Letter to André Weil from January, 1967
In January of 1967, while he was at Princeton University,
Langlands wrote a letter of 17 handwritten 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 `
In order to help follow in the facsimile, page numbers like [2] have been inserted. Covering notes accompanying the letter to Weil
Langlands' comments:
In reply to a question asked by many: there was no written reply from Weil. Letter to Serre
Langlands spent 196768 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
Langlands' comments:
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 Two letters to Roger Howe
The second letter
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, SpringerVerlag, 1970.
Langlands' comments:
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
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 reexamination of the
Hecke theory, Jacquet and I had developed a theory
for
Although specific attention is drawn in the lecture to the case that
The question about elliptic curves appearing toward the end of §7 is nothing
but a supplement to the conjecture of TaniyamaShimuraWeil, 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
The observation about 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 TausskyTodd (volume 61 (1998), pp. 231250).
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:
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 ShimuraTaniyama conjecture and not having any more precise concept of what is now known as the JacquetLanglands 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 ShimuraTaniyama 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.
