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# Functoriality

written in 1967, to appear 2011
[ handwritten.pdf ]  ,   [ weil1.pdf ]

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.

Langlands 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.

1967
[ langlands-hc-ps.pdf ]  ,   [ l0.jpg ]
Written in 2010

Langlands 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.

Lectures in modern analysis and applications III, Lecture Notes in Mathematics 170 1970

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., in honor of Solomon Bochner: Lectures in modern analysis and applications III, Lecture Notes in Mathematics 170, Springer-Verlag, 1970.

Langlands 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 GL(n) are important because they admit anistropic R-forms. The coefficients of automorphic L-functions attached to groups anisotropic over 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 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 GL(2) came ε-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 ε-factors exist. One achievement of the following year, accomplished in collaboration with Jacquet, was a complete proof of the correspondence between automorphic forms on 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 l-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 61 -19 1998

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 61 (1998), pp. 231-250).

This is a short note written to illustrate some examples of how the conjectures worked out in very explicit examples.

Langlands 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.

Proceedings of the Gibbs symposium 1989
Edinburgh conference on automorphic forms, AMS 1997

# Letters

## Letter to Serre

March 15, 1968

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 GL2. This correspondence was later expanded to the Deligne-Langlands conjecture, proven eventually by Kazhdan and Lusztig.

Langlands Comments:This letter is a response to a question of Serre about the gamma-factors appearing in the functional equations of automorphic L-functions. Fortunately Serre's letter to me was accompanied by several reprints, among them apparently the paper Groupes de Lie l-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 l-adic representations associated to elliptic curves whose j-invariant was not integral in the pertinent local field. The special representations of GL(2) corresponded to these l-adic representations. The connection between non-semisimple l-adic representations and various kinds of special representations is now generally accepted. The theorem of Kazhdan-Lusztig is a striking example.

## Letter to Deligne

March 31, 1974

Langlands 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 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 ε-factor; the role of the special representation, at first for 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 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 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 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 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 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 GL(2) together with later work by Arthur and Clozel for GL(n) has played a major role in the fusion of the general theory of automorphic forms on one hand and the study of l-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 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 π 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 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 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 L2(G(F)G(AF). 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" GA onto the "motivic Galois group" GM, but only as a group over 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 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 GL(1) to the motivic Galois group. Something similar is available for automorphic representations: the automorphic representations of 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" GA exists. These two homomorphisms, of an abelian group TA into GA and of GA to TA allow the introduction of various twistings of any homomorphism of the automorphic Galois group GA to the motivic Galois group GM.

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 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 C. One is very quickly led to ask, what will be the function of a richer theory over Q, in which the relation with automorphic representations appears, to the theory over 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 l-adic motivation is certainly somewhat embarrassing! I was trying to express whatever understanding I had of the l-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 l-adic representations correspond to a pair (ψ,Y).

The loose phrase "such that ψ(σ) is semi-simple if σ 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(σ) is semi-simple. I neglected, moreover, to state that the residual characteristic of the local field F is supposed prime to l. Anyhow, the purpose is clearly to explain to myself --- not to the recipient --- how an l-adic representation leads to a pair φ, 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 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.

March 24, 1974

## Letter to Roger Howe

February 23, 1975
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