Correspondence leading to the book written with Jacquet

Robert P. Langlands
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First tests

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.

School of Mathematics: