First tests and first consequences of functoriality
Author's comments: Some surprise has been expressed that the notes of Jacquet-Langlands have been placed in the same section as the notes on the \(\epsilon\)-factor. There is a good reason for this. Although the notion of functoriality had been introduced in the original letter to Weil, there were few arguments apart from aesthetic ones to justify it. So it was urgent to make a more cogent case. One tool lay at hand, the Hecke theory, in its original form and in the more precise form created by Weil.
The theory as developed in terms of representation theory, both local and global, suggested the existence of the \(\epsilon\)-factors in the context of Galois representations. Moreover the existence of these factors was an essential ingredient in the application of the converse theorem, as formulated in the Jacquet-Langlands notes, to establish that the Artin conjecture in its original form could be valid for two-dimensional representations only if the stronger version was also valid, that to every two-dimensional complex representation of the Galois group was associated, as predicted by functoriality, an automorphic form on \(\mathrm{GL}(2)\). Thus the proof of the existence of the \(\epsilon\)-factor and the use of the converse theorem to provide solid evidence for functoriality are for me intimately linked.
It is this confirmation of functoriality in its relation to the Artin conjecture and the introduction of the local correspondence that is, in my view, one of the two principal contributions of Jacquet-Langlands to a clearer, more mature formulation of functoriality and to a more solidly based confidence in its validity. The other is the formulation of the correspondence between automorphic forms on \(\mathrm{GL}(2)\) and on the multiplicative group of a quaternion algebra. This correspondence as such was not new and had appeared in work of Eichler and of Shimizu, but not in complete generality, not with the necessary precision, and not in both a local and global form. With this correspondence well in hand, the special role of quasi-split groups in functoriality became clear, as it had not been before.
There was one letter to Weil on the Hecke theory in an adelic, group-theoretic context and another, later letter to Jacquet. Although called letters, they were long and written in the form of essays, intended, perhaps, as first drafts of papers. They were, however, handwritten documents, not intended for publication. In particular, no attention was given to problems of typing or typesetting. The present typed version comforms as closely as possible to the original handwritten letter.
So far as I can tell from the evidence available, the first letter was written in two parts, chapters 2 through 5 in Princeton in late spring or early summer of 1967 and chapters 1, 6, and 7 in Ankara, presumably in August and September. There is an acknowledgement from Weil extant, dated Sept. 20 and a substantial difference in the quality of the xerox copies of the two parts.
The first letter was originally intended as a response to a question of Weil, who was having trouble extending his original paper on the Hecke theory to fields with complex primes, but it began to take on a different shape as the possibility for verifying some simple consequences of an earlier letter, on what is now referred to as functoriality, presented itself. In that letter the suggestions were entirely global, whereas in the published lecture Problems in the theory of automorphic forms the global conjectures had local counterparts. It was the study of \(\mathrm{GL}(2)\) that first permitted some confidence in the local conjectures.
The first letter did not fully deal with the nonarchimedean places. This was not possible until at some point during the year in Ankara I stumbled across, in the university library and purely by accident as I was idly thumbing through various journals, the article of Kirillov that contained the notion referred to in the notes of Jacquet-Langlands as the Kirillov model. With the Kirillov model in hand, it was possible to develop a complete local theory even at the nonarchimedean places. This is explained in the second letter. The date of this second letter can be inferred from the collection of short notes to Jacquet, as can the approximate date for my first acquaintance with the Kirillov paper. These letters, as well as two letters to Harish-Chandra and one to Deligne, document -- for those curious about such matters -- the path to the conviction, far from immediate, that there were more representations over fields of residual characteristic two than at first expected. I myself was surprised to discover, on reading the long letter to Jacquet, that as late as January, 1968 I still thought that the Plancherel formula for \(\mathrm{GL}(2)\) for such fields would not demand any more representations than for fields of odd residual characteristic. Lemma 5.2 of that letter, for which the proof was supposed to come later, is not, as we know very well today, correct for residual characteristic two.
Real conviction in the matter demanded the existence of the local \(\epsilon\)-factor for Artin \(L\)-functions and, as appears from the letters to Harish-Chandra and Deligne, this took some time to establish.
There is little in the two long letters that does not appear in Jacquet-Langlands, except the proofs, which are more naive than many of those appearing in those notes and to which I am sentimentally attached. That is the main reason for including the letters in this collection. The others are included principally to establish the sequence of events. I have taken the liberty of correcting a number of grammatical errors in the letter to Deligne.
Editorial comments:
- Although a part of these notes have circulated as a rather bulky preprint, they remained, for reasons to be described, incomplete, and even the parts completed were never all typed.
- (2024-01-02): Dwork's thesis referred to below is available here: http://alpha.math.uga.edu/~lorenz/articles.html
Author's comments: One project that was formulated after writing the letter to Weil and that was suggested by his 1967 paper on the Hecke theory was to establish a representation-theoretic form of it and to acquire thereby a clearer notion of the implications of the conjectures. In particular, I suppose although I have no clear memories, it was only after writing the letter that the possibility of local forms of the conjectures, over the reals, the complexes, and nonarchimedean fields, presented themselves. As the theory for \(\mathrm{GL}(2)\) worked itself out, with precise product formulas for the factor appearing in the functional equation, it became clear that, as a consequence of the conjectures in the form they were taking, there would have to be a similar product formula for the analogous factor in the theory of Artin \(L\)-functions.
My office in Ankara was next to that of Cahit Arf, and when I mentioned the question to him, he drew my attention to a paper of Hasse that had appeared in a journal not widely read, the Acta Salmanticensia of 1954. He fortunately had a reprint. So I could begin to think seriously about the matter. The critical idea came in April 1968 in a hotel room in Izmir, where I had gone to deliver a lecture. It was the understanding that all identities needed were consequences of four basic ones, formulated in the notes as the four main lemmas. Once this is understood and basic facts about Gauss sums are understood, as in the papers of Lamprecht and Davenport-Hasse, three of these four identities are not so difficult to establish. The second main lemma turned out, on the other hand, to be a major obstacle. Fortunately while leafing idly through journals in the library, either in Ankara or later in New Haven (I no longer remember), I came across Dwork's paper in which the first and the second main lemmas were proved. Dwork had indeed tried to establish a product formula for what has come to be called the \(\epsilon\)-factor but, without the insight that came from the adelic form of the Hecke theory and the conjectured relations of that to Artin \(L\)-functions, did not appreciate the need to introduce the factor \(\lambda(E/F,\psi_F)\) in condition (iii) of Theorem A. So he fell short of the goal, but fortunately not before he had established these two lemmas, which are indeed far more than lemmas, the proof of the second being a magnificent tour de force of \(p\)-adic analysis. Unfortunately he did not publish a proof, and the only material I had available when writing these notes was the thesis of K. Lakkis which reproduced Dwork's arguments, but only up to sign, and this is of course not enough. Nonetheless although many of the calculations are there, I was never able to work my way through them or put them in a form that was at all publishable. What I put down on paper from my attempts to understand the arguments of Dwork as reproduced by Lakkis is included here as fragmentary Chapters 12 and 13. They are included for what they are worth. Chapter 10, in which the proof of the first main lemma is completed, is also missing. Either it was never written or was misplaced. In any case, the material of Chapters 7, 8, and 9 at hand, the proof of the first main lemma is neither long nor difficult. With the exception of Chapters 10, 11, and 12, and perhaps some easy material that was to have been included in Chapters 8 and 9, the notes are complete. The proof is complete if one accepts the two lemmas of Dwork. Whether the complete proofs, which certainly existed, appeared in his thesis, I do not know, nor do I know whether his notes are still extant.
I abandoned my attempt to prepare a complete manuscript when Deligne observed that it is an easy matter to reverse the arguments and to proceed from the existence of the global \(\epsilon\)-factor, known to exist since Artin introduced the \(L\)-functions, to the existence of the local factors. It suffices to be clearly aware of their defining properties. Since these had escaped a mathematician of Dwork's quality, they cannot be regarded as manifest, or in the words of an eminent French mathematician "peu de chose"! Perhaps he was misled once again by partisan sentiments.
What of any possible use remains of the arguments here? First of all a general lemma about the structure of relations between induced representations of nilpotent groups that is conceivably of interest beyond the purposes of these notes, but that has never, so far as I know, found application elsewhere. Perhaps of more importance: although the local proof, which could be reconstructed from Dwork's notes and the material here, is far too long, a global proof of a local lemma is also not satisfactory. So the problem of finding a satisfactory local proof remains open.
The local \(\epsilon\)-factor is often incorporated into characterizations of the local correspondence for \(\mathrm{GL}(n)\). This is also unsatisfactory. The only real criterion for deciding whether a local correspondence is correct is that it be compatible firstly with the global correspondence and secondly with localization for representations of the Galois groups on one hand and automorphic representations on the other. Such a local correspondence established, the existence of the \(\epsilon\)-factor is immediate. At present, however, all aspects of the theory are rudimentary and inchoate. What may ultimately happen---I am not inclined to predictions in the matter---is that the existence of the local correspondence and of the \(\epsilon\)-factor will be established simultaneously, and that some of the arguments of these notes will reappear, but supplemented with information about the representations of \(\mathrm{GL}(n)\) over nonarchimedean fields.
I stress that these notes were written about 1970. I have not examined them in the intervening years with any care. There may be slips of the pen and even small mathematical errors.
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