Correspondence leading to
the book Jacquet-Langlands
The book written with Jacquet
Introductory comment by Langlands
Correspondence leading to the book written with Jacquet
Next comes a collection of letters leading up to the Springer Lecture Notes written with Jacquet.
A second letter to Weil
A long letter to Jacquet
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
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
Real conviction in the matter demanded the existence of the
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.
The book itself
Langlands' Notes on Artin
These notes, although representing a huge amount of work, remained incomplete, and although widely distributed were never published.
One project that was formulated after writing the letter to Weil and that was suggested by
his 1957 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
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, as I discovered
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
-factor but, without
the insight that came from the adelic form of the
Hecke theory and the conjectured
relations of that to Artin
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 -factor, known to exist since Artin introduced the
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 -factor
is often incorporated into characterizations of the local correspondence
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.
On Artin's L-functions
This was originally published in 1970 as volume 56 in the Rice University Studies.
Automorphic forms on GL(2)
This is the text of a talk delivered at the International Congress of Mathematicians in Nice, 1970. First published in Actes du Congrès International des Mathématiciens. Gauthiers-Villars, Paris, 1971.