Shimura varieties

Modular functions of one variable II, Lecture Notes in Mathematics, 349, 1973
MathReview: 50:7095
Mathematical developments arising from Hilbert's problems, Proceedings of Symposia in Pure Mathematics, 28, 1974
MathReview: 55:10426
Canadian Journal of Mathematics, 29, 1977
MathReview: 58:16528
Canadian Mathematical Society Selecta, 2, 1995
Canadian Journal of Mathematics, 31, 1979
MathReview: 82h:10042
Proceedings of the 1977 A.M.S. conference on L functions and automorphic representations, 1977
MathReview: 83f:12010

Journées de Géométrie Algébrique de Rennes, Astérisques, 65, 1979
MathReview: 82c:10039

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

Journal für die reine und angewandte Mathematik, 366, 1986
MathReview: 87k:11066
Journal für die reine und angewandte Mathematik, 378, 1987
MathReview: 88i:11036

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

The mathematical heritage of Hermann Weyl, Proceedings of Symposia in Pure Mathematics, AMS, 48, 1987
MathReview: 90e:11076


Letter to Lang

[ rl-lang.1.pdf ]

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

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

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

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

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

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

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

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

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

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

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

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

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

Letter to Lang

[ rl-lang.2.pdf ]

Appendix to previous letter

[ lang-appendix.pdf ]

Letter to Gee

[ rl-gee.pdf ]

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

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

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

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