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The work of Robert Langlands


Letter to André Weil from January, 1967

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.

In reply to a question asked by many: there was no written reply from Weil.

Letter to Serre

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.

Two letters to Roger Howe

Problems in the Theory of Automorphic Forms

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 -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 -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.

The representation theory of abelian algebraic groups

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).

A little bit of number theory

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.

Representation theory - its rise and its role in number theory

This first appeared in Proceedings of the Gibbs symposium of 1989, published by the A. M. S. in 1990.

Where stands functoriality today?

This is the written version of a talk presented at the Edinburgh conference on automorphic forms, published by the AMS in 1997.