Emil Artin and beyond---Class field theory and L-functions, written in 1967, appeared in volume in 2015, 165--173, European Mathematical Society

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.

Emil Artin and beyond---Class field theory and \(L\)-functions, 2015, written 2010, 175--209, European Mathematical Society

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,"

Lectures in modern analysis and applications III, Lecture Notes in Mathematics, 170, 1970
MathSciNet Review: 302614

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.

Pacific Journal of Mathematics, 181, 1997, 231-250
MathSciNet Review: 1610871

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.

Edinburgh conference on automorphic forms, AMS, 1997
MathSciNet Review: 1476510


Letter to Serre

March 15, 1968

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.

[ letter-to-serre-1968-03-15_rpl.pdf ]

Letter to Deligne

March 31, 1974

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.

[ letter-to-deligne-1974-03-31_rpl.pdf ]

Letter to Roger Howe

March 24, 1974
[ letter-to-howe-1974-03-24_rpl.pdf ]

Letter to Roger Howe

February 23, 1975
[ letter-to-howe-1975-02-23_rpl.pdf ]

Two Messages to Mueller and Volpato

December 22, 2013, December 27, 2013

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.

[ two-messages-to-mueller-volpato-2013-12_rpl.pdf ]