Letter to Deligne

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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