Goresky-MacPherson, Laumon and, finally and successfully, by Ngo Bao Chau. The proof of the lemma, at first formulated for \(p\)-adic fields, passes through a proof of equivalence of the \(p\)-adic lemma with a similar lemma for power-series fields over finite fields, an equivalence that has, apparently, some element of mathematical logic in it, but which was proved by hand by Waldspurger in a marvelous tour-de-force and a proof for power-series fields that entails, in the work of Laumon and Ngo, a global argument for curves over finite fields. It is worthwhile to mention in passing that, so far as I understand, a precise definition of the transfer factor is essential to the argument. This precise definition was only possible thanks to the very careful analysis of Harish-Chandra's theory of nonabelian harmonic analysis in Shelstad's treatment of the transfer over archimedean fields.

Nevertheless, it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of Shimura varieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the Grothendieck-Lefschetz formula. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years. I, at least, was tremendously discouraged, and I think the severe limitations created by its lack also influenced one or two others to be more circumspect and less enterprising. Not only was their success circumscribed but also the recognition they received. I hope that with the fundamental lemma at hand we will see in the coming years great progress both with functoriality and with the general theory of Shimura varieties.

The second third is taken up with a first discussion of possible proofs of the conjectured equality. It is not easy to follow and no longer, as far as I can see, of much interest. By 1972, at the time of the Antwerp lecture on the Eichler-Shimura relation and related matters, I had already begun to use a comparison of the trace formula with the Grothendieck-Lefschetz fixed point formula, a method that has been developed in general, at first by me, later, more deeply, with much better results, by Kottwitz. Both of us were handicapped by the lack of the fundamental lemma. Now that it is available, I hope that these methods will be taken up again. I believe that they still offer the best prospects for a complete and systematic treatment of the zeta-functions of Shimura varieties and their relation to automorphic \(L\)-functions, at least of the unramified factors.

The full comparison of the associated Galois representations with the automorphic representations will require, in addition to a full development of endoscopy, more sophisticated algebraic geometry and algebraic geometry. I have never thought about these matters in any serious way.

The last third of the letter is a discussion of the complex cohomology of Shimura varieties, Matsushima's theory, and Blattner's conjecture for the discrete series and their relation to each other. One major question raised, but only implicitly, by the letter was not discussed: whether indeed, if \(\pi_\infty\otimes\pi_f\) is an automorphic representation with \(\pi_\infty\) in the discrete series and if \(\pi'_\infty\) is a second element of the discrete series, the representation \(\pi'_\infty\otimes\pi_\infty\) is also an automorphic representation and whether it occurs with the same multiplicity? This would now be recognized as a question about endoscopy and global \(L\)-packets, but at the time was formulated more elementarily. As aleady observed, these questions were considered at the very first only for \(\mathrm{SL}(2)\), in part by me in conversation and correspondence with Labesse, and then by Shelstad for real groups, and it was only slowly that the theory reached even the stage of the 1980 lectures. Then, aside from substantial but largely unnoticed progress by Kottwitz, it languished for almost twenty years.

The second letter is somewhat more technical. It establishes, in a somewhat informal manner, that the ideas in the first, including the use of the automorphic \(L\)-functions introduced in my letter to Weil, are compatible with the behavior of the \(\Gamma\)-factors proposed a couple of years earlier by Serre. Its core is a relation, whose proof is preserved in my personal notes and included here but was not sent, so far as I can see, with the letter. Both the letters and the appendix suggest a very industrious young man.

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Author's comments: The Galois representations attached in the context of Shimura varieties to certain automorphic representations usually correspond---under the correspondence of the "Langlands program"---not to the representation to which they are attached but to some twist of it by a central character. There is no reason---logical or aesthetical---that it should be otherwise. Nevertheless a casual search of the literature will probably reveal that a number of authors were troubled by it and attempted---on their own initiative---to revise the definitions. This has led to a certain amount of confusion. My own feeling is that the parametrization at infinity described in "On the Classification of Irreducible Representations of Real Algebraic Groups", depending as it does on curious properties of one-half the sum of positive roots is a kind of miracle, so that it is rash to tamper with it without very, very good reasons.

Of course, the twist, perhaps not so different from that arising in the theory of perverse sheaves, causes, as that does, annoying mnemonic difficulties. Although I have never made the effort, I suppose they can be overcome by thinking the whole matter through carefully.

The letter, written under hurried circumstances in a hotel room, is not so clear as it might be. Indeed, as written it is downright confusing. What is important to underline is that the representation \(\pi'\) is the one with the right properties at infinity, namely it is associated to representations of the local Weil group that are algebraic on the multiplicative group of the complex numbers, a subgroup of each of the local Weil groups. It also, at least for the group of two-by-two matrices, and presumably in general, at the other local places conforms to the local correspondence of the "Langlands program". On the other hand, \(\pi'\) is different from \(\pi\), the representation yielding the complex cohomology on whose \(\ell\)-adic counterpart \(\sigma\) is realized. (The equality \(\pi'_p=\sigma_p\) is nonsense! What I meant was that the Galois representation \(\sigma_p\) and the local component \(\pi'_p\) of the automorphic representation \(\pi\) correspond.) There is, so far as I can see, no harm in this.

I have had occasion since writing this note to look at Clozel's paper. It appears in the first volume of the proceedings of the Ann Arbor conference on Automorphic forms, Shimura varieties and L-functions. Clozel is aware of the difference between \(\pi\) and \(\pi'\) but draws different conclusions from this than I do, at least at the moment. I discuss this and related matters in the comments on the 1974 letter to Deligne.

Author's comments: Problems of endoscopy first arose as I began the study of Shimura varieties in Bonn during the academic year 1970/71. I reflected on them for a long time, in part in collaboration with Labesse, in part in collaboration with Shelstad. I presented a fairly mature form of my reflections in the Paris lectures, Les débuts d'une formule des traces stable, in which the presence of a major obstacle, overcome considerably later through the efforts of a number of mathematicians, in particular Waldspurger and Ngô, was clearly described. It was labelled the fundamental lemma. I continued trying to find a proof during the eighties, but was no doubt discouraged and searching for a diversion.

I turned, at some point, although not completely, away from automorphic forms to mathematical questions of an altogether different kind, about matters that I had not considered since I was an undergraduate, studying, as I recall, among others the book of Lanczos on The variational principles of mechanics and the book of Whitham on Linear and nonlinear waves, but also less classical aspects of mathematics physics, among them the biography of Einstein, Subtle is the Lord... by Pais, which is of course a serious introduction to modern physics.

By good luck, I fell into conversation one afternoon with the physicist Giovanni Gallavotti, who was visiting the School of Mathematics at the Institute for the academic year 1984-85 as a participant in a special program in mathematical physics that I had, I recall, suggested and encouraged, but only as a spectator. He explained to me, as we strolled through the meadow that surrounded the Institute, the problem of renormalization. I was, in spite of my ignorance, fascinated by it and tried to learn more. Renormalization is, of course, a term that I had seen before in connection with quantum field theory, but that was an area with which I had no familiarity. I learned from Gallavotti that it was, in particular, associated to dynamical systems in an infinite number of variables with a fixed point that had only a finite number of expanding directions and in which the eigenvalues of the linearized transformation, all but a finite number less than \(1\) in absolute value, tended to \(0\). So for me, a mathematician and not a mathematical physicist, and certainly not a physicist, the immediate problem became the construction of such systems together with a proof that they possessed this property.

It was not of course that no examples were available. What was missing was a clear mathematical definition of systems of this type, of their fixed points, and proofs of the desired structure. Since we are dealing, at first, with infinite-dimensional spaces with no precise definition, thus with no obvious coordinate system and no exact notion of admissible points, some reflection about the basic notions was---and remains---necessary. The simplest example seemed, after some time spent with the pertinent literature, to be percolation. I bought and studied with some care the book of Kesten, Percolation theory for mathematicians, which had appeared not long before, in 1982. The possibility slowly occurred to me that the crossing probabilities studied with considerable success by Kesten might yield the desired fixed point. Some care has to be taken with this statement, because there could be---and is---a continuous family of fixed points. In the present context they are defined by symmetries that manifest themselves in the form of conformal invariance, to which I shall return.

My first experiment with this possibility, which had, so far as I know, not been earlier examined in the literature, was to construct a finite-dimensional model of percolation in the paper Finite models for percolation, in which the presentation has, thanks to my co-author Marc-André Lafortune, at the time an undergraduate at the Université de Montréal and much more practiced with computers than I, considerable elegance.

Later on, in a discussion with Yvan Saint-Aubin, a colleague at the same university, I explained my developing views on the crossing probabilities, thus that they could serve as the coordinates of the fixed point. This, correctly interpreted, means that the crossing probabilities are universal. They are not absolutely universal, only universal for systems subject to certain constraints, in particular to the same constraints on symmetry, for example translational invariance and an appropriate reflection-symmetry. Saint-Aubin's first reaction was sceptical, a justified scepticism. So we decided to examine the question numerically. The strategy to be used was, for various reasons, largely in the hands of Saint-Aubin, who had had much more experience with computers than I. We were joined by a student and by a colleague. The experiments and their results are clearly stated in the abstract to the paper On the universality of crossing probabilities in two-dimensional percolation. I give it here because it was not reproduced with the paper.

`Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of width \(a\) and height \(b\) are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratio \(a/b\). In the limits set by the sample size and by the conventions and on the range of the ratio \(a/b\) measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.´

The models are chosen so that the crossing probabilities in all models would have considerable symmetry, indeed as it turned out a rotational symmetry and, of course, translational symmetry. Symmetry was, however, not our main concern. That was universality. However, I had already as we prepared our results for publication discussed them with Michael Aizenman and Thomas Spencer. Aizenman then suggested the possibility of conformal invariance, which we began to test immediately. The conclusions are presented in a subsequent paper, Conformal invariance for two-dimensional percolation . This second paper had a more immediate influence than the first, above all on Oded Schramm, who unfortunately lost his life not many years later in a climbing accident. As I remember the one conversation I had with him, he mentioned that this paper played a role in his creation of the theory of Schramm-Loewner Evolution.

Both papers are mentioned, but only incidentally, in the laudatios for the two Fields medals related to conformal invariance and percolation: the one by Charles Newman for Wendelin Werner and the one by Harry Kesten for Stanislaw Smirnov. The one by Kesten is misleading and misleading in an important, although presumably unintended, way. Kesten refers to a sentence in the first paper, "Conversations with Michael Aizenman have greatly clarified our views as to the nature of the universality manifested by the crossing probabilities, and our understanding of their invariance under various transformations of the curves defining the event E. In particular, they have suggested a number of conjectures to which we shall return in a later paper, in which the modifications required for models with less symmetry than those treated here will also be discussed." In the second, the statement "Conversations with Michael Aizenman after the data were in hand greatly clarified for us their nature. In particular he suggested that these crossing probabilities would be conformally invariant." Kesten, however, refers to the first sentence alone and deduces from it the following conclusion, taken from his laudatio, "... we did not specify what it means that the scaling limit exists and is conformally invariant. It seems that M. Aizenman (see [13], bottom of p. 556) was the first to express this as a requirement about the scaling limit of crossing probabilities." The reference is to the first paper. In other words, Kesten confounds the problem of conformal invariance with that of the existence of an essentially unique scaling limit and this confounds, in my view, two, even three, problems of quite different depth. The existence of the scaling limit had been proved by Kesten himself; the essential uniqueness is related to the very broad collection of problems raised by renormalization; conformal invariance will be, if my intuition is correct, relatively easy to deduce in general from a theory of renormalization and its fixed points, a theory that will necessarily include universality, if that can be established. The probabilists are very attached to conformal invariance and why not? The Scramm-Loewner theory is very elegant and the results of Smirnov significant. It is nevertheless renormalization that is the broader issue and deserved to be discussed in at least one of the two laudatios.

The view of critical phenomena revealed to me by the initial conversation with Gallavotti as associated to infinite-dimensional dynamical systems and their fixed-points was tremendously appealing and percolation appeared to be a promising place to begin. The questions themselves are, however, omnipresent in modern physics: in statistical physics, in quantum field theory, in fluid dynamics. It is, I found, difficult, even impossible, for a mathematician, at least this mathematician, to find his bearings in the variety of approximations, guesses, and intuitions, and insights available, some more, some less convincing. It is hard to know how or where to begin to think about all the questions that appear.

Even for percolation, a space to contain the fixed-points was needed, as well as possible transformations (the dynamical system) to distinguish the pertinent fixed points. So there is a transition to be made, from the lattices with probabilities of occupation to the dynamical system, in this instance, the passage to an object defined by crossings. Even with Kesten's theorem in hand it is not entirely evident what to do. The theorem takes us a long way, but the universality must still be proved. My intuition, which is not supported by much evidence, is that a proof of universality will inevitably contain in a more or less clear manner a proof of conformal invariance. It is because of this intuition, which I expect to be more widely applicable and valid not just for percolation, that I find it more than unfortunate that Kesten has confounded universality with conformal invariance.

The following section, the one on mathematical physics, contains little, perhaps nothing, of interest. I tried for some time to acquire a useable understanding of the various domains of physics and mathematical physics that would lead to some concrete definitions and questions that permitted, on the one hand, the formulation of real mathematical theorems or even conjectures and that, on the other, were relevant to the insights of the mathematical physicists or to an understanding of the natural phenomena of, say, fluid dynamics. I made a real effort and learned quite a bit, but not enough. I had always hoped to return to these questions, not with any hope of accomplishing something but just to educate myself. This appears less and less likely.

Author's comments: There are two papers on the Bethe Ansatz, but the work is far from complete. I have always wanted to return not only to the algebraic geometrical arguments initiated in the second paper, which seem to me of considerable intrinsic mathematical interest as algebraic geometry, but also to the notion of Wellenkomplex and the Puiseux expansions introduced briefly at the end of the first paper. So far, I have not found the time.

Author's comments: This paper was prepared for a meeting in Bialowieza that I was unable at the last minute to attend. It has appeared in the proceedings of that conference, Twenty years of Bialowieza: A mathematical anthology. The paper was intended as a beginning. Several years of work, largely numerical and very often in collaboration, on percolation and the Ising model were an attempt on my part to get a handle on what was for me their mathematically fascinating aspect, referred to as renormalization: the observed behavior of large systems for which repeated re-scaling is possible can be described by a very small number of parameters, and the convergence under re-scaling of the values of these parameters to their limits is extremely rapid. I have never found in the literature or discovered on my own any method of any general promise for defining these parameters or for demonstrating their properties. I had hoped when writing this paper to have within my grasp some promising ideas. I thought about them, but either I did not think long enough or hard enough, or they were worth less than I thought. Whatever it was, my attention has been distracted for several years by other matters, every bit as difficult and intractable as renormalization, so that I have not been able to return to it. This was likely no loss to science, but I regret it, for the mathematical questions are in my view of profound interest. I still harbor a little hope that in the coming years I can return to them.

Editorial comment: The original version of this paper was posted on 2018-04-18. It was last updated on 2022-12-19.

Editorial comments: This is an English translation of the first paper in this section, Об аналитическом виде геометрической теории автоморфных форм. The author would like readers to know that the translation is a little rough.

The first version was posted on 2020-06-03. This latest version was posted on 2022-12-19.

Editorial Comments: This interview originally appeared in the September 2018 issue of the Newsletter of the European Mathematical Society and was reprinted in the April 2019 issue of the Notices of the American Mathematical Society. The interviewers were Bjørn Ian Dundas and Christian Skau.

A bust of André Weil sculpted by Charlotte Langlands
(now located in the mathematics common room at the Institute)

Solomon Bochner in his office at Rice University
(Photograph courtesy of William Veech and the Rice University Archives, Woodson Research Center)

Portrait engraved by van Schooten the younger, editor and translator of the Latin edition of La géometrie.
Descartes said of it, "La barbe & les habits ne ressemblent aucunement."
(From the Rosenwald Collection at the Institute in Princeton)

Author's comments: This note contains a few recollections of a year I spent in Turkey in 1967/68, where my office was adjacent to that of Cahit Arf, known, among other things, for the Hasse-Arf theorem and the Arf invariant. It was he who referred me -- as I was first attempting to define local \(\epsilon\)-factors for Artin \(L\)-functions -- to the paper of Hasse published in the Acta Salmanticensia. Hasse's paper was my first introduction to the methods that had already been introduced for calculating and comparing the \(\epsilon\)-factors. The note was published in the winter number of Ali Nesin's Matematik Dünyası for the year 2004.

Author's comments: The following brief discourse was delivered in Erlangen in October, 2004, on the occasion of the award of the Karl Georg Christian von Staudt-Preis to Günter Harder. It does not do justice to his many contributions to mathematics, but does attempt to express my great admiration of him and my great respect for the passion and the tenacity with which he continues to reflect on what seem to me some of the central problems of the modern theory of numbers.

At one or two points in the text there are references to diagrams. The diagrams are hardly necessary for a mathematically experienced reader and are not included.

Author's comments: The text on the genesis and gestation of functoriality was for an informal lecture at the Tata Institute of Fundamental Research in Bombay delivered on short notice at the suggestion of Venkataraman. It has been suggested to me that the first four pages, a brief summary of the development of the theory of automorphic forms before 1960, roughly as it affected my initial reflections, manage to be simultaneously trite and eccentric and might be best omitted. The reader is free to do so, but the purpose of posting this text is to record the lecture, not to improve it.

Although I attach some importance to the historical origins of the theory, even to my own understanding of them, and am not entirely persuaded that all contemporary readers will be fully aware of the mathematical climate in the early 1960's or of the various strands in the number theory of the nineteenth and twentieth century, it may be best to consider the doubtful first four pages simply as remarks in the way of warm-up.

On the other hand, in retrospect, one perhaps curious feature of my mathematical education and development is that I never studied elementary number theory, either on my own or formally, and some obvious things never appealed to me. Although I mention Weyl's book on algebraic number theory in the lecture, I do not confess that it was only on reading it that I began to appreciate the beauty of the law of quadratic reciprocity to which I had earlier attached no importance. I like now to think that I would have greatly benefited from an introduction to Gauss's Disquisitiones by a perceptive teacher at the beginning of my career. So I may, after all, having been trying to convey autobiographical information even in the warm-up.

There is a specific point, perhaps not entirely devoid of interest, that did not occur to me at the time of the Mumbai lecture. It can be more or less squared with the recollections there of lectures at Yale in 1967. At Yale, I listed on the blackboard the representations \(\rho\) of the \(L\)-group for which \(L(s,\pi,\rho) \) could be analytically continued by what is now known as the Langlands-Shahidi method. I had calculated them case by case. It was Jacques Tits who immediately observed that these were the representations on the unipotent radical of the Levi subgroup of the dual group associated to the Levi subgroup of \(G\) defining the pertinent Eisenstein series. So I was perhaps not so reticent at Yale as I suggested in Mumbai.

Author's comments: The article is an exercise in the reading of mathematics from earlier times. An explanation of Descartes's solution of the problem of Pappus as included in the appendix "La géométrie" to "Discours de la méthode" and an explanation of a solution to another form of the same problem by Fermat, described briefly in a letter included in his collected works, are taken as an occasion to compare the mathematical styles of the two men and to observe their mutual debt to Apollonius as well as the differences in their depth of understanding of his work. The purpose is not scientific or historic; it is simply to encourage the private reading of classical mathematics by those who like me have no special knowledge of the history of mathematics.

This article appeared on pages 54-61 of the mathematical magazine Matematik Dünyası, No. 2, 2005. The magazine (as mentioned above, edited by Ali Nesin), is published in Istanbul and has a wide circulation among mathematics students and teachers in Turkey.

Editorial comments: Istanbul, 2004

Editorial comments: Orta Doğu Teknik Üniversitesi, Kasım, 2004

Author's comments: This review comes with a supplement (footnote) that contains the comments of several leading specialists and will be much more useful to the potential reader of the book, whether a novice or a specialist, than the review itself.

Author's comments: The following three documents were composed on the receipt of the Shaw Prize in 2007. They have appeared or will appear in publications of the Shaw Foundation. The first two, a very brief autobiography and a slightly longer memoir, need no explanation. They are informal. So is the third, an attempt to explain clearly the nature of what is often referred to as the Langlands program. It is still uncertain that I succeeded, partly because the program remains in good part just that---a program and not a mature theory, although I have been cheered by the reaction to this essay, which contains a dangerously large speculative element.

Programs, if that is the correct word, are a combination of insights, partial results and conceptual constructions, whose value depends to a very large extent on the quality and vigor of those who understand their purpose and are convinced of their pertinence. This particular program has found many friends, to whom its strength is due, but in some respects it went against the grain of a good number mathematicians. There are elements in it sufficiently foreign to them that even with good will they were unable to grasp its nature. The misunderstanding and resistance that ensue can, I hope, be overcome by patience and time. The occasional frank discussion or candid admission of cherished, even if somewhat veiled, goals may also be of some value. I hope that this essay is blunt enough to be clear, but not so blunt that it is ineffective.

Author's comments: The following article appeared in novembre, 2007 in the popular scientific review, Pour la Science, but in a version slightly revised by the editors and their consultants for expository purposes and with diagrams added. I am sure the revised form is indeed easier for a layman to understand, but some assessments were added that are not mine. Rather than interfere with the editors' difficult task of turning arcane material into something meaningful to their readers, I let the revised version stand. I wished, nevertheless, to make my own version available as well. There are many imaginative diagrams in the version published and I urge the curious reader to look at it if possible. Unfortunately, it does not seem to be available on line.

I am grateful to Claude Levesque for a close, critical reading of the article.

Authors's comments: These are my responses to questions of Farzin Barekat, a graduate student at the University of British Columbia, where I was an undergraduate student for four years and a graduate student for one year. The questions and my responses were transmitted electronically. An abbreviated version of my responses will be published in a newsletter of the mathematics department of the university. The longer version is likely to be of interest only to a very limited circle, those curious about undergraduate mathematics education in Canada in the fifties of the last century.

"Cityscape sunset" by Zoe Isabelle Côté (with the permission of the artist)

Editorial comments: This text was prepared as a complement to a lecture at the Conference on Beauty held at the Notre Dame Institute for Advanced Study in January, 2010.

Author's comments: The book La formule des traces tordue needed no introduction from me, but I did write it at the authors' request, in part because I was troubled by the circumstances of its appearance. I have, I believe, as a mathematician led a much richer intellectual life than the circumstances of my childhood would have normally permitted. So I am distressed by the diminishing possibilities of our profession and cannot always resist expressing my uneasiness and disappointment in a somewhat dyspeptic voice.

Although I do not fully understand the nature of the difficulties surrounding the book's publication, presumably all related to the ever increasing reluctance, perhaps even refusal, of publishers to accept technical books in the vernacular, I do know that in the end its publication, not in France, not by a Quebec editor, but by the American Mathematical Society was possible largely, perhaps only, thanks to the fortunate, and probably rare, circumstance that the then president of the Society was a francophile and was willing to use his good offices with the Society's publication branch. Although grateful to the president, I found the incident a sad sign of a serious intellectual decay, not in the USA but in Europe---and perhaps in Quebec as well.

A final comment. There are very many mathematicians who have made important contributions in recent decades to the analytic theory of automorphic forms. Those of James Arthur and of Jean-Loup Waldspurger are I believe, even among these, outstanding. I do not think that they have received from the mathematical community the recognition they deserve.

Author's comments: This essay, entitled "A letter to Turkish readers from Robert Langlands" appears in the Turkish translation of the extremely popular book "Love and Math" by Edward Frenkel. It does not refer to the book but comments in an informal manner on my relations with Turkey and Turkish mathematicians. The two photographs were taken by the Turkish authorities when we applied for a residence permit for a year's stay in 1967/68. I am very fond of the one with the children.

Editorial Comments: This essay was written in 2016 and published in August 2018 in the book Langlands 纲领和他的数学世界. (Langlands program and his mathematical world.) The essay appeared in English and was also translated into Chinese.