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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.
The letter
Covering notes accompanying the letter to Weil
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.
The letter
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
The first letter
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, Springer-Verlag, 1970.
The lecture
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).
The booklet
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.
The note
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.
The paper
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.
The note
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Facsimiles of pages
of the original letter and the note from Harish-Chandra.